Content-Type: multipart/mixed; boundary="-------------0105110510408" This is a multi-part message in MIME format. ---------------0105110510408 Content-Type: text/plain; name="01-175.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-175.keywords" period function, monotonicity, isochronicity, Lienard equation, polynomial systems ---------------0105110510408 Content-Type: application/postscript; name="PerLienard.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="PerLienard.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: C:\Users\chouikha\Lienard\PerLienard.dvi %%Pages: 25 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: C:\TEX\BIN\WIN32\dvips.exe %+ C:\Users\chouikha\Lienard\PerLienard.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2001.05.11:1207 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def 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b(that)e Fo(f)11 b Fp(\(0\))29 b(=)h Fo(g)t Fp(\(0\))e(=)i(0)p Fo(;)17 b(g)1530 3289 y Fy(0)1552 3325 y Fp(\(0\))29 b Fo(>)h Fp(0)p Fo(;)j Fq(the)h(origin)e Fp(0)h Fq(is)g(an)h(iso)s(c)m (hronous)g(cen)m(ter)h(if)324 3446 y(and)d(only)g(if)1262 3605 y Fo(g)t Fp(\()p Fo(x)p Fp(\))27 b(=)h Fo(g)1626 3564 y Fy(0)1648 3605 y Fp(\(0\))p Fo(x)22 b Fp(+)1982 3537 y(1)p 1958 3582 95 4 v 1958 3673 a Fo(x)2013 3644 y Fu(3)2063 3605 y Fp([)2090 3488 y Fk(Z)2173 3514 y Fw(x)2137 3676 y Fu(0)2234 3605 y Fo(sf)11 b Fp(\()p Fo(s)p Fp(\))p Fo(ds)p Fp(])2585 3564 y Fu(2)324 3798 y Fq(These)33 b(h)m(yp)s(otheses)h(in)e(particular)d(imp)s(ose)i(the)h (strong)g(condition)62 b Fo(g)2966 3762 y Fy(00)3008 3798 y Fp(\(0\))27 b(=)h(0)p Fq(.)j(This)324 3919 y(one)42 b(will)d(b)s(e)k(the)f(cen)m(tral)f(p)s(oin)m(t)g(of)h(our)g (discussion,)i(where)f(one)f(will)e(attempt)h(to)324 4039 y(o)m(v)m(ercome)33 b(this)f(h)m(yp)s(othesis.)470 4280 y(Concerning)38 b(the)h(ab)s(o)m(v)m(e)f(problem,)g(the)h(in)m (teresting)e(question)h(that)g(remains)f(is)324 4400 y(the)28 b(one)g(to)f(kno)m(w)i(if)d(the)i(only)f(iso)s(c)m(hronous)h (cen)m(ters)h(for)e(Lienard)g(systems)i(are)f(those)324 4521 y(giv)m(en)35 b(b)m(y)h([S],)g(as)f(b)s(eliev)m(e)h(certain)e (authors.)52 b(F)-8 b(or)35 b(the)g(momen)m(t,)g(one)h(not)f(kno)m(w)h (ex-)324 4641 y(amples)31 b(of)h(other)h(iso)s(c)m(hronous)g(systems.) 470 4882 y(This)d(pap)s(er)h(is)e(organized)h(as)g(follo)m(ws.)42 b(In)30 b(the)h(second)g(part)f(w)m(e)i(examine)d(mono-)324 5002 y(tonicit)m(y)f(conditions)f(of)i(the)g(p)s(erio)s(d)e(function)h (for)h(the)g(Lienard)f(equation)g(where)i(the)1918 5251 y(2)p eop %%Page: 3 3 3 2 bop 324 548 a Fq(functions)32 b Fo(f)42 b Fq(and)32 b Fo(g)j Fq(are)d(suc)m(h)i(that)d Fo(f)39 b Fj(2)28 b Fo(C)1955 512 y Fu(1)1994 548 y Fp(\()p Fo(J)n(;)17 b(R)q Fp(\))32 b Fq(and)g Fo(g)f Fj(2)d Fo(C)2711 512 y Fu(3)2750 548 y Fp(\()p Fo(J)n(;)17 b(R)q Fp(\))32 b Fq(on)g(the)g(in)m(ter-)324 668 y(v)-5 b(al)32 b Fo(J)39 b Fp(=)29 b(\()p Fo(a;)17 b(b)p Fp(\))p Fo(;)115 b(a)30 b(<)f Fp(0)g Fo(<)h(b)k Fq(and)g(satisfying)e(some)i(conditions)e(so)i (that)g(the)g(origin)324 789 y Fp(0)e Fq(is)h(a)g(cen)m(ter)h(of)e (this)h(equation.)44 b(It)33 b(implies)e(in)h(particular)64 b Fo(g)2701 753 y Fy(0)2724 789 y Fp(\(0\))28 b Fo(>)g Fp(0)p Fo(:)33 b Fq(The)h(p)s(erio)s(d)324 909 y Fo(T)41 b Fj(\021)28 b Fo(T)14 b Fp(\()p Fo(\015)5 b Fp(\))65 b Fq(where)h Fo(\015)38 b Fq(is)32 b(a)g(cycle)h(surrounding)65 b Fp(0)32 b Fq(is)g(de\034ned.)470 1150 y(W)-8 b(e)33 b(pro)m(v)m(e)h(that)1068 1396 y Fo(g)1119 1355 y Fy(0)1142 1396 y Fp(\(0\))p Fo(g)1318 1355 y Fu(\(3\))1411 1396 y Fp(\()p Fo(x)p Fp(\))22 b Fj(\000)1674 1329 y Fp(5)p 1674 1373 49 4 v 1674 1465 a(3)1733 1396 y Fo(g)1784 1355 y Fy(00)1825 1344 y Fu(2)1865 1396 y Fp(\()p Fo(x)p Fp(\))g Fj(\000)2128 1329 y Fp(2)p 2128 1373 V 2128 1465 a(3)2186 1396 y Fo(f)2245 1355 y Fy(0)2269 1344 y Fu(2)2308 1396 y Fp(\()p Fo(x)p Fp(\))p Fo(g)2490 1355 y Fy(0)2513 1396 y Fp(\(0\))27 b Fj(6)p Fp(=)h(0)324 1629 y Fq(implies)20 b(the)k(monotonicit)m(y)d(of)i(the)g(p)s(erio)s(d)g(function)45 b Fo(T)60 b Fq(in)23 b(a)g(neigh)m(b)s(orho)s(o)s(d)e(of)i Fp(0)g Fq(.)40 b(In)324 1749 y(particular,)22 b Fo(g)843 1713 y Fy(0)866 1749 y Fp(\(0\))p Fo(g)1042 1713 y Fu(\(3\))1135 1749 y Fp(\(0\))q Fj(\000)1349 1710 y Fu(5)p 1349 1726 36 4 v 1349 1783 a(3)1393 1749 y Fo(g)1444 1713 y Fy(00)1486 1702 y Fu(2)1526 1749 y Fp(\(0\))q Fj(\000)1740 1710 y Fu(2)p 1740 1726 V 1740 1783 a(3)1784 1749 y Fo(f)1843 1713 y Fy(0)1866 1702 y Fu(2)1905 1749 y Fp(\(0\))28 b(=)f(0)44 b Fq(and)g Fo(f)2514 1713 y Fy(0)2537 1749 y Fp(\(0\))p Fo(g)2713 1713 y Fy(00)2754 1749 y Fp(\(0\))q Fj(\000)q Fo(g)3009 1713 y Fy(0)3031 1749 y Fp(\(0\))p Fo(f)3215 1713 y Fy(00)3257 1749 y Fp(\(0\))27 b(=)h(0)324 1869 y Fq(are)k(necessary)j(conditions)d(for)g(the)h(cen)m(ter)g Fp(0)g Fq(to)f(b)s(e)h(iso)s(c)m(hronous.)324 1990 y(When)g Fo(g)j Fq(is)31 b(of)h(class)h Fo(C)1205 1954 y Fu(3)1244 1990 y Fq(,)g(our)f(criteria)e(app)s(ears)j(in)f(a)g(certain)g (direction)f(to)h(b)s(e)g(more)324 2110 y(general)e(than)i(those)g(giv) m(en)f(b)m(y)h(Sabatini)e([S])h(for)g(the)h(p)s(erio)s(d)e(function)h (to)g(b)s(e)g(mono-)324 2231 y(tonic.)324 2351 y(In)e(particular,)f(if) g Fo(f)38 b Fj(\021)28 b Fp(0)p Fq(,)i(w)m(e)g(\034nd)g(the)f (condition)e Fo(g)2270 2315 y Fy(0)2293 2351 y Fp(\(0\))p Fo(g)2469 2315 y Fu(\(3\))2562 2351 y Fp(\(0\))15 b Fj(\000)2804 2312 y Fu(5)p 2804 2328 V 2804 2385 a(3)2849 2351 y Fo(g)2900 2315 y Fy(00)2942 2304 y Fu(2)2982 2351 y Fp(\(0\))27 b Fj(6)p Fp(=)g(0)i Fq(whic)m(h)324 2471 y(implies)e(the)j(monotonicit) m(y)e(of)h(the)h(p)s(erio)s(d)f(for)g(the)h(conserv)-5 b(ativ)m(e)31 b(case.)43 b(This)30 b(agrees)324 2592 y(with)i(Sc)m(haaf)h(condition,)e([R])1385 2844 y Fo(g)1436 2802 y Fy(0)1459 2844 y Fp(\()p Fo(x)p Fp(\))p Fo(g)1641 2802 y Fu(\(3\))1735 2844 y Fp(\()p Fo(x)p Fp(\))22 b Fj(\000)1998 2776 y Fp(5)p 1998 2820 49 4 v 1998 2912 a(3)2057 2844 y 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b(the)h(p)s(erio)s(d)e(of)h(the)h(conserv)-5 b(ativ)m(e)34 b(asso)s(ciated)e(system)i(has)e(to)h(b)s(e)f (decreasing.)470 4882 y(W)-8 b(e)36 b(also)e(notice)i(that)f(under)h (restrictiv)m(e)g(h)m(yp)s(otheses,)i(the)e(function)f Fo(g)t Fp(\()p Fo(x)p Fp(\))g Fq(is)g(of)324 5002 y(class)f Fo(C)631 4966 y Fw(k)673 5002 y Fo(;)51 b(k)32 b Fj(\025)f Fp(3)i Fq(and)h Fo(f)40 b Fj(2)30 b Fo(C)1476 4966 y Fu(2)1516 5002 y Fq(,)k(it)f(is)g(p)s(ossible)g(to)g(extend)i(some)f (results)g(of)f([S].)h(F)-8 b(or)1918 5251 y(3)p eop %%Page: 4 4 4 3 bop 324 548 a Fq(that,)32 b(one)h(has)g(to)f(supp)s(ose)739 766 y Fo(g)790 724 y Fy(00)831 766 y Fp(\(0\))c(=)f(0)p Fo(;)114 b(f)1336 724 y Fy(00)1378 766 y Fp(\(0\))28 b(=)f(0)98 b Fg(and)107 b Fo(g)2094 724 y Fy(0)2116 766 y Fp(\()p Fo(x)p Fp(\))p Fo(g)2298 724 y Fu(\(3\))2392 766 y Fp(\()p Fo(x)p Fp(\))23 b Fj(\000)2655 698 y Fp(2)p 2655 742 49 4 v 2655 834 a(3)2714 766 y Fo(f)2773 724 y Fy(0)2796 714 y Fu(2)2836 766 y Fp(\()p Fo(x)p Fp(\))k Fj(6)p 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b(the)g(function)f Fo(g)39 b Fh(satis\034es)134 b Fo(xg)t Fp(\()p Fo(x)p Fp(\))28 b Fo(>)g Fp(0)35 b Fh(for)g Fo(x)29 b Fj(2)g Fo(J)9 b Fh(,)324 789 y(then)34 b(for)h(e)-5 b(ach)34 b(fol)5 b(lowing)34 b(c)-5 b(ondition)324 909 y(\(i\))34 b(-)70 b Fo(g)31 b Fj(2)d Fo(C)822 873 y Fu(1)861 909 y Fp(\()p Fo(J)n(;)17 b(R)q Fp(\))p Fh(,)35 b(and)69 b Fo(g)1448 873 y Fy(0)1471 909 y Fp(\(0\))27 b Fo(>)g Fp(0)p Fh(.)324 1029 y(\(ii\))34 b(-)70 b Fo(g)31 b Fj(2)d Fo(C)852 993 y Fu(2)891 1029 y Fp(\()p Fo(J)n(;)17 b(R)q Fp(\))p Fh(,)35 b(and)69 b Fo(g)1478 993 y Fy(0)1500 1029 y Fp(\(0\))28 b Fo(>)f Fp(0)35 b Fh(and)69 b Fo(g)2115 993 y Fy(00)2157 1029 y Fp(\(0\))27 b(=)h(0)p Fh(,)324 1150 y(we)34 b(have)843 1515 y Fg(if)997 1447 y Fo(d)p 969 1492 107 4 v 969 1583 a(dx)1085 1515 y Fp(\()1136 1447 y Fo(g)p 1133 1492 56 4 v 1133 1583 a(x)1198 1515 y Fp(\))h Fg(is)1366 1394 y Fk(\032)1479 1455 y Fg(p)-5 b(ositive)42 b(for)j Fo(x)28 b Fj(2)h Fo(J)44 b Fg(and)g Fo(x)28 b(<)g Fp(0)p Fo(;)1479 1575 y 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y(example)k(\(2-2\))g(is)h(v)m(eri\034ed,)i(let)d(us)i (pro)m(v)m(e)g(that)f Fo(T)2221 3853 y Fu(1)2295 3838 y Fq(is)g(strictly)f(increasing)h(on)g Fp(\(0)p Fo(;)17 b(b)p Fp(\))p Fq(.)324 3959 y(F)-8 b(ollo)m(wing)22 b([O],)k(consider)f Fo(p)j Fp(=)1500 3919 y Fw(r)1534 3896 y Ff(0)p 1500 3935 57 4 v 1511 3993 a Fw(r)1566 3959 y Fq(,)f(where)f Fp(0)i Fo(<)f(r)k(<)c(r)2299 3922 y Fy(0)2350 3959 y Fo(<)g(b)p Fq(.)42 b(De\034ne)27 b(the)f(new)g(function)1625 4243 y Fo(h)1681 4258 y Fw(p)1721 4243 y Fp(\()p Fo(x)p Fp(\))i(=)1993 4176 y Fo(g)t Fp(\()p Fo(px)p Fp(\))p 1993 4220 231 4 v 2084 4311 a Fo(p)2234 4243 y(:)324 4524 y Fq(By)33 b(\(2-2\),)e(for)h Fo(x)c Fj(2)h Fp(\(0)p Fo(;)1212 4485 y Fw(b)p 1209 4501 36 4 v 1209 4558 a(p)1254 4524 y Fp(\))k Fq(w)m(e)g(ha)m(v)m(e)1247 4821 y Fo(g)t Fp(\()p Fo(x)p Fp(\))27 b(=)1569 4754 y Fo(g)t Fp(\()p Fo(x)p Fp(\))p 1569 4798 182 4 v 1632 4890 a Fo(x)1761 4821 y(x)h(>)1957 4754 y(g)t Fp(\()p Fo(px)p Fp(\))p 1957 4798 231 4 v 2021 4890 a Fo(px)2198 4821 y(x)g Fp(=)g Fo(h)2441 4836 y Fw(p)2480 4821 y Fp(\()p Fo(x)p Fp(\))p Fo(:)1918 5251 y Fq(5)p eop %%Page: 6 6 6 5 bop 324 548 a Fq(Let)33 b Fo(I)8 b Fp(\()p Fo(x)p Fp(\))27 b(=)812 477 y Fk(R)867 503 y Fw(x)851 573 y Fu(0)928 548 y Fo(h)984 563 y Fw(p)1024 548 y Fp(\()p Fo(u)p Fp(\))p Fo(du)p Fq(,)k(w)m(e)i(then)g(ha)m(v)m(e)h Fo(G)p Fp(\()p Fo(r)s Fp(\))22 b Fj(\000)h Fo(G)p Fp(\()p Fo(x)p Fp(\))28 b Fo(>)f(I)8 b Fp(\()p Fo(r)s Fp(\))22 b Fj(\000)g Fo(I)8 b Fp(\()p Fo(x)p Fp(\))p Fo(:)33 b Fq(W)-8 b(e)33 b(obtain)561 817 y Fo(T)618 832 y Fu(1)658 817 y Fp(\()p Fo(r)s Fp(\))27 b Fo(<)912 730 y Fj(p)p 995 730 49 4 v 87 x Fp(2)1060 699 y Fk(Z)1143 726 y Fw(r)1106 888 y Fu(0)1446 749 y Fo(dx)p 1208 793 584 4 v 1208 810 a Fk(q)p 1291 810 501 4 v 100 x Fo(I)8 b Fp(\()p Fo(r)1427 881 y Fy(0)1450 910 y Fp(\))22 b Fj(\000)g Fo(I)8 b Fp(\()p Fo(x)p Fp(\))1829 817 y(=)1933 730 y Fj(p)p 2016 730 49 4 v 87 x Fp(2)2081 699 y Fk(Z)2164 726 y Fw(r)2127 888 y Fu(0)2518 749 y Fo(dx)p 2229 793 685 4 v 2229 810 a Fk(q)p 2312 810 602 4 v 100 x Fo(G)p Fp(\()p Fo(r)2474 881 y Fy(0)2496 910 y Fp(\))22 b Fj(\000)h Fo(G)p Fp(\()p Fo(px)p Fp(\))2951 817 y(=)k Fo(T)3111 832 y Fu(1)3151 817 y Fp(\()p Fo(r)3236 775 y Fy(0)3259 817 y Fp(\))p Fo(:)324 1135 y Fq(Then)37 b Fo(T)639 1150 y Fu(1)716 1135 y Fq(is)f(strictly)g(increasing)f(on)i Fp(\(0)p Fo(;)17 b(b)p Fp(\))p Fo(:)36 b Fq(By)h(similarit)m(y)c(,)38 b(w)m(e)g(pro)m(v)m(e)f(that)g(under)324 1255 y(h)m(yp)s(othesis)c (\(2-2\))1309 1425 y Fo(T)1366 1440 y Fu(2)1406 1425 y Fp(\()p Fo(s)p Fp(\))27 b(=)1659 1338 y Fj(p)p 1742 1338 49 4 v 87 x Fp(2)1807 1308 y Fk(Z)1890 1334 y Fw(r)1853 1496 y Fu(0)2207 1358 y Fo(dx)p 1955 1402 612 4 v 1955 1418 a Fk(q)p 2038 1418 529 4 v 100 x Fo(G)p Fp(\()p Fo(s)p Fp(\))22 b Fj(\000)g Fo(G)p Fp(\()p Fo(x)p Fp(\))324 1700 y Fq(is)32 b(strictly)g(decreasing)h(on)f Fp(\()p Fo(a;)17 b Fp(0\))p Fq(.)43 b(The)33 b(p)s(erio)s(d)f(function)g Fo(T)46 b Fq(is)32 b(ob)m(viously)1502 1916 y Fo(T)14 b Fp(\()p Fo(r)s 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b(e)g(over,)41 b(e)-5 b(ach)39 b(of)g(these)h(c)-5 b(onditions)39 b(implies)g(that)h (the)g(p)-5 b(erio)g(d)39 b(function)h Fo(T)14 b Fp(\()p Fo(r)s Fp(\))39 b Fh(of)324 4718 y(\(2.3\))34 b(is)g(strictly)i(incr)-5 b(e)g(asing)33 b(for)i Fp(0)27 b Fo(<)h(r)i(<)e(r)2017 4733 y Fu(1)2056 4718 y Fh(.)425 4984 y Fo(xg)531 4942 y Fy(00)574 4984 y Fp(\()p Fo(x)p Fp(\))g Fo(>)f Fp(0)100 b Fg(implies)106 b Fo(g)1442 4942 y Fu(2)1481 4984 y Fp(\()p Fo(x)p Fp(\))22 b Fj(\000)h Fp(2)p Fo(G)p Fp(\()p Fo(x)p Fp(\))p Fo(g)2042 4942 y Fy(0)2065 4984 y Fp(\()p Fo(x)p Fp(\))28 b Fo(<)f Fp(0)100 b Fg(implies)106 b Fo(x)p Fp(\()3013 4916 y Fo(d)p 2985 4960 107 4 v 2985 5052 a(dx)3114 4916 y(g)p 3112 4960 56 4 v 3112 5052 a(x)3177 4984 y Fp(\)\))28 b Fo(>)f Fp(0)p Fo(:)1918 5251 y Fq(8)p eop %%Page: 9 9 9 8 bop 324 548 a Fh(Mor)-5 b(e)g(over,)41 b(e)-5 b(ach)39 b(of)g(these)h(c)-5 b(onditions)39 b(implies)g(that)h(the)g(p)-5 b(erio)g(d)39 b(function)h Fo(T)14 b Fp(\()p Fo(r)s Fp(\))39 b Fh(of)324 668 y(\(2.3\))34 b(is)g(strictly)i(de)-5 b(cr)g(e)g(asing)33 b(for)i Fp(0)27 b Fo(<)h(r)i(<)e(r)2022 683 y Fu(1)2061 668 y Fh(.)324 789 y(A)35 b(ne)-5 b(c)g(essary)34 b(c)-5 b(ondition)34 b(to)h(have)f(any)h(of)f(these)h(c)-5 b(onditions)33 b(is)i Fo(g)2799 753 y Fy(00)2841 789 y Fp(\(0\))27 b(=)h(0)34 b Fh(.)324 909 y(If)g(we)g(supp)-5 b(ose)35 b(in)f(addition)g Fo(g)t Fp(\()p Fo(x)p Fp(\))h Fh(is)f(of)h(class)f Fo(C)2174 873 y Fu(3)2213 909 y Fh(,)h(then)739 1129 y Fo(g)790 1088 y Fu(\(3\))884 1129 y Fp(\()p Fo(x)p Fp(\))28 b Fo(<)f Fp(0\()p Fo(>)g Fp(0\))35 b Fg(and)45 b Fo(g)1709 1088 y Fy(00)1750 1129 y Fp(\(0\))28 b(=)f(0)35 b Fg(implies)42 b Fo(xg)2538 1088 y Fy(00)2580 1129 y Fp(\()p Fo(x)p Fp(\))28 b Fo(<)f Fp(0\()p Fo(>)h Fp(0\))p Fo(:)470 1569 y Fi(Pro)s(of)227 b Fq(Notice)32 b(that,)h(since)g Fo(G)p Fp(\()p Fo(x)p Fp(\))28 b Fo(>)f Fp(0)33 b Fq(for)f Fo(x)c Fj(6)p Fp(=)f(0)33 b Fq(and)1168 1771 y Fo(d)p 1141 1815 107 4 v 1141 1907 a(dx)1257 1839 y Fp([)p Fo(g)1335 1797 y Fu(2)1373 1839 y Fp(\()p Fo(x)p Fp(\))23 b Fj(\000)f Fp(2)p Fo(G)p Fp(\()p Fo(x)p Fp(\))p Fo(g)1934 1797 y Fy(0)1957 1839 y Fp(\()p Fo(x)p Fp(\)])28 b(=)g(2)p Fo(G)p Fp(\()p Fo(x)p Fp(\))p Fo(g)2555 1797 y Fy(00)2596 1839 y Fp(\()p Fo(x)p Fp(\))p Fo(;)324 2094 y Fq(then)62 b Fp(2)p Fo(xG)p Fp(\()p Fo(x)p Fp(\))p Fo(g)938 2058 y Fy(00)980 2094 y Fp(\()p Fo(x)p Fp(\))28 b Fo(<)g Fp(0)j(\()p Fq(r)p Fo(esp:)c(>)h Fp(0\))62 b Fq(implies)d Fo(g)2257 2058 y Fu(2)2296 2094 y Fp(\()p Fo(x)p Fp(\))19 b Fj(\000)g Fp(2)p Fo(G)p Fp(\()p Fo(x)p Fp(\))p Fo(g)2850 2058 y Fy(0)2873 2094 y Fp(\()p Fo(x)p Fp(\))28 b Fo(>)f Fp(0)k(\()p Fq(r)p Fo(esp:)d(<)324 2214 y Fp(0\))p Fq(.)37 b(Because)77 b Fo(x)976 2175 y Fw(d)p 956 2191 77 4 v 956 2248 a(dx)1042 2214 y Fp([)p Fo(g)1120 2178 y Fu(2)1159 2214 y Fp(\()p Fo(x)p Fp(\))22 b Fj(\000)h Fp(2)p Fo(G)p Fp(\()p Fo(x)p Fp(\))p Fo(g)1720 2178 y Fy(0)1743 2214 y Fp(\()p Fo(x)p Fp(\)])75 b Fq(and)g Fo(xg)2314 2178 y Fy(00)2356 2214 y Fp(\()p Fo(x)p Fp(\))38 b Fq(ha)m(v)m(e)h(the)f(same)g(sign.)58 b(So,)324 2335 y(w)m(e)33 b(ha)m(v)m(e)h(pro)m(v)m(ed)g(the)f(t)m(w)m(o)g(\034rst)g (implications.)324 2455 y(The)24 b(second)h(implications)44 b Fo(g)1435 2419 y Fu(2)1474 2455 y Fp(\()p Fo(x)p Fp(\))t Fj(\000)t Fp(2)p Fo(G)p Fp(\()p Fo(x)p Fp(\))p Fo(g)1998 2419 y Fy(0)2021 2455 y Fp(\()p Fo(x)p Fp(\))28 b Fo(>)f Fp(0)d(\()p Fq(r)p Fo(esp:)k(<)f Fp(0\))c Fg(implies)31 b Fo(x)p Fp(\()3294 2416 y Fw(d)p 3274 2432 V 3274 2489 a(dx)3372 2412 y(g)p 3370 2432 40 4 v 3370 2489 a(x)3420 2455 y Fp(\))d Fo(<)324 2575 y Fp(0)33 b(\()p Fq(r)p Fo(esp:)c(>)h Fp(0\))66 b Fq(ha)m(v)m(e)35 b(b)s(een)f(pro)m(v)m(ed)h (b)m(y)g(Rothe)e(\(see)i([R])e(Prop)s(osition)f(4)h(p.)47 b(138,)33 b(ac-)324 2696 y(cording)f(his)g(notations:)42 b Fo(h)1345 2654 y Fu(+)1345 2717 y(3)1432 2696 y Fj(\032)28 b Fo(h)1593 2654 y Fu(+)1593 2717 y(2)1653 2696 y Fq(\).)470 2936 y(Remark)k(that)g(conditions)624 3156 y Fo(xg)730 3115 y Fy(00)773 3156 y Fp(\()p Fo(x)p Fp(\))c Fo(<)f Fp(0)32 b Fg(r)-5 b(esp)6 b Fo(:)28 b(>)f Fp(0)98 b Fg(and)107 b Fo(g)1911 3115 y Fu(2)1950 3156 y Fp(\()p Fo(x)p Fp(\))22 b Fj(\000)h Fp(2)p Fo(G)p Fp(\()p Fo(x)p Fp(\))p Fo(g)2511 3115 y Fy(0)2533 3156 y Fp(\()p Fo(x)p Fp(\))28 b Fo(<)g Fp(0)k Fg(r)-5 b(esp)6 b Fo(:)28 b(>)f Fp(0)324 3387 y Fq(indep)s(enden)m(tly)g(imply)52 b Fo(H)1325 3402 y Fu(0)1364 3387 y Fp(\()p Fo(x)p Fp(\))28 b(=)f Fo(g)t Fp(\()p Fo(x)p Fp(\))1808 3338 y Fu(2)1858 3387 y Fp(+)1979 3340 y Fw(g)2015 3317 y Ff(00)2056 3340 y Fu(\(0\))p 1954 3364 219 4 v 1954 3428 a(3)p Fw(g)2025 3409 y Ff(0)2047 3428 y Fu(\(0\))2137 3399 y Fd(2)2182 3387 y Fo(g)t Fp(\()p Fo(x)p Fp(\))2364 3338 y Fu(3)2413 3387 y Fj(\000)10 b Fp(2)p Fo(G)p Fp(\()p Fo(x)p Fp(\))p Fo(g)2808 3351 y Fy(0)2831 3387 y Fp(\()p Fo(x)p Fp(\))28 b Fj(\025)h Fp(0)p Fo(;)43 b Fp(\()p Fg(r)-5 b(esp)6 b Fo(:)27 b Fj(\024)324 3528 y Fp(0\))41 b Fq(and)h(imply)e(the)i(p)s(erio)s(d)e (function)i(increasing)e(\(resp.)72 b(decreasing\),)44 b(see)f([C-W])324 3648 y(Corollary)31 b(2.3.)324 3769 y(On)h(the)h(other)g(hand,)g(condition)63 b Fo(g)1691 3733 y Fu(\(3\))1785 3769 y Fp(\()p Fo(x)p Fp(\))28 b Fo(<)g Fp(0\()p Fo(>)f Fp(0\))32 b Fq(a)p Fo(nd)g(g)2598 3733 y Fy(00)2640 3769 y Fp(\(0\))27 b(=)h(0)k Fq(implies)415 4111 y Fo(H)496 4126 y Fu(3)536 4111 y Fp(\()p Fo(x)p Fp(\))c(=)f(5)p Fo(g)898 4075 y Fy(00)940 4064 y Fu(2)979 4111 y Fp(\()p Fo(x)p Fp(\))c Fj(\000)f Fp(3)p Fo(g)1332 4075 y Fy(0)1355 4111 y Fp(\()p Fo(x)p Fp(\))p Fo(g)1537 4075 y Fu(\(3\))1630 4111 y Fp(\()p Fo(x)p Fp(\))28 b Fo(>)g Fp(0\()p Fo(<)f Fp(0\))p Fo(;)83 b Fg(for)206 b Fo(x)28 b Fj(2)g Fp(\()p Fo(a;)17 b(b)p Fp(\))p Fo(;)g(x)28 b Fj(6)p Fp(=)f(0)p Fo(:)3344 4109 y Fp(\()p Fj(C)3434 4124 y Fu(3)3474 4109 y Fp(\))3437 4230 y Fq(\(8\))470 4350 y(holds,)32 b(whic)m(h)h(itself)f(implies)e Fp(\()p Fj(C)1689 4365 y Fu(0)1729 4350 y Fp(\))p Fq(.)324 4470 y(W)-8 b(e)45 b(pro)s(ceed)h(as)f(in)f(Prop)s(osition)f(1,)48 b(let)c(the)h(function)89 b Fo( )t Fp(\()p Fo(x)p Fp(\))49 b(=)g Fo(x)p Fp(\()3063 4431 y Fw(d)p 3043 4447 77 4 v 3043 4505 a(dx)3141 4427 y(g)p 3139 4447 40 4 v 3139 4505 a(x)3189 4470 y Fq(,)c(By)g(the)324 4591 y(l'Hopital's)30 b(rule)i(w)m(e)i(\034nd)1330 4855 y Fp(2)e Fo(l)r(im)1560 4870 y Fw(x)p Fy(!)p Fu(0)1711 4855 y Fp(\()1787 4787 y Fo(d)p 1759 4831 107 4 v 1759 4923 a(dx)1885 4787 y( )p 1885 4831 67 4 v 1891 4923 a(x)1962 4855 y Fp(\))27 b(=)h Fo(g)2182 4813 y Fy(00)2224 4855 y Fp(\(0\))f(=)h(0)p Fo(:)1918 5251 y Fq(9)p eop %%Page: 10 10 10 9 bop 324 548 a Fq(W)-8 b(e)33 b(ha)m(v)m(e)h(seen)f(that)g Fo( )t Fp(\()p Fo(x)p Fp(\))g Fq(is)674 799 y Fo( )741 758 y Fy(00)784 799 y Fp(\()p Fo(x)p Fp(\))28 b(=)f Fo(g)1097 758 y Fu(\(3\))1191 799 y Fp(\()p Fo(x)p Fp(\))22 b Fj(\000)1454 732 y Fo( )1521 695 y Fy(0)1544 732 y Fp(\()p Fo(x)p Fp(\))p 1454 776 222 4 v 1537 867 a Fo(x)1708 799 y Fp(+)1816 732 y Fo( )t Fp(\()p Fo(x)p Fp(\))p 1816 776 199 4 v 1867 867 a Fo(x)1922 838 y Fu(2)2052 799 y Fp(=)27 b Fo(g)2206 758 y Fu(\(3\))2300 799 y Fp(\()p Fo(x)p Fp(\))22 b Fj(\000)2563 732 y Fo(g)2614 695 y Fy(00)2656 732 y Fp(\()p Fo(x)p Fp(\))p 2563 776 225 4 v 2647 867 a Fo(x)2819 799 y Fp(+)g(2)2976 732 y Fo( )t Fp(\()p Fo(x)p Fp(\))p 2976 776 199 4 v 3027 867 a Fo(x)3082 838 y Fu(2)3184 799 y Fo(:)470 1020 y Fq(W)-8 b(e)33 b(ha)m(v)m(e)h(also)d(calculated) 1139 1246 y Fo(l)r(im)1288 1261 y Fw(x)p Fy(!)p Fu(0)1439 1246 y Fo( )1506 1205 y Fy(00)1548 1246 y Fp(\()p Fo(x)p Fp(\))d(=)g Fo(g)1862 1205 y Fu(\(3\))1955 1246 y Fp(\(0\))22 b Fj(\000)h Fo(l)r(im)2351 1261 y Fw(x)p Fy(!)p Fu(0)2511 1179 y Fo( )t Fp(\()p Fo(x)p Fp(\))p 2511 1223 V 2563 1314 a Fo(x)2618 1286 y Fu(2)2719 1246 y Fo(:)324 1484 y Fq(So,)32 b(notice)g(that)h Fo(g)1036 1448 y Fu(\(3\))1129 1484 y Fp(\(0\))g Fq(has)f(the)h(same)g(sign)f(as)h(the)g(function)f Fo( )t Fp(\()p Fo(x)p Fp(\))27 b(=)h Fo(x)p Fp(\()3196 1445 y Fw(d)p 3176 1461 77 4 v 3176 1519 a(dx)3274 1441 y(g)p 3272 1461 40 4 v 3272 1519 a(x)3322 1484 y Fp(\))470 1605 y Fq(Th)m(us,)43 b Fo(g)804 1568 y Fu(\(3\))898 1605 y Fp(\()p Fo(x)p Fp(\))d Fo(<)f Fp(0\()p Fo(>)h Fp(0\))f Fg(and)49 b Fo(g)1768 1568 y Fy(00)1810 1605 y Fp(\(0\))39 b(=)h(0)f Fg(implies)47 b Fo(xg)2631 1568 y Fy(00)2673 1605 y Fp(\()p Fo(x)p Fp(\))40 b Fo(<)f Fp(0\()p Fo(>)h Fp(0\))p Fo(;)f Fq(whic)m(h)324 1725 y(implies)62 b Fo(x)p Fp(\()811 1686 y Fw(d)p 790 1702 77 4 v 790 1759 a(dx)889 1682 y(g)p 887 1702 40 4 v 887 1759 a(x)936 1725 y Fp(\)\))28 b Fo(<)f Fp(0\()p Fo(>)h Fp(0\))p Fq(.)470 1966 y(F)-8 b(rom)30 b(the)h(latter)f(remark,)i(if)61 b Fo(g)1685 1930 y Fu(\(3\))1778 1966 y Fp(\(0\))28 b Fj(6)p Fp(=)f(0)k Fq(the)h(function)62 b Fo(x)p Fp(\()2816 1927 y Fw(d)p 2796 1943 77 4 v 2796 2000 a(dx)2894 1923 y(g)p 2892 1943 40 4 v 2892 2000 a(x)2942 1966 y Fp(\))31 b Fq(has)g(the)h(same)324 2086 y(sign)g(as)65 b Fo(g)728 2050 y Fu(\(3\))822 2086 y Fp(\(0\))p Fo(;)32 b Fq(w)m(e)h(ma)m(y)g (deduce)h(the)f(follo)m(wing.)324 2371 y Fe(2.3)130 b(Corrolary)45 b(1)519 2556 y Fh(L)-5 b(et)34 b(a)f(function)67 b Fo(g)31 b Fj(2)d Fo(C)1434 2520 y Fw(k)1477 2556 y Fp(\()p Fo(J)n(;)17 b(R)q Fp(\))p Fo(;)50 b(k)31 b Fj(\025)d Fp(2)33 b Fh(wher)-5 b(e)33 b Fo(J)43 b Fh(is)33 b(an)g(interval)h(c)-5 b(ontaining)32 b Fp(0)324 2677 y Fh(such)j(that)71 b Fo(g)t Fp(\(0\))27 b(=)i Fo(g)1140 2640 y Fy(00)1182 2677 y Fp(\(0\))f(=)g(0)p Fo(;)52 b(g)1618 2640 y Fu(\(3\))1712 2677 y Fp(\(0\))28 b Fj(6)p Fp(=)g(0)36 b Fh(and)70 b Fo(g)2330 2640 y Fy(0)2352 2677 y Fp(\(0\))29 b Fo(>)f Fp(0)p Fh(.)35 b(Supp)-5 b(ose)35 b(in)g(addition)324 2797 y Fo(xg)t Fp(\()p Fo(x)p Fp(\))27 b Fo(>)h Fp(0)35 b Fh(for)69 b Fo(x)28 b Fj(2)g Fo(J)9 b Fh(.)35 b(Then)f(we)h(have)501 3138 y Fg(if)56 b Fo(g)668 3097 y Fu(\(3\))762 3138 y Fp(\(0\))27 b Fo(<)h Fp(0)34 b Fg(and)1338 3070 y Fo(d)p 1311 3115 107 4 v 1311 3206 a(dx)1427 3138 y Fp(\()1478 3070 y Fo(g)p 1475 3115 56 4 v 1475 3206 a(x)1540 3138 y Fp(\))h Fg(is)1707 3017 y Fk(\032)1821 3078 y Fg(p)-5 b(ositive)42 b(for)j Fo(x)28 b Fj(2)g Fo(J)44 b Fp(and)35 b Fo(x)29 b(<)e Fp(0)p Fo(;)1821 3198 y Fg(ne)-5 b(gative)42 b(for)j Fo(x)28 b Fj(2)g Fo(J)44 b Fg(and)h Fo(x)28 b(>)g Fp(0)p Fo(;)3437 3138 y Fq(\(9\))324 3376 y Fh(then)k(the)g(p)-5 b(erio)g(d)32 b(function)g Fo(T)46 b Fh(of)32 b(\(2.1\))f(is)i (strictly)f(incr)-5 b(e)g(asing)31 b(in)h(a)h(neighb)-5 b(orho)g(o)g(d)30 b(of)324 3496 y Fp(0)p Fh(;)518 3736 y Fg(if)55 b Fo(g)684 3695 y Fu(\(3\))778 3736 y Fp(\(0\))27 b Fo(>)h Fp(0)34 b Fg(and)1355 3669 y Fo(d)p 1327 3713 107 4 v 1327 3805 a(dx)1443 3736 y Fp(\()1494 3669 y Fo(g)p 1491 3713 56 4 v 1491 3805 a(x)1556 3736 y Fp(\))h Fg(is)1724 3615 y Fk(\032)1837 3676 y Fg(p)-5 b(ositive)42 b(for)j Fo(x)29 b Fj(2)f Fo(J)44 b Fp(and)35 b Fo(x)28 b(>)f Fp(0)p Fo(;)1837 3797 y Fg(ne)-5 b(gative)42 b(for)11 b Fo(x)28 b Fj(2)g Fo(J)44 b Fp(and)35 b Fo(x)28 b(<)f Fp(0)p Fo(;)3388 3736 y Fq(\(10\))324 3975 y Fh(then)32 b(the)f(p)-5 b(erio)g(d)32 b(function)f Fo(T)46 b Fh(of)32 b(\(2.1\))f(is)g(strictly)i(de)-5 b(cr)g(e)g(asing)31 b(in)g(a)h(neighb)-5 b(orho)g(o)g(d)30 b(of)324 4095 y Fp(0)p Fh(.)324 4663 y Fr(3)155 b(Equation)53 b(of)f(Lienard)i(t)l (yp)t(e)324 4882 y Fq(Let)33 b(the)f(equation)1445 5002 y Fo(x)1500 4961 y Fy(00)1565 5002 y Fp(+)22 b Fo(f)11 b Fp(\()p Fo(x)p Fp(\))p Fo(x)1908 4961 y Fy(0)1954 5002 y Fp(+)22 b Fo(g)t Fp(\()p Fo(x)p Fp(\))27 b(=)h(0)p Fo(;)947 b Fq(\(11\))1894 5251 y(10)p eop %%Page: 11 11 11 10 bop 470 548 a Fq(suc)m(h)36 b(that)e Fo(g)t Fp(\(0\))c(=)h(0)p Fq(.)50 b(Th)m(us,)36 b Fo(x)c Fj(\021)g Fp(0)i Fq(is)g(a)g(trivial)e (solution,)i(and)g(the)h(origin)e(is)h(a)324 668 y(singular)d(p)s(oin)m (t)g(of)h(the)h(equiv)-5 b(alen)m(t)33 b(system)1391 765 y Fk(\032)1470 825 y Fo(x)1525 789 y Fy(0)1577 825 y Fp(=)27 b Fj(\000)p Fo(y)1470 945 y(y)1522 909 y Fy(0)1572 945 y Fp(=)h Fo(g)t Fp(\()p Fo(x)p Fp(\))22 b Fj(\000)g Fo(f)11 b Fp(\()p Fo(x)p Fp(\))p Fo(y)3388 886 y Fq(\(12\))470 1109 y(Supp)s(ose)49 b Fp(0)e Fq(is)h(a)f(cen)m(ter)i(of)f(\(3.1\),)j (let)c Fo(\015)2092 1124 y Fu(0)2179 1109 y Fq(the)h Fh(c)-5 b(entr)g(al)48 b(r)-5 b(e)g(gion)p Fq(,)51 b(b)s(e)d(the)g(op)s (en)324 1229 y(connected)d(set)f(co)m(v)m(ered)h(with)e(cycles)h (surrounded)g(the)g(cen)m(ter)h Fp(0)p Fq(.)75 b(This)43 b(p)s(erio)s(dic)324 1350 y(tra)5 b(jectories)28 b(ma)m(y)h(b)s(e)g (parametrized)e(for)h(exemple)h(b)m(y)h(c)m(ho)s(osing)e(their)g (initial)d(v)-5 b(alues)324 1470 y(in)32 b(the)h(segmen)m(t)g Fp(\(0)p Fo(;)17 b(\031)t Fp(\))31 b Fq(on)i(the)g Fo(x)g Fq(-)f(axis.)324 1590 y(Let)46 b Fo(T)66 b Fp(:)51 b Fo(\015)764 1605 y Fu(0)854 1590 y Fj(!)h Fo(R)47 b Fq(,)g(b)s(e)f(the) h(function)f(de\034ned)i(b)m(y)f(asso)s(ciating)e(to)h(ev)m(ery)i(p)s (oin)m(t)324 1711 y Fp(\()p Fo(x;)17 b Fp(0\))29 b Fj(2)h Fo(\015)724 1726 y Fu(0)797 1711 y Fq(the)k(minim)m(um)d(p)s(erio)s(d)h (of)i(the)g(tra)5 b(jectory)34 b(starting)f(at)g Fp(\()p Fo(x;)17 b Fp(0\))p Fq(,)34 b(to)g(reac)m(h)324 1831 y(the)f(negativ)m(e)g(x-axis.)78 b Fo(T)47 b Fq(is)32 b(the)i(p)s(erio)s(d)d(function)i(and)g(is)g(constan)m(t)g(on)g 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b(In)36 b(particular,)e(existence,)j(b) s(oundeness,)h(uniqueness,)g(m)m(ultiplicit)m(y)324 2674 y(of)32 b(p)s(erio)s(dic)f(solutions)g(and)i(related)f(questions)h(w)m (ere)h(debated.)324 2794 y(M.)g(Sabatini)e([S])i(in)m(terested)h(in)e (the)h(monotonicit)m(y)e(of)h(the)h(p)s(erio)s(d)f(function)g Fo(T)48 b Fq(of)33 b(a)324 2914 y(cen)m(ter)h Fp(0)e Fq(of)g(\(3.1\),)g(or)g(in)g(the)h(iso)s(c)m(hronicit)m(y)e(of)i Fp(0)p Fq(.)324 3035 y(Supp)s(ose)c(the)f(functions)g Fo(f)39 b Fq(and)28 b Fo(g)k Fq(are)c(assumed)g(to)g(b)s(e)h Fo(C)2497 2999 y Fu(2)2564 3035 y Fq(functions)f(on)g(an)g(in)m(terv)-5 b(al)324 3155 y Fo(J)45 b Fq(whic)m(h)36 b(satisfy)72 b Fo(f)11 b Fp(\(0\))33 b(=)h Fo(g)t Fp(\(0\))e(=)i(0)p Fo(;)52 b(g)1873 3119 y Fy(0)1896 3155 y Fp(\(0\))33 b Fo(>)g Fp(0)p Fo(:)j Fq(These)i(assumptions)e(ensure)h(the)324 3276 y(origin)30 b Fp(0)i Fq(to)h(b)s(e)f(a)h(cen)m(ter,)h(so)e(the)h (p)s(erio)s(d)f(function)g Fo(T)46 b Fq(is)32 b(de\034ned.)324 3396 y(Since)24 b(m)m(ultiplication)19 b(of)k(the)h(system)h(of)e (equations)h(\(3.2\))g(b)m(y)g Fo(\013)2707 3360 y Fy(\000)p Fu(1)p Fw(=)p Fu(2)2896 3396 y Fq(do)s(es)g(not)g(c)m(hange)324 3516 y(the)35 b(nature)f(\(of)g(monotonicit)m(y\))e(of)i(the)h(p)s (erio)s(d)e(but)i(only)f(c)m(hanges)h(eac)m(h)g(p)s(erio)s(d)f(b)m(y) 324 3637 y(a)e(constan)m(t)h(m)m(ultiple.)470 3878 y(More)24 b(precisely)-8 b(,)26 b(for)d(an)m(y)h(p)s(ositiv)m(e)f(real)g(n)m(um)m (b)s(er)h Fo(\013)48 b Fq(equation)24 b(\(3.1\))f(is)g(equiv)-5 b(alen)m(t)324 3998 y(to)1268 4138 y Fo(X)1357 4097 y Fy(00)1421 4138 y Fp(+)1577 4070 y(1)p 1529 4115 146 4 v 1529 4135 a Fj(p)p 1612 4135 63 4 v 71 x Fo(\013)1685 4138 y(f)11 b Fp(\()p Fo(X)d Fp(\))p Fo(X)1998 4097 y Fy(0)2042 4138 y Fp(+)2157 4070 y(1)p 2150 4115 V 2150 4206 a Fo(\013)2223 4138 y(g)t Fp(\()p Fo(X)g Fp(\))26 b(=)i(0)324 4343 y Fq(b)m(y)33 b(the)g(scaling)1607 4463 y Fo(x)p Fp(\()p Fo(t)p Fp(\))c(=)e Fo(X)8 b Fp(\()2032 4387 y Fj(p)p 2115 4387 V 76 x Fo(\013)q(t)p Fp(\))p Fo(:)470 4618 y Fq(W)-8 b(e)33 b(are)g(led)f(to)g(a)g(system)i(of)e (the)h(form)1179 4806 y Fk(\()1263 4890 y Fo(x)1318 4854 y Fy(0)1369 4890 y Fp(=)27 b Fj(\000)1637 4850 y Fu(1)p 1559 4866 191 4 v 1559 4927 a Fw(\013)1604 4907 y Ff(\000)p Fd(1)p Fc(=)p Fd(2)1760 4890 y Fo(y)1263 5010 y(y)1315 4974 y Fy(0)1365 5010 y Fp(=)1556 4971 y Fu(1)p 1478 4987 V 1478 5047 a Fw(\013)1523 5027 y Ff(\000)p Fd(1)p Fc(=)p Fd(2)1678 5010 y Fo(g)t Fp(\()p Fo(x)p Fp(\))22 b Fj(\000)2069 4971 y Fu(1)p 1992 4987 V 1992 5047 a Fw(\013)2037 5027 y Ff(\000)p Fd(1)p Fc(=)p Fd(2)2192 5010 y Fo(f)11 b Fp(\()p Fo(x)p Fp(\))p Fo(y)3388 4952 y Fq(\(13\))1894 5251 y(11)p eop %%Page: 12 12 12 11 bop 470 548 a Fq(In)37 b(fact)f(it)g(is)g(equiv)-5 b(alen)m(t)37 b(to)f(another)g(system)i(more)e(con)m(v)m(enien)m(t)i (to)e(study)-8 b(,)39 b(W)-8 b(e)324 668 y(ma)m(y)32 b(sho)m(w)i(the)f(follo)m(wing)c(whic)m(h)34 b(agrees)f(with)f(Lemma)f (2)h(of)g([S])h(for)f Fo(g)3032 632 y Fy(0)3055 668 y Fp(\(0\))27 b(=)h(1)470 909 y Fi(Lemma)37 b(1)227 b Fh(Supp)-5 b(ose)33 b Fo(f)5 b(;)17 b(g)38 b Fh(ar)-5 b(e)34 b(c)-5 b(ontinuous)34 b(functions)g(of)h(class)e Fo(C)3205 873 y Fw(k)3248 909 y Fo(;)51 b(k)31 b Fj(\025)d Fp(1)324 1029 y Fh(on)48 b Fo(J)9 b Fh(,)52 b(an)d(interval)f(c)-5 b(ontaining)48 b Fp(0)g Fh(and)g Fo(f)11 b Fp(\(0\))53 b(=)h(0)p Fh(.)86 b(L)-5 b(et)49 b(the)g(function)f Fo(C)7 b Fp(\()p Fo(x)p Fp(\))54 b(=)390 1111 y Fu(1)p 334 1127 149 4 v 334 1184 a Fw(g)370 1165 y Ff(0)392 1184 y Fu(\(0\))492 1150 y Fo(g)t Fp(\()p Fo(x)p Fp(\))22 b Fj(\000)899 1111 y Fu(1)p 806 1127 223 4 v 806 1184 a Fw(g)842 1165 y Ff(0)864 1184 y Fu(\(0\))p Fw(x)993 1165 y Fd(3)1038 1150 y Fp([)1065 1079 y Fk(R)1121 1105 y Fw(x)1105 1175 y Fu(0)1181 1150 y Fo(sf)11 b Fp(\()p Fo(s)p Fp(\))p Fo(ds)p Fp(])1532 1114 y Fu(2)1571 1150 y Fh(.)45 b(Then)34 b(the)g(system)627 1298 y Fk(8)627 1372 y(>)627 1397 y(<)627 1547 y(>)627 1572 y(:)718 1384 y Fo(x)773 1348 y Fy(0)824 1384 y Fp(=)28 b Fo(y)d Fj(\000)1228 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y(at)j Fp(0)g Fq(and)h(its)f(deriv)-5 b(ativ)m(e)32 b(tak)m(es)i(the)f(v)-5 b(alue)2033 2483 y Fw(f)2074 2460 y Ff(0)2097 2483 y Fu(\(0\))p 1977 2506 267 4 v 1977 2584 a(3)2012 2518 y Fj(p)p 2095 2518 149 4 v 66 x Fw(g)2131 2565 y Ff(0)2153 2584 y Fu(\(0\))2253 2530 y Fo(:)324 2696 y Fq(Moreo)m(v)m(er,)30 b(the)e(function)e Fo(C)7 b Fp(\()p Fo(x)p Fp(\))28 b Fq(is)f(ob)m(viously)g(di\033eren)m (tiable)f(and)i(w)m(e)g(get)g Fo(C)3207 2660 y Fy(0)3230 2696 y Fp(\(0\))f(=)h(1)p Fq(.)324 2817 y(A)k(calculation)e(giv)m(es) 411 3075 y Fo(C)488 3034 y Fy(00)530 3075 y Fp(\(0\))e(=)796 3008 y Fo(g)847 2972 y Fy(00)889 3008 y Fp(\(0\))p 796 3052 218 4 v 806 3143 a Fo(g)857 3115 y Fy(0)879 3143 y Fp(\(0\))1046 3075 y Fj(\000)1254 3008 y Fp(2)p 1155 3052 247 4 v 1155 3143 a(3)p Fo(g)1255 3115 y Fy(0)1277 3143 y Fp(\(0\))1412 3075 y Fo(f)11 b Fp(\(0\))p Fo(f)1655 3034 y Fy(0)1677 3075 y Fp(\(0\))98 b Fg(and)107 b Fo(C)2239 3034 y Fy(000)2300 3075 y Fp(\(0\))28 b(=)2566 3008 y Fo(g)2617 2972 y 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b(to)g(\(3.1\))g(b)m(y)h (scaling)f(the)h(time,)e(here)66 b Fo(\013)28 b Fp(=)g Fo(g)2822 4644 y Fy(0)2844 4680 y Fp(\(0\))p Fq(.)k(This)h(giv)m(es) 1552 4912 y Fo(x)p Fp(\()1769 4845 y Fo(\034)p 1655 4889 282 4 v 1655 4906 a Fk(q)p 1738 4906 199 4 v 100 x Fo(g)1789 4977 y Fy(0)1812 5006 y Fp(\(0\))1946 4912 y(\))28 b(=)f Fo(X)8 b Fp(\()p Fo(\034)j Fp(\))1894 5251 y Fq(12)p eop %%Page: 13 13 13 12 bop 324 548 a Fq(w)m(e)40 b(then)g(obtain)e(b)m(y)i(di\033eren)m (tiating)d(with)i(resp)s(ect)h(to)f Fo(\034)11 b Fq(,)41 b(and)f(since)f Fo(f)50 b Fq(and)39 b Fo(g)k Fq(are)324 668 y(indep)s(enden)m(t)33 b(on)g Fo(t)p Fq(,)g(w)m(e)g(ha)m(v)m(e)h Fo(f)11 b Fp(\()p Fo(x)p Fp(\))28 b(=)f Fo(f)11 b Fp(\()p Fo(X)d Fp(\))32 b Fq(and)h Fo(g)t Fp(\()p Fo(x)p Fp(\))27 b(=)h Fo(g)t Fp(\()p Fo(X)8 b Fp(\))1423 909 y Fo(X)1512 868 y Fy(00)1582 909 y Fp(=)28 b Fj(\000)p Fo(g)t Fp(\()p Fo(X)8 b Fp(\))21 b Fj(\000)i Fo(f)11 b Fp(\()p Fo(X)d Fp(\))p Fo(X)2413 868 y Fy(0)2435 909 y Fo(:)470 1183 y Fq(The)35 b(follo)m(wing)c(result)j(sp)s(eci\034es)h(the)f(b)s(eha)m (vior)g(of)f(the)i(p)s(erio)s(d)d(function)i(for)f(the)324 1303 y(Lienard)28 b(system)i(in)e(the)h(neigh)m(b)s(orho)s(o)s(d)e(of)i (the)g(cen)m(ter)h(0.)42 b(W)-8 b(e)29 b(ha)m(v)m(e)h(need)g(nev)m (erthe-)324 1424 y(less)37 b(h)m(yp)s(otheses,)i(to)d(kno)m(w)74 b Fo(f)47 b Fq(is)36 b Fo(C)1728 1388 y Fu(1)1804 1424 y Fq(and)h Fo(g)i Fq(is)d Fo(C)2263 1388 y Fu(3)2302 1424 y Fq(.)56 b(It)36 b(allo)m(ws)f(of)h(some)g(to)g(deduct)324 1544 y(sev)m(eral)44 b(in)m(teresting)g(corollaries.)75 b(Notably)-8 b(,)47 b(simple)42 b(conditions)h(of)h(monotonicit)m(y)324 1665 y(and)32 b(to)h(test)g(in)f(a)g(quic)m(k)h(w)m(a)m(y)h(the)f(iso)s (c)m(hronicit)m(y)e(of)h(the)h(cen)m(ter.)470 1885 y Fi(Theorem)49 b(1.)270 b Fh(L)-5 b(et)44 b(a)h(function)f Fo(f)56 b Fj(2)45 b Fo(C)2271 1848 y Fu(1)2311 1885 y Fp(\([)p Fo(a;)17 b(b)p Fp(]\))45 b Fh(,)h(and)e Fo(g)49 b Fj(2)d Fo(C)3182 1848 y Fu(3)3221 1885 y Fp(\([)p Fo(a;)17 b(b)p Fp(]\))45 b Fh(,)324 2005 y(verifying)34 b Fo(f)11 b Fp(\(0\))27 b(=)h Fo(g)t Fp(\(0\))e(=)i(0)p Fo(;)17 b(g)1491 1969 y Fy(0)1513 2005 y Fp(\(0\))27 b Fo(>)h Fp(0)34 b Fh(and)g(let)h(the)g(origin)f Fp(0)h Fh(b)-5 b(e)35 b(a)f(c)-5 b(enter)35 b(of)1445 2225 y Fo(x)1500 2184 y Fy(00)1565 2225 y Fp(+)22 b Fo(f)11 b Fp(\()p Fo(x)p Fp(\))p Fo(x)1908 2184 y Fy(0)1954 2225 y Fp(+)22 b Fo(g)t Fp(\()p Fo(x)p Fp(\))27 b(=)h(0)p Fo(:)324 2445 y Fh(Then,)992 2604 y Fg(if)120 b Fo(g)1223 2563 y Fy(0)1246 2604 y Fp(\(0\))p Fo(g)1422 2563 y Fu(\(3\))1515 2604 y Fp(\(0\))22 b Fj(\000)1772 2537 y Fp(5)p 1772 2581 49 4 v 1772 2673 a(3)1830 2604 y Fo(g)1881 2563 y Fy(00)p Fu(2)1958 2604 y Fp(\(0\))g Fj(\000)2215 2537 y Fp(2)p 2215 2581 V 2215 2673 a(3)2273 2604 y Fo(f)2332 2563 y Fy(0)p Fu(2)2391 2604 y Fp(\(0\))p Fo(g)2567 2563 y Fy(0)2589 2604 y Fp(\(0\))27 b Fj(6)p Fp(=)h(0)494 b Fq(\(15\))470 2803 y Fh(the)33 b(p)-5 b(erio)g(d)32 b(function)h Fo(T)47 b Fh(of)32 b(a)h(p)-5 b(erio)g(dic)32 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3875 a(3)2273 3807 y Fo(f)2332 3766 y Fy(0)p Fu(2)2391 3807 y Fp(\(0\))p Fo(g)2567 3766 y Fy(0)2589 3807 y Fp(\(0\))27 b Fo(>)h Fp(0)324 4052 y Fh(then)34 b Fo(T)49 b Fh(is)35 b(de)-5 b(cr)g(e)g(asing)33 b(in)i(a)f(neighb)-5 b(orho)g(o)g(d)33 b(of)i Fp(0)p Fh(.)470 4392 y Fi(Pro)s(of)227 b Fq(T)-8 b(o)32 b(establish)g (necessary)j(conditions,)c(w)m(e)i(giv)m(e)f(for)g(that)g(an)g(expan-) 324 4513 y(sion)41 b(of)h(the)g(p)s(erio)s(d)f(function)g(near)i(the)f (cen)m(ter.)73 b(W)-8 b(e)43 b(will)c(use)k(implicit)38 b(function)324 4633 y(tec)m(hniques.)324 4753 y(Since)h(the)h(origin)c (is)j(a)g(cen)m(ter,)j(orbits)d(solutions)f(starting)g(on)h(the)g(p)s (ositiv)m(e)g(x-axis)324 4874 y(m)m(ust)32 b(b)s(e)h(closed.)44 b(Let)65 b Fo(x)p Fp(\()p Fo(t;)17 b(c)p Fp(\))p Fo(;)g(y)t Fp(\()p Fo(t;)g(c)p Fp(\))64 b Fq(b)s(e)33 b(a)f(solution)f(other)i (than)f(the)h(origin)d(of)1894 5251 y(13)p eop %%Page: 14 14 14 13 bop 1134 670 a Fk(8)1134 745 y(<)1134 894 y(:)1225 763 y Fo(x)1280 726 y Fy(0)1331 763 y Fp(=)28 b Fj(\000)1512 663 y Fk(q)p 1595 663 199 4 v 100 x Fo(g)1646 734 y Fy(0)1669 763 y Fp(\(0\))o Fo(y)1225 899 y(y)1277 863 y Fy(0)1327 899 y Fp(=)1538 860 y Fu(1)p 1440 876 232 4 v 1440 887 a Fj(p)p 1523 887 149 4 v 67 x Fw(g)1559 935 y Ff(0)1582 954 y Fu(\(0\))1682 899 y Fo(g)t Fp(\()p Fo(x)p Fp(\))21 b Fj(\000)2093 860 y Fu(1)p 1995 876 232 4 v 1995 887 a Fj(p)p 2078 887 149 4 v 67 x Fw(g)2114 935 y Ff(0)2136 954 y Fu(\(0\))2237 899 y Fo(f)11 b Fp(\()p Fo(x)p Fp(\))p Fo(y)3388 844 y Fq(\(16\))470 1165 y(whic)m(h)34 b(tak)m(e)h(the)f(v)-5 b(alue)66 b Fo(x)p Fp(\(0)p Fo(;)17 b(c)p Fp(\))29 b(=)h Fo(c;)50 b(y)t Fp(\(0)p Fo(;)17 b(c)p Fp(\))28 b(=)h(0)34 b Fq(at)66 b Fo(t)30 b Fp(=)f(0)p Fo(:)34 b Fq(Let)g(us)g(supp)s(ose)h Fo(c)324 1286 y Fq(is)c(a)g(p)s(ositiv)m(e)f(constan)m(t)j(closed)e(to) g Fp(0)p Fq(.)43 b(After)31 b(a)g(certain)g(time)f(closed)i(to)3177 1246 y Fu(2)p Fw(\031)p 3100 1262 232 4 v 3100 1274 a Fj(p)p 3183 1274 149 4 v 66 x Fw(g)3219 1321 y Ff(0)3241 1340 y Fu(\(0\))3404 1286 y Fq(this)324 1446 y(solution)d(will)f(go)i (around)g(the)h(origin)d(and)i(will)e(again)h(in)m(tersect)i(the)g(p)s (ositiv)m(e)f(x-axis)324 1567 y(at)64 b Fo(x)p Fp(\()p Fo(T)8 b(;)17 b(c)p Fp(\))p Fq(.)33 b(Consider)g(the)g(follo)m(wing)d (functions)i(dep)s(endan)m(t)i(on)f Fo(c)p Fq(,)1076 1787 y Fo(\036)p Fp(\()p Fo(T)8 b(;)17 b(c)p Fp(\))27 b(=)h Fo(x)p Fp(\()p Fo(T)8 b(;)17 b(c)p Fp(\))22 b Fj(\000)h Fo(c;)114 b( )t Fp(\()p Fo(T)8 b(;)17 b(c)p Fp(\))28 b(=)f Fo(y)t Fp(\()p Fo(T)8 b(;)17 b(c)p Fp(\))p Fo(:)324 2007 y Fq(W)-8 b(e)32 b(will)e(solv)m(e)65 b Fo( )t Fp(\()p Fo(T)8 b(;)17 b(c)p Fp(\))27 b(=)h(0)k Fq(for)f Fo(T)42 b Fp(=)27 b Fo(T)14 b Fp(\()p Fo(c)p Fp(\))64 b Fq(a)32 b(function)f(on)h Fo(c)g Fq(small.)41 b(Th)m(us,)66 b Fo(\036)32 b Fq(is)g(a)324 2127 y(function)h(on)h Fo(c)p Fq(.)47 b(Let)68 b Fp(\010\()p Fo(c)p Fp(\))31 b(=)f Fo(\036)p Fp(\()p Fo(T)14 b Fp(\()p Fo(c)p Fp(\))p Fo(;)j(c)p Fp(\))p Fq(.)32 b(W)-8 b(e)35 b(\034nd)f(that)g(the)g(p)s(osition)e(of) i(return)g(is)324 2247 y Fo(x)28 b Fp(=)f Fo(c)c Fp(+)f(\010\()p Fo(c)p Fp(\))p Fo(:)65 b Fq(Th)m(us,)34 b(the)f(orbit)f(is)g(closed)g (if)g(and)g(only)g(if)1745 2467 y Fp(\010\()p Fo(c)p Fp(\))c(=)g(0)p Fo(:)324 2687 y Fq(W)-8 b(e)29 b(will)e(kno)m(w)j(the)f (b)s(eha)m(vior)g(of)57 b Fp(\010\()p Fo(c)p Fp(\))h Fq(when)30 b Fo(c)f Fq(tends)h(to)f Fp(0)p Fq(,)g(in)f(calculating)f (its)h(\034rst)324 2808 y(deriv)-5 b(ativ)m(es)33 b(at)h Fp(0)p Fq(.)f(A)m(t)h(\034rst,)g(w)m(e)h(ha)m(v)m(e)68 b Fp(\010\(0\))30 b(=)f(0)p Fo(;)50 b(T)14 b Fp(\(0\))29 b(=)2688 2769 y Fu(2)p Fw(\031)p 2612 2785 232 4 v 2612 2796 a Fj(p)p 2695 2796 149 4 v 66 x Fw(g)2731 2843 y Ff(0)2753 2862 y Fu(\(0\))2920 2808 y Fq(and)34 b(the)g(partial)324 2969 y(deriv)-5 b(ativ)m(es)32 b(of)g Fo(\036)h Fq(and)f Fo( )37 b Fq(are)682 3228 y Fo(\036)740 3243 y Fw(T)795 3228 y Fp(\(0)p Fo(;)17 b Fp(0\))27 b(=)g Fo(x)1198 3187 y Fy(0)1222 3228 y Fp(\()1357 3161 y(2)p Fo(\031)p 1270 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Fp(0\))p Fo(:)470 3877 y Fq(The)34 b(index)e Fo(c)h Fq(or)f Fo(T)46 b Fq(denote)34 b(di\033eren)m(tiation)c(with)i(resp)s(ect)i(to)e Fo(c)h Fq(or)f Fo(T)14 b Fq(.)324 3998 y(The)33 b(deriv)-5 b(ativ)m(es)32 b Fo(x)1064 4013 y Fw(c)1099 3998 y Fp(\()p Fo(t;)17 b(c)p Fp(\))32 b Fq(and)h Fo(y)1566 4013 y Fw(c)1600 3998 y Fp(\()p Fo(t;)17 b(c)p Fp(\))32 b Fq(with)g(resp)s(ect)h(to)f Fo(t)g Fq(satisfy)g(the)h(Lienard)f(sys-)324 4118 y(tem)694 4240 y Fk(8)694 4315 y(<)694 4464 y(:)784 4333 y Fo(x)839 4297 y Fy(0)839 4357 y Fw(c)902 4333 y Fp(=)c Fj(\000)1083 4233 y Fk(q)p 1166 4233 V 100 x Fo(g)1217 4304 y Fy(0)1240 4333 y Fp(\(0\))p Fo(y)1413 4348 y Fw(c)784 4469 y Fo(y)836 4433 y Fy(0)832 4494 y Fw(c)894 4469 y Fp(=)1106 4430 y Fu(1)p 1008 4446 232 4 v 1008 4457 a Fj(p)p 1091 4457 149 4 v 67 x Fw(g)1127 4505 y Ff(0)1149 4524 y Fu(\(0\))1249 4469 y Fo(g)1300 4433 y Fy(0)1323 4469 y Fp(\()p Fo(x)p Fp(\))p Fo(x)1509 4484 y Fw(c)1567 4469 y Fj(\000)1774 4430 y Fu(1)p 1676 4446 232 4 v 1676 4457 a Fj(p)p 1759 4457 149 4 v 67 x Fw(g)1795 4505 y Ff(0)1818 4524 y Fu(\(0\))1918 4469 y Fo(f)11 b Fp(\()p Fo(x)p Fp(\))p Fo(y)2156 4484 y Fw(c)2212 4469 y Fj(\000)2420 4430 y Fu(1)p 2322 4446 232 4 v 2322 4457 a Fj(p)p 2405 4457 149 4 v 67 x Fw(g)2441 4505 y Ff(0)2463 4524 y Fu(\(0\))2563 4469 y Fo(f)2622 4433 y Fy(0)2645 4469 y Fp(\()p Fo(x)p Fp(\))p Fo(x)2831 4484 y Fw(c)2867 4469 y Fo(y)3388 4414 y Fq(\(17\))470 4735 y(with)40 b(initial)d(conditions)80 b Fo(x)1560 4750 y Fw(c)1595 4735 y Fp(\(0)p Fo(;)17 b(c)p Fp(\))40 b(=)h(1)p Fo(;)17 b(y)2104 4750 y Fw(c)2138 4735 y Fp(\(0)p Fo(;)g(c)p Fp(\))40 b(=)h(0)p Fq(.)67 b(A)m(ccording)41 b(to)f(our)g(h)m(y-)324 4856 y(p)s(otheses,)34 b Fo(g)t Fp(\(0\))26 b(=)i Fo(f)11 b Fp(\(0\))27 b(=)h(0)p Fq(,)k(If)g(w)m(e)i (set)f Fo(c)28 b Fp(=)f(0)33 b Fq(the)g(system)g(b)s(ecomes)1894 5251 y(14)p eop %%Page: 15 15 15 14 bop 1328 657 a Fk(8)1328 732 y(<)1328 881 y(:)1418 757 y Fo(x)1473 720 y Fy(0)1473 781 y Fw(c)1508 757 y Fp(\()p Fo(t;)17 b Fp(0\))28 b(=)f Fj(\000)1920 657 y Fk(q)p 2004 657 199 4 v 2004 757 a Fo(g)2055 728 y Fy(0)2077 757 y Fp(\(0\))p Fo(y)2250 772 y Fw(c)1418 918 y Fo(y)1470 881 y Fy(0)1466 942 y Fw(c)1500 918 y Fp(\()p Fo(t;)17 b Fp(0\))28 b(=)1835 818 y Fk(q)p 1918 818 V 100 x Fo(g)1969 889 y Fy(0)1992 918 y Fp(\(0\))o Fo(x)2171 933 y Fw(c)3388 832 y Fq(\(18\))470 1172 y(It)22 b(implies)d(in)i(particular,)44 b Fo(x)1533 1187 y Fw(c)1568 1172 y Fp(\()p Fo(t;)17 b Fp(0\))27 b(=)h(cos\()2071 1072 y Fk(q)p 2154 1072 V 100 x Fo(g)2205 1144 y Fy(0)2228 1172 y Fp(\(0\))p Fo(t)p Fp(\))43 b Fq(and)h Fo(y)2718 1187 y Fw(c)2752 1172 y Fp(\()p Fo(t;)17 b Fp(0\))27 b(=)h(sin)o(\()3244 1072 y Fk(q)p 3327 1072 V 100 x Fo(g)3378 1144 y Fy(0)3401 1172 y Fp(\(0\))p Fo(t)p Fp(\))p Fo(:)324 1301 y Fq(Th)m(us,)34 b Fo(\036)p Fp(\(0)p Fo(;)17 b Fp(0\))26 b(=)i(0)65 b Fq(and)g Fo( )t Fp(\(0)p Fo(;)17 b Fp(0\))27 b(=)g(0)p Fq(.)324 1422 y(W)-8 b(e)33 b(no)m(w)g(calculate)e(the)i(second)h (deriv)-5 b(ativ)m(es)612 1655 y Fo(\036)670 1670 y Fw(T)10 b(T)776 1655 y Fp(\(0)p Fo(;)17 b Fp(0\))26 b(=)i Fo(x)1179 1614 y Fy(00)1222 1655 y Fp(\()1357 1588 y(2)p Fo(\031)p 1270 1632 282 4 v 1270 1649 a Fk(q)p 1353 1649 199 4 v 100 x Fo(g)1404 1720 y Fy(0)1426 1749 y Fp(\(0\))1561 1655 y Fo(;)17 b Fp(0\))27 b(=)h(0)p Fo(;)114 b( )2076 1670 y Fw(T)10 b(T)2182 1655 y Fp(\(0)p Fo(;)17 b Fp(0\))27 b(=)g Fo(y)2582 1614 y Fy(00)2624 1655 y Fp(\()2759 1588 y(2)p Fo(\031)p 2672 1632 282 4 v 2672 1649 a Fk(q)p 2755 1649 199 4 v 100 x Fo(g)2806 1720 y Fy(0)2828 1749 y Fp(\(0\))2963 1655 y Fo(;)17 b Fp(0\))27 b(=)h(0)526 2028 y Fo(\036)584 2043 y Fw(T)10 b(c)669 2028 y Fp(\(0)p Fo(;)17 b Fp(0\))27 b(=)g Fo(x)1072 1987 y Fy(0)1072 2053 y Fw(c)1107 2028 y Fp(\()1242 1961 y(2)p Fo(\031)p 1155 2005 282 4 v 1155 2022 a Fk(q)p 1238 2022 199 4 v 100 x Fo(g)1289 2093 y Fy(0)1312 2122 y Fp(\(0\))1447 2028 y Fo(;)17 b Fp(0\))27 b(=)g(0)p Fo(;)114 b( )1961 2043 y Fw(T)10 b(c)2047 2028 y Fp(\(0)p Fo(;)17 b Fp(0\))27 b(=)h Fo(y)2448 1987 y Fy(0)2444 2053 y Fw(c)2478 2028 y Fp(\()2613 1961 y(2)p Fo(\031)p 2526 2005 282 4 v 2526 2022 a Fk(q)p 2609 2022 199 4 v 100 x Fo(g)2660 2093 y Fy(0)2682 2122 y Fp(\(0\))2817 2028 y Fo(;)17 b Fp(0\))27 b(=)3079 1923 y Fk(q)p 3162 1923 V 105 x Fo(g)3213 2000 y Fy(0)3235 2028 y Fp(\(0\))798 2367 y Fo(\036)856 2382 y Fw(cc)921 2367 y Fp(\(0)p Fo(;)17 b Fp(0\))27 b(=)g Fo(x)1324 2382 y Fw(cc)1390 2367 y Fp(\()1525 2300 y(2)p Fo(\031)p 1438 2344 282 4 v 1438 2361 a Fk(q)p 1521 2361 199 4 v 100 x Fo(g)1572 2432 y Fy(0)1595 2461 y Fp(\(0\))1729 2367 y Fo(;)17 b Fp(0\))p Fo(;)114 b( )2064 2382 y Fw(cc)2130 2367 y Fp(\(0)p Fo(;)17 b Fp(0\))26 b(=)i Fo(y)2526 2382 y Fw(cc)2591 2367 y Fp(\()2725 2300 y(2)p Fo(\031)p 2639 2344 282 4 v 2639 2361 a Fk(q)p 2722 2361 199 4 v 100 x Fo(g)2773 2432 y Fy(0)2795 2461 y Fp(\(0\))2930 2367 y Fo(;)17 b Fp(0\))p Fo(:)470 2635 y Fq(The)34 b(deriv)-5 b(ativ)m(es)32 b(satisfy)h(the)g(system)324 2770 y Fk(8)324 2845 y(<)324 2995 y(:)414 2863 y Fo(x)469 2827 y Fy(0)469 2888 y Fw(cc)563 2863 y Fp(=)27 b Fj(\000)743 2763 y Fk(q)p 827 2763 V 827 2863 a Fo(g)878 2834 y Fy(0)900 2863 y Fp(\(0\))p Fo(y)1073 2878 y Fw(cc)414 3000 y Fo(y)466 2963 y Fy(0)462 3024 y Fw(cc)555 3000 y Fp(=)766 2960 y Fu(1)p 668 2976 232 4 v 668 2988 a Fj(p)p 751 2988 149 4 v 66 x Fw(g)787 3035 y Ff(0)810 3054 y Fu(\(0\))910 3000 y Fp([[)p Fo(g)1015 2963 y Fy(0)1037 3000 y Fp(\()p Fo(x)p Fp(\))c Fj(\000)g Fo(f)1350 2963 y Fy(0)1373 3000 y Fp(\()p Fo(x)p Fp(\))p Fo(y)t Fp(])p Fo(x)1638 3015 y Fw(cc)1725 3000 y Fp(+)f([)p Fo(g)1901 2963 y Fy(00)1943 3000 y Fp(\()p Fo(x)p Fp(\))g Fj(\000)h Fo(f)2255 2963 y Fy(00)2297 3000 y Fp(\()p Fo(x)p Fp(\))p Fo(y)t Fp(]\()p Fo(x)2600 3015 y Fw(c)2635 3000 y Fp(\))2673 2963 y Fu(2)2734 3000 y Fj(\000)g Fo(f)11 b Fp(\()p Fo(x)p Fp(\))p Fo(y)3072 3015 y Fw(cc)3159 3000 y Fj(\000)22 b Fp(2)p Fo(f)3366 2963 y Fy(0)3389 3000 y Fp(\()p Fo(x)p Fp(\))p Fo(x)3575 3015 y Fw(c)3611 3000 y Fo(y)3659 3015 y Fw(c)3693 3000 y Fp(])470 3254 y Fq(Setting)39 b(no)m(w)79 b Fo(c)40 b Fp(=)f(0)79 b Fq(in)38 b(the)i(preceding)g(system,)i(according)d(to)g (ab)s(o)m(v)m(e)h(in)m(tial)324 3374 y(conditions)31 b(w)m(e)j(get)324 3598 y Fk(8)324 3673 y(>)324 3697 y(<)324 3847 y(>)324 3872 y(:)414 3703 y Fo(x)469 3667 y Fy(0)469 3728 y Fw(cc)563 3703 y Fp(=)27 b Fj(\000)743 3603 y Fk(q)p 827 3603 199 4 v 827 3703 a Fo(g)878 3674 y Fy(0)900 3703 y Fp(\(0\))p Fo(y)1073 3718 y Fw(cc)414 3864 y Fo(y)466 3828 y Fy(0)462 3889 y Fw(cc)555 3864 y Fp(=)766 3825 y Fu(1)p 668 3841 232 4 v 668 3852 a Fj(p)p 751 3852 149 4 v 67 x Fw(g)787 3900 y Ff(0)810 3919 y Fu(\(0\))910 3864 y Fp([)p Fo(g)988 3828 y Fy(0)1010 3864 y Fp(\(0\))p Fo(x)1190 3879 y Fw(cc)1278 3864 y Fp(+)22 b([)p Fo(g)1454 3828 y Fy(00)1496 3864 y Fp(\(0\)\(cos\()1827 3764 y Fk(q)p 1910 3764 199 4 v 100 x Fo(g)1961 3835 y Fy(0)1983 3864 y Fp(\(0\))p Fo(t)p Fp(\)\))2219 3828 y Fu(2)2281 3864 y Fj(\000)g Fp(2)p Fo(f)2488 3828 y Fy(0)2511 3864 y Fp(\(0\))17 b(cos\()2821 3764 y Fk(q)p 2904 3764 V 100 x Fo(g)2955 3835 y Fy(0)2978 3864 y Fp(\(0\))o Fo(t)p Fp(\))g(sin\()3350 3764 y Fk(q)p 3433 3764 V 100 x Fo(g)3484 3835 y Fy(0)3506 3864 y Fp(\(0\))p Fo(t)p Fp(\)])324 4118 y Fq(The)33 b(solution)e(of)h(the)h(latter)f(system)h(is)460 4348 y Fk(8)460 4422 y(>)460 4447 y(>)460 4472 y(>)460 4497 y(>)460 4522 y(>)460 4547 y(>)460 4572 y(>)460 4597 y(>)460 4622 y(<)460 4771 y(>)460 4796 y(>)460 4821 y(>)460 4846 y(>)460 4871 y(>)460 4896 y(>)460 4920 y(>)460 4945 y(>)460 4970 y(:)551 4447 y Fo(x)606 4411 y Fy(0)606 4472 y Fw(cc)699 4447 y Fp(=)929 4408 y Fu(1)p 813 4424 267 4 v 813 4502 a(6)848 4435 y Fj(p)p 931 4435 149 4 v 67 x Fw(g)967 4483 y Ff(0)989 4502 y Fu(\(0\))1090 4447 y Fp([)p Fj(\000)p Fp(3)p Fo(g)1294 4411 y Fy(00)1336 4447 y Fp(\(0\))21 b(+)h Fo(g)1631 4411 y Fy(00)1673 4447 y Fp(\(0\))17 b(cos\()1983 4347 y Fk(q)p 2066 4347 199 4 v 100 x Fo(g)2117 4419 y Fy(0)2140 4447 y Fp(\(0\))o Fo(t)p Fp(\))22 b(+)g(4)p Fo(f)2565 4411 y Fy(0)2588 4447 y Fp(\(0\))17 b(sin)o(\()2887 4347 y Fk(q)p 2970 4347 V 100 x Fo(g)3021 4419 y Fy(0)3044 4447 y Fp(\(0\))o Fo(t)p Fp(\)+)746 4646 y Fo(g)797 4610 y Fy(00)839 4646 y Fp(\(0\))g(cos)o(\(2)1197 4546 y Fk(q)p 1280 4546 V 100 x Fo(g)1331 4617 y Fy(0)1354 4646 y Fp(\(0\))o Fo(t)p Fp(\))23 b Fj(\000)f Fp(2)p Fo(f)1781 4610 y Fy(0)1804 4646 y Fp(\(0\))17 b(sin)o(\(2)2152 4546 y Fk(q)p 2235 4546 V 100 x Fo(g)2286 4617 y Fy(0)2308 4646 y Fp(\(0\))p Fo(t)p Fp(\)])551 4807 y Fo(y)603 4771 y Fy(0)599 4832 y Fw(cc)691 4807 y Fp(=)921 4768 y Fu(1)p 805 4784 267 4 v 805 4862 a(3)840 4795 y Fj(p)p 923 4795 149 4 v 67 x Fw(g)959 4843 y Ff(0)982 4862 y Fu(\(0\))1082 4807 y Fp([[)p Fo(g)1187 4771 y Fy(00)1229 4807 y Fp(\(0\))g(sin)o(\()1528 4707 y Fk(q)p 1611 4707 199 4 v 100 x Fo(g)1662 4778 y Fy(0)1684 4807 y Fp(\(0\))p Fo(t)p Fp(\))22 b Fj(\000)h Fp(2)p Fo(f)2112 4771 y Fy(0)2134 4807 y Fp(\(0\))17 b(cos\()2444 4707 y Fk(q)p 2527 4707 V 100 x Fo(g)2578 4778 y Fy(0)2601 4807 y Fp(\(0\))p Fo(t)p Fp(\)+)746 5006 y Fo(g)797 4970 y Fy(00)839 5006 y Fp(\(0\))g(sin)o(\(2)1187 4906 y Fk(q)p 1269 4906 V 1269 5006 a Fo(g)1320 4977 y Fy(0)1343 5006 y Fp(\(0\))p Fo(t)p Fp(\))22 b(+)g(2)p Fo(f)1769 4970 y Fy(0)1792 5006 y Fp(\(0\))17 b(cos\()2102 4906 y Fk(q)p 2185 4906 V 100 x Fo(g)2236 4977 y Fy(0)2258 5006 y Fp(\(0\))p Fo(t)p Fp(\)])1894 5251 y Fq(15)p eop %%Page: 16 16 16 15 bop 470 548 a Fq(W)-8 b(e)33 b(deduce)h(from)d(this)i(the)g(v)-5 b(alues)1444 740 y Fo(\036)1502 755 y Fw(cc)1567 740 y Fp(\(0)p Fo(;)17 b Fp(0\))27 b(=)g Fo( )1978 755 y Fw(cc)2044 740 y Fp(\(0)p Fo(;)17 b Fp(0\))27 b(=)h(0)324 932 y Fq(By)38 b(similar)c(metho)s(d)i(w)m(e)j(establish)e(the)g(v)-5 b(alue)37 b(of)g(the)h(third)e(deriv)-5 b(ativ)m(es.)58 b(W)-8 b(e)38 b(\034nd)324 1052 y(in)32 b(particular)381 1294 y Fo(\036)439 1309 y Fw(T)10 b(cc)555 1294 y Fp(\(0)p Fo(;)17 b Fp(0\))27 b(=)h Fo(x)959 1252 y Fy(0)959 1318 y Fw(cc)1024 1294 y Fp(\()1159 1226 y(2)p Fo(\031)p 1072 1270 282 4 v 1072 1287 a Fk(q)p 1155 1287 199 4 v 100 x Fo(g)1206 1358 y Fy(0)1229 1387 y Fp(\(0\))1363 1294 y Fo(;)17 b Fp(0\))28 b(=)f(0)p Fo(;)212 b( )1976 1309 y Fw(T)10 b(cc)2092 1294 y Fp(\(0)p Fo(;)17 b Fp(0\))27 b(=)h Fo(y)2493 1252 y Fy(0)2489 1318 y Fw(cc)2553 1294 y Fp(\()2688 1226 y(2)p Fo(\031)p 2601 1270 282 4 v 2601 1287 a Fk(q)p 2684 1287 199 4 v 100 x Fo(g)2735 1358 y Fy(0)2758 1387 y Fp(\(0\))2892 1294 y Fo(;)17 b Fp(0\))28 b(=)3220 1226 y Fo(g)3271 1190 y Fy(00)3313 1226 y Fp(\(0\))p 3164 1270 331 4 v 3164 1387 a(2)3213 1287 y Fk(q)p 3296 1287 199 4 v 100 x Fo(g)3347 1358 y Fy(0)3369 1387 y Fp(\(0\))425 1724 y Fo(\036)483 1739 y Fw(T)10 b(T)g(c)619 1724 y Fp(\(0)p Fo(;)17 b Fp(0\))27 b(=)g Fo(x)1022 1683 y Fy(00)1022 1749 y Fw(c)1065 1724 y Fp(\()1200 1657 y(2)p Fo(\031)p 1113 1701 282 4 v 1113 1718 a Fk(q)p 1196 1718 199 4 v 100 x Fo(g)1247 1789 y Fy(0)1270 1818 y Fp(\(0\))1404 1724 y Fo(;)17 b Fp(0\))27 b(=)h Fj(\000)p Fo(g)1794 1683 y Fy(0)1817 1724 y Fp(\(0\))p Fo(;)211 b( )2243 1739 y Fw(T)10 b(T)g(c)2380 1724 y Fp(\(0)p Fo(;)17 b Fp(0\))27 b(=)h Fo(y)2781 1683 y Fy(00)2777 1749 y Fw(c)2822 1724 y Fp([)2946 1657 y(2)p Fo(\031)p 2859 1701 282 4 v 2859 1718 a Fk(q)p 2942 1718 199 4 v 100 x Fo(g)2993 1789 y Fy(0)3016 1818 y Fp(\(0\))3150 1724 y Fo(;)17 b Fp(0\))27 b(=)h(0)569 2166 y Fo(\036)627 2181 y Fw(ccc)723 2166 y Fp(\(0)p Fo(;)17 b Fp(0\))27 b(=)g Fo(x)1126 2181 y Fw(ccc)1223 2166 y Fp(\()1357 2098 y(2)p Fo(\031)p 1271 2142 282 4 v 1271 2159 a Fk(q)p 1354 2159 199 4 v 100 x Fo(g)1405 2230 y Fy(0)1427 2259 y Fp(\(0\))1562 2166 y Fo(;)17 b Fp(0\))27 b(=)1944 2098 y(3)p Fo(\031)p 1833 2142 331 4 v 1833 2259 a Fp(2)1882 2159 y Fk(q)p 1965 2159 199 4 v 100 x Fo(g)2016 2230 y Fy(0)2039 2259 y Fp(\(0\))2173 2166 y([)2225 2098 y Fo(g)2276 2062 y Fy(00)2318 2098 y Fp(\(0\))p 2210 2142 247 4 v 2210 2234 a(2)p Fo(g)2310 2205 y Fy(0)2333 2234 y Fp(\(0\))2539 2098 y Fo(f)2598 2062 y Fy(0)2621 2098 y Fp(\(0\))p 2477 2142 331 4 v 2477 2259 a(2)2526 2159 y Fk(q)p 2609 2159 199 4 v 100 x Fo(g)2660 2230 y 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a(6)p Fo(g)2787 4952 y Fy(0)2810 4981 y Fp(\(0\))2955 4912 y(])p Fo(c)3024 4871 y Fu(2)3086 4912 y Fp(+)g Fo(o)p Fp(\()p Fo(c)3311 4871 y Fu(3)3350 4912 y Fp(\))1894 5251 y Fq(16)p eop %%Page: 17 17 17 16 bop 324 548 a Fq(whic)m(h)33 b(leads)f(to)g(the)h(necessary)i (condition)c(and)i(giv)m(es)g(the)g(theorem.)470 767 y(W)-8 b(e)33 b(obtain)e(in)h(particular)f(this)h(consequence)470 986 y Fi(Corollary)53 b(2)243 b Fh(L)-5 b(et)49 b(the)g(c)-5 b(onservative)47 b(e)-5 b(quation)97 b Fo(x)2713 949 y Fy(00)2788 986 y Fp(+)33 b Fo(g)t Fp(\()p Fo(x)p Fp(\))53 b(=)g(0)97 b Fh(and)324 1106 y(the)42 b(Lienar)-5 b(d)41 b(e)-5 b(quations)42 b Fo(x)1352 1070 y Fy(00)1422 1106 y Fj(\006)28 b Fo(f)11 b Fp(\()p Fo(x)p Fp(\))p Fo(x)1772 1070 y Fy(0)1823 1106 y Fp(+)28 b Fo(g)t Fp(\()p Fo(x)p Fp(\))40 b(=)h(0)p Fh(.)h(Suppp)-5 b(ose)41 b(the)h(two)g(functions)324 1226 y Fo(f)g Fj(2)31 b Fo(C)588 1190 y Fu(1)628 1226 y Fp(\([)p Fo(a;)17 b(b)p Fp(]\))37 b Fh(,)g(and)f Fo(g)e Fj(2)e Fo(C)1445 1190 y Fu(3)1484 1226 y Fp(\([)p Fo(a;)17 b(b)p Fp(]\))37 b Fh(,)g(verifying)f Fo(f)11 b Fp(\(0\))31 b(=)g Fo(g)t Fp(\(0\))f(=)h(0)p Fo(;)17 b(g)3037 1190 y Fy(0)3059 1226 y Fp(\(0\))31 b Fo(>)g Fp(0)36 b Fh(and)324 1347 y(let)f(the)g(origin)f Fp(0)g Fh(b)-5 b(e)35 b(a)g(c)-5 b(enter)34 b(for)h(these)f(e)-5 b(quations.)324 1467 y(If)38 b(the)h(p)-5 b(erio)g(d)38 b(function)h(of)78 b Fo(x)1491 1431 y Fy(00)1559 1467 y Fp(+)25 b Fo(g)t Fp(\()p Fo(x)p Fp(\))34 b(=)i(0)77 b Fh(is)39 b(incr)-5 b(e)g(asing)38 b(in)g(a)h(neighb)-5 b(orho)g(o)g(d)37 b(of)324 1588 y Fp(0)p Fh(,)d(then)g(the)g(p)-5 b(erio)g(d)33 b(function)h(of)68 b Fo(x)1689 1551 y Fy(00)1752 1588 y Fj(\006)21 b Fo(f)11 b Fp(\()p Fo(x)p Fp(\))p Fo(x)2095 1551 y Fy(0)2140 1588 y Fp(+)20 b Fo(g)t Fp(\()p Fo(x)p Fp(\))27 b(=)h(0)34 b Fh(is)g(also)f(incr)-5 b(e)g(asing)33 b(in)h(a)324 1708 y(neighb)-5 b(orho)g(o)g(d)33 b(of)h Fp(0)p Fh(.)470 1927 y Fq(As)e(a)f(consequence)j(of)c(Theorem)h(1,)h(w) m(e)g(ma)m(y)f(deduce)h(the)g(follo)m(wing)c(whic)m(h)k(also)324 2047 y(precises)h(results)g(of)f([S])h(.)470 2266 y Fi(Prop)s(osition) 42 b(3)234 b Fh(L)-5 b(et)41 b(a)f(function)g Fo(f)49 b Fj(2)38 b Fo(C)2303 2230 y Fu(2)2343 2266 y Fp(\([)p Fo(a;)17 b(b)p Fp(]\))40 b Fh(,)i(and)e Fo(g)h Fj(2)e Fo(C)3186 2230 y Fu(3)3225 2266 y Fp(\([)p Fo(a;)17 b(b)p Fp(]\))41 b Fh(,)324 2386 y(verifying)35 b Fo(f)11 b Fp(\(0\))30 b(=)g Fo(g)t Fp(\(0\))g(=)g(0)p Fo(;)17 b(g)1503 2350 y Fy(0)1525 2386 y Fp(\(0\))30 b Fo(>)g Fp(0)36 b Fh(and)g(let)g(the)h(origin)e Fp(0)h Fh(b)-5 b(e)36 b(a)g(c)-5 b(enter)36 b(of)g(\(3.1\))324 2507 y(and)42 b(in)g(addition)g(supp)-5 b(ose)42 b(that)h Fo(g)1659 2471 y Fy(00)1701 2507 y Fp(\(0\))f(=)g Fo(f)2045 2471 y Fy(00)2087 2507 y Fp(\(0\))g(=)g(0)p Fh(.)68 b(L)-5 b(et)86 b Fo(C)7 b Fp(\()p Fo(x)p Fp(\))42 b(=)3173 2467 y Fu(1)p 3116 2483 149 4 v 3116 2541 a Fw(g)3152 2522 y Ff(0)3174 2541 y Fu(\(0\))3275 2507 y Fo(g)t Fp(\()p Fo(x)p Fp(\))27 b Fj(\000)428 2609 y 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b(onc)g(ave)41 b(for)11 b Fo(x)28 b Fj(2)g Fo(J)44 b Fp(and)35 b Fo(x)28 b(<)f Fp(0)p Fo(;)3388 3443 y Fq(\(20\))324 3697 y Fh(then)34 b Fo(T)49 b Fh(is)35 b(de)-5 b(cr)g(e)g(asing)33 b(in)i(a)f(neighb)-5 b(orho)g(o)g(d)33 b(of)i Fp(0)p Fh(;)1295 3968 y Fg(if)1449 3901 y Fo(d)1500 3865 y Fu(2)p 1421 3945 146 4 v 1421 4037 a Fo(dx)1527 4008 y Fu(2)1576 3968 y Fo(C)7 b Fp(\()p Fo(x)p Fp(\))28 b Fj(\021)h Fp(0)34 b Fg(for)46 b Fo(x)28 b Fj(2)g Fo(J)n(;)964 b Fq(\(21\))324 4211 y Fh(then)34 b Fo(T)49 b Fh(is)35 b(c)-5 b(onstant)34 b(in)h(a)f(neighb)-5 b(orho)g(o)g(d)33 b(of)i Fp(0)p Fh(.)470 4452 y Fi(Pro)s(of)224 b Fq(After)29 b(c)m(hange)h(in)e(p)s (olar)g(co)s(ordinates)g Fp(\()p Fo(r)m(;)17 b(\022)s Fp(\))p Fq(,)30 b(w)m(e)g(obtain)e(the)i(follo)m(w-)324 4572 y(ing)h(equiv)-5 b(alen)m(t)33 b(system)g(to)f(\(3.4\))633 4703 y Fk(8)633 4778 y(>)633 4803 y(<)633 4952 y(>)633 4977 y(:)723 4813 y Fo(r)770 4777 y Fy(0)821 4813 y Fp(=)924 4713 y Fk(q)p 1007 4713 199 4 v 100 x Fo(g)1058 4784 y Fy(0)1081 4813 y Fp(\(0\))p Fo(r)19 b Fp(cos)e Fo(\022)j Fp(sin)c Fo(\022)25 b Fj(\000)e Fo(r)s(\014)6 b Fp(\()p Fo(r)19 b Fp(cos)e Fo(\022)s Fp(\))22 b Fj(\000)h Fp(sin)16 b Fo(\022)36 b(C)7 b Fp(\()p Fo(r)19 b Fp(cos)e Fo(\022)s Fp(\))723 4965 y Fo(\022)771 4929 y Fy(0)822 4965 y Fp(=)28 b Fj(\000)1111 4926 y Fu(1)p 1013 4942 232 4 v 1013 4953 a Fj(p)p 1096 4953 149 4 v 67 x Fw(g)1132 5001 y Ff(0)1155 5020 y Fu(\(0\))1271 4965 y Fp(cos)1402 4929 y Fu(2)1458 4965 y Fo(\022)d Fj(\000)e Fp(sin)1748 4925 y Fu(2)1804 4965 y Fo(\022)i Fj(\000)1984 4919 y Fu(cos)12 b Fw(\022)r(C)5 b Fu(\()p Fw(r)14 b Fu(cos)e Fw(\022)r Fu(\))p Fy(\000)p Fw(C)2531 4895 y Ff(0)2554 4919 y Fu(\(0\)\()p Fw(r)j Fu(cos)d Fw(\022)r Fu(\))p 1984 4942 902 4 v 2418 4999 a Fw(r)3388 4902 y Fq(\(22\))1894 5251 y(17)p eop %%Page: 18 18 18 17 bop 470 548 a Fq(W)-8 b(e)33 b(also)f(observ)m(e)i(that)e(from)g (system)h(\(3.1\))f(w)m(e)i(get)1263 762 y Fo(r)1310 721 y Fu(2)1349 762 y Fo(\022)1397 721 y Fy(0)1448 762 y Fp(=)27 b Fo(r)1598 721 y Fu(2)1638 762 y Fo(!)t Fp(\()p Fo(r)m(;)17 b(\022)s Fp(\))27 b(=)g Fj(\000)p Fo(xC)7 b Fp(\()p Fo(x)p Fp(\))23 b Fj(\000)g Fo(y)2557 721 y Fu(2)2596 762 y Fo(;)324 977 y Fq(here)j Fo(T)14 b Fp(\()p Fo(r)s Fp(\))27 b(=)853 906 y Fk(R)892 1002 y Fu([0)p Fw(;)p Fu(2)p Fw(\031)r Fu(])1095 938 y Fw(d\022)p 1095 954 72 4 v 1107 1011 a(!)1176 977 y Fq(.)e(By)h(Theorem)g(1)f(of)g ([S],)h(it)f(is)g(su\036cien)m(t)h(to)f(pro)m(v)m(e)i(for)e(example)324 1123 y(that)30 b(h)m(yp)s(othesis)h(\(3.9\))f(implies)d(that)1789 1077 y Fw(\016)r(!)r Fu(\()p Fw(r)n(;\022)r Fu(\))p 1789 1100 220 4 v 1865 1158 a Fw(\016)r(r)2046 1123 y Fj(\024)h Fp(0)i Fq(for)g(almost)e(all)g(v)-5 b(alues)30 b Fo(\022)h Fj(2)d Fp([0)p Fo(;)17 b Fp(2)p Fo(\031)t Fp(])p Fq(.)470 1244 y(A)33 b(calculus)f(giv)m(es)921 1440 y Fo(\016)t(!)t Fp(\()p Fo(r)m(;)17 b(\022)s Fp(\))p 921 1484 321 4 v 1035 1576 a Fo(\016)t(r)1279 1508 y Fp(=)28 b Fj(\000)1494 1440 y Fo(\016)p 1470 1484 94 4 v 1470 1576 a(\016)t(r)1584 1440 y Fp([cos)17 b Fo(\022)s Fp(\()p Fo(C)7 b Fp(\()p Fo(r)19 b Fp(cos)f Fo(\022)s Fp(\))k Fj(\000)g Fo(C)2454 1404 y Fy(0)2478 1440 y Fp(\(0\)\()p Fo(r)d Fp(cos)e Fo(\022)s Fp(\)])p 1584 1484 1381 4 v 2251 1576 a Fo(r)1114 1762 y Fp(=)28 b Fo(r)19 b Fp(cos)1412 1721 y Fu(2)1468 1762 y Fo(\022)35 b(C)1625 1721 y Fy(0)1649 1762 y Fp(\()p Fo(r)19 b Fp(cos)e Fo(\022)s Fp(\))22 b Fj(\000)h Fp(cos)17 b Fo(\022)36 b(C)7 b Fp(\()p Fo(r)19 b Fp(cos)e Fo(\022)s Fp(\))p Fo(:)470 1934 y Fq(W)-8 b(e)33 b(then)g(obtain)926 2116 y Fo(\016)t(!)t Fp(\()p Fo(r)m(;)17 b(\022)s Fp(\))p 926 2160 321 4 v 1039 2251 a Fo(\016)t(r)1284 2183 y Fp(=)1397 2116 y Fo(xC)7 b Fp(\()p Fo(x)p Fp(\))23 b Fj(\000)g Fo(x)1838 2079 y Fu(2)1877 2116 y Fo(C)1954 2079 y Fy(0)1978 2116 y Fp(\()p Fo(x)p Fp(\))p 1397 2160 712 4 v 1507 2251 a(\()p Fo(x)1600 2223 y Fu(2)1662 2251 y Fp(+)f Fo(y)1812 2223 y Fu(2)1851 2251 y Fp(\))1889 2223 y Fu(3)p Fw(=)p Fu(2)2147 2183 y Fp(=)27 b Fo(x)2315 2116 y(C)7 b Fp(\()p 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2687 2738 a Fw(\016)r(r)2868 2703 y Fj(\024)c Fp(0)p Fq(.)324 2824 y(By)33 b(the)g(same)f(w)m(a)m(y)-8 b(,)33 b(w)m(e)h(pro)m(v)m(e)g (condition)d Fo(xC)2072 2787 y Fy(00)2114 2824 y Fp(\()p Fo(x)p Fp(\))d Fj(\025)h Fp(0)j Fq(for)g Fo(x)c Fj(2)g Fo(J)41 b Fq(whic)m(h)33 b(is)f(equiv)-5 b(a-)324 2953 y(len)m(t)32 b(to)g(h)m(yp)s(othesis)i(\(3.6\))e(implies)1690 2907 y Fw(\016)r(!)r Fu(\()p Fw(r)n(;\022)r Fu(\))p 1690 2930 V 1766 2988 a Fw(\016)r(r)1947 2953 y Fo(>)27 b Fp(0)p Fq(.)324 3074 y(W)-8 b(e)33 b(then)g(pro)m(v)m(e)h(the)f (functions)1575 3034 y Fw(\016)r(!)p 1575 3050 80 4 v 1581 3108 a(\016)r(r)1697 3074 y Fq(and)g Fo(xC)2019 3037 y Fy(00)2062 3074 y Fp(\()p Fo(x)p Fp(\))g Fq(ha)m(v)m(e)h(a)e (same)g(sign.)324 3194 y(In)h(fact,)f(w)m(e)i(can)e(that)h(with)f(a)g (other)h(manner.)43 b(A)33 b(calculation)d(giv)m(es)411 3458 y Fo(C)488 3417 y Fy(00)530 3458 y Fp(\(0\))e(=)796 3390 y Fo(g)847 3354 y Fy(00)889 3390 y Fp(\(0\))p 796 3435 218 4 v 806 3526 a Fo(g)857 3497 y Fy(0)879 3526 y Fp(\(0\))1046 3458 y Fj(\000)1254 3390 y Fp(2)p 1155 3435 247 4 v 1155 3526 a(3)p Fo(g)1255 3497 y Fy(0)1277 3526 y Fp(\(0\))1412 3458 y Fo(f)11 b Fp(\(0\))p Fo(f)1655 3417 y Fy(0)1677 3458 y Fp(\(0\))98 b Fg(and)107 b Fo(C)2239 3417 y Fy(000)2300 3458 y Fp(\(0\))28 b(=)2566 3390 y Fo(g)2617 3354 y Fy(000)2678 3390 y Fp(\(0\))p 2566 3435 237 4 v 2585 3526 a Fo(g)2636 3497 y Fy(0)2659 3526 y Fp(\(0\))2835 3458 y Fj(\000)3044 3390 y Fp(2)p 2944 3435 247 4 v 2944 3526 a(3)p Fo(g)3044 3497 y Fy(0)3067 3526 y Fp(\(0\))3201 3458 y Fo(f)3260 3417 y Fy(0)3283 3458 y Fp(\(0\))3408 3417 y Fu(2)3448 3458 y Fo(:)324 3728 y Fq(Then)k(,)g(w)m(e)g(necessarely)h(ha)m(v)m(e)63 b Fo(C)1610 3692 y Fy(00)1653 3728 y Fp(\(0\))27 b(=)h(0)p Fq(,)j(since)62 b Fo(xC)2416 3692 y Fy(00)2459 3728 y Fp(\()p Fo(x)p Fp(\))28 b Fj(6)p Fp(=)g(0)62 b Fq(\(if)30 b Fo(x)e Fj(6)p Fp(=)f(0)p Fq(\))k(implies)324 3848 y(the)e (monotonicit)m(y)d(of)i(the)h(p)s(erio)s(d.)41 b(Th)m(us,)60 b Fo(xC)2117 3812 y Fy(00)2160 3848 y Fp(\()p Fo(x)p Fp(\))28 b Fq(and)58 b Fo(g)2585 3812 y Fy(000)2645 3848 y Fp(\(0\))14 b Fj(\000)2885 3809 y Fu(2)p 2885 3825 36 4 v 2885 3883 a(3)2931 3848 y Fo(f)2990 3812 y Fy(0)3013 3848 y Fp(\(0\))3138 3812 y Fu(2)3205 3848 y Fq(ha)m(v)m(e)30 b(the)324 3969 y(same)35 b(sign)f(in)g(a)h(neigh)m(b)s(orho)s(o)s(d)e (of)i Fp(0)p Fq(.)50 b(This)35 b(determines)g(the)h(monotonicit)m(y)d (of)h(the)324 4089 y(p)s(erio)s(d)d(function.)324 4210 y(Also,)h(condition)f Fo(f)1058 4173 y Fy(00)1100 4210 y Fp(\(0\))c(=)h(0)k Fq(app)s(ears)h(to)f(b)s(e)h(necessary)i(b)m(y)e (the)g(follo)m(wing)d(Lemma.)470 4547 y Fi(Lemma)40 b(2)230 b Fh(L)-5 b(et)38 b Fo(f)43 b Fj(2)33 b Fo(C)1594 4511 y Fu(2)1633 4547 y Fp(\([)p Fo(a;)17 b(b)p Fp(]\))38 b Fh(,)g(and)e Fo(g)g Fj(2)d Fo(C)2455 4511 y Fu(3)2494 4547 y Fp(\([)p Fo(a;)17 b(b)p Fp(]\))38 b Fh(,)g(verifying)e Fo(f)11 b Fp(\(0\))32 b(=)324 4667 y Fo(g)t Fp(\(0\))40 b(=)i(0)p Fo(;)17 b(g)802 4631 y Fy(0)824 4667 y Fp(\(0\))41 b Fo(>)g Fp(0)h Fh(and)g(let)g(the)h(Lienar)-5 b(d)41 b(e)-5 b(quation)42 b(\(3.1\).)66 b(Then,)43 b(a)f(ne)-5 b(c)g(essary)324 4788 y(c)g(ondition)34 b(for)g(the)h(origin)f Fp(0)h Fh(to)g(b)-5 b(e)34 b(a)h(c)-5 b(enter)35 b(of)f(\(3.1\))g(is) 1343 5002 y Fo(f)1402 4961 y Fy(0)1425 5002 y Fp(\(0\))p Fo(g)1601 4961 y Fy(00)1643 5002 y Fp(\(0\))22 b Fj(\000)g Fp(2)p Fo(g)1989 4961 y Fy(0)2012 5002 y Fp(\(0\))p Fo(f)2196 4961 y Fy(00)2237 5002 y Fp(\(0\))28 b(=)f(0)1894 5251 y Fq(18)p eop %%Page: 19 19 19 18 bop 470 548 a Fi(Pro)s(of)195 b Fq(In)34 b(the)g(pro)s(of)f(of)g (Theorem)h(1,)f(w)m(e)i(ha)m(v)m(e)g(giv)m(en)f(the)g(b)s(eha)m(vior)f (of)g(the)324 668 y(p)s(erio)s(d)40 b(near)i(the)f(origin.)110 b Fp(\010\()p Fo(c)p Fp(\))43 b Fj(6)p Fp(=)f(0)f Fq(for)g(small)e Fo(c)j Fq(means)f(there)h(will)d(b)s(e)j(no)f(an)m(y)324 789 y(closed)i(orbit)f(in)h(a)g(neigh)m(b)s(orho)s(o)s(d)f(of)h Fp(0)p Fq(.)76 b(Indeed,)47 b Fo(c)30 b Fp(+)f(\010\()p Fo(c)p Fp(\))44 b Fq(corresp)s(onds)g(to)f(the)324 909 y(p)s(osition)30 b(of)j(return)g(to)f(the)h(x-axis.)43 b(Since,)33 b(w)m(e)g(ha)m(v)m(e)h(seen)557 1176 y Fp(\010\()p Fo(c)p Fp(\))28 b(=)g Fo(\036)p Fp(\()p Fo(T)14 b Fp(\()p Fo(c)p Fp(\))p Fo(;)j(c)p Fp(\))27 b(=)g Fj(\000)1503 1109 y Fo(g)1554 1073 y Fy(0)1577 1109 y Fp(\(0\))p 1503 1153 199 4 v 1578 1244 a(2)1712 1176 y(\()p Fo(T)36 b Fj(\000)2039 1109 y Fp(2)p Fo(\031)p 1952 1153 282 4 v 1952 1169 a Fk(q)p 2035 1169 199 4 v 100 x Fo(g)2086 1241 y Fy(0)2109 1269 y Fp(\(0\))2243 1176 y(\))2281 1135 y Fu(2)2321 1176 y Fo(c)22 b Fp(+)2493 1109 y(1)p 2493 1153 49 4 v 2493 1244 a(6)2552 1176 y Fo(\036)2610 1191 y Fw(ccc)2705 1176 y Fp(\(0)p Fo(;)17 b Fp(0\))p Fo(c)2965 1135 y Fu(3)3026 1176 y Fp(+)22 b Fo(o)p Fp(\()p Fo(c)3251 1135 y Fu(3)3290 1176 y Fp(\))324 1496 y Fq(and)32 b(according)g(the)h(expansion)g(of)f Fo(T)14 b Fp(\()p Fo(c)p Fp(\))p Fq(,)32 b(w)m(e)i(get)1360 1758 y Fp(\010\()p Fo(c)p Fp(\))28 b(=)1690 1690 y(1)p 1690 1735 V 1690 1826 a(6)1749 1758 y Fo(\036)1807 1773 y Fw(ccc)1902 1758 y Fp(\(0)p Fo(;)17 b Fp(0\))p Fo(c)2162 1717 y Fu(3)2223 1758 y Fp(+)22 b Fo(o)p Fp(\()p Fo(c)2448 1717 y Fu(3)2487 1758 y Fp(\))470 2000 y Fq(Th)m(us,)34 b(a)f(necessary)h(condition)d (for)h(the)h(origin)e(to)h(b)s(e)h(a)f(cen)m(ter)i(is)e(:)855 2271 y Fo(\036)913 2286 y Fw(ccc)1009 2271 y Fp(\(0)p Fo(;)17 b Fp(0\))27 b(=)1478 2204 y(3)p Fo(\031)p 1367 2248 331 4 v 1367 2365 a Fp(2)1416 2264 y Fk(q)p 1499 2264 199 4 v 101 x Fo(g)1550 2336 y Fy(0)1573 2365 y Fp(\(0\))1707 2271 y([)1759 2204 y Fo(g)1810 2168 y Fy(00)1852 2204 y Fp(\(0\))p 1744 2248 247 4 v 1744 2339 a(2)p Fo(g)1844 2311 y Fy(0)1867 2339 y Fp(\(0\))2073 2204 y Fo(f)2132 2168 y Fy(0)2155 2204 y Fp(\(0\))p 2011 2248 331 4 v 2011 2365 a(2)2060 2264 y Fk(q)p 2143 2264 199 4 v 101 x Fo(g)2194 2336 y Fy(0)2217 2365 y Fp(\(0\))2373 2271 y Fj(\000)2535 2204 y Fo(f)2594 2168 y Fy(00)2636 2204 y Fp(\(0\))p 2483 2248 331 4 v 2483 2365 a(2)2532 2264 y Fk(q)p 2615 2264 199 4 v 101 x Fo(g)2666 2336 y Fy(0)2688 2365 y Fp(\(0\))2823 2271 y(])h(=)f(0)470 2540 y Fq(The)34 b(lemma)c(is)i(then)h(pro)m(v)m(ed.)470 2759 y(Notice)46 b(that)f(b)m(y)i(de\034nition)e(of)91 b Fo(C)7 b Fp(\()p Fo(x)p Fp(\))p Fo(;)92 b(f)11 b Fp(\()p Fo(x)p Fp(\))51 b Fj(\021)g Fp(0)91 b Fq(implies)f Fo(C)7 b Fp(\()p Fo(x)p Fp(\))50 b Fj(\021)3412 2713 y Fw(g)r Fu(\()p Fw(x)p Fu(\))p 3403 2736 149 4 v 3403 2793 a Fw(g)3439 2775 y Ff(0)3462 2793 y Fu(\(0\))324 2879 y Fq(and)34 b(Corollary)f(2)h (reduces)i(to)e(the)h(ab)s(o)m(v)m(e)g(Prop)s(osition)d(1.)49 b(Remark)34 b(also)f(condition)324 3000 y Fo(g)375 2964 y Fy(00)417 3000 y Fp(\(0\))d(=)g(0)k Fq(app)s(ears)h(to)f(b)s(e)g (necessary)j(to)d(study)h(the)g(v)-5 b(arious)33 b(monotonicit)m(y)g (condi-)324 3120 y(tions)f(of)g(the)h(p)s(erio)s(d)e(function.)470 3361 y(The)40 b(follo)m(wing)c(whic)m(h)j(ma)m(y)f(b)s(e)h(deduce)h (from)e(Theorem)h(1)f(is)g(particularly)f(in-)324 3481 y(teresting)470 3700 y Fi(Corollary)53 b(3)243 b Fh(L)-5 b(et)49 b(the)g(c)-5 b(onservative)47 b(e)-5 b(quation)97 b Fo(x)2713 3664 y Fy(00)2788 3700 y Fp(+)33 b Fo(g)t Fp(\()p Fo(x)p Fp(\))53 b(=)g(0)97 b Fh(and)324 3820 y(the)42 b(Lienar)-5 b(d)41 b(e)-5 b(quations)42 b Fo(x)1352 3784 y Fy(00)1422 3820 y Fj(\006)28 b Fo(f)11 b Fp(\()p Fo(x)p Fp(\))p Fo(x)1772 3784 y Fy(0)1823 3820 y Fp(+)28 b Fo(g)t Fp(\()p Fo(x)p Fp(\))40 b(=)h(0)p Fh(.)h(Suppp)-5 b(ose)41 b(the)h(two)g(functions)324 3941 y Fo(f)g Fj(2)31 b Fo(C)588 3905 y Fu(1)628 3941 y Fp(\([)p Fo(a;)17 b(b)p Fp(]\))37 b Fh(,)g(and)f Fo(g)e Fj(2)e Fo(C)1445 3905 y Fu(3)1484 3941 y Fp(\([)p Fo(a;)17 b(b)p Fp(]\))37 b Fh(,)g(verifying)f Fo(f)11 b Fp(\(0\))31 b(=)g Fo(g)t Fp(\(0\))f(=)h(0)p Fo(;)17 b(g)3037 3905 y Fy(0)3059 3941 y Fp(\(0\))31 b Fo(>)g Fp(0)36 b Fh(and)324 4061 y(let)f(the)g(origin)f Fp(0)g Fh(b)-5 b(e)35 b(a)g(c)-5 b(enter)34 b(of)h(these)f(e)-5 b(quations.)324 4182 y(If)83 b Fo(g)526 4145 y Fy(00)568 4182 y Fp(\(0\))40 b Fj(6)p Fp(=)g(0)84 b Fh(and)41 b(the)h(c)-5 b(enter)41 b Fp(0)h Fh(of)83 b Fo(x)1955 4145 y Fy(00)2025 4182 y Fj(\006)28 b Fo(f)11 b Fp(\()p Fo(x)p Fp(\))p Fo(x)2375 4145 y Fy(0)2426 4182 y Fp(+)27 b Fo(g)t Fp(\()p Fo(x)p Fp(\))40 b(=)h(0)g Fh(is)h(iso)-5 b(chr)g(onous)324 4302 y(then)42 b(the)g(p)-5 b(erio)g(d)41 b(function)84 b Fo(T)98 b Fh(of)84 b Fo(x)1821 4266 y Fy(00)1891 4302 y Fp(+)27 b Fo(g)t Fp(\()p Fo(x)p Fp(\))41 b(=)g(0)84 b Fh(is)42 b(strictly)g(de)-5 b(cr)g(e)g(asing)41 b(in)h(a)324 4422 y(neighb)-5 b(orho)g(o)g(d)33 b(of)h Fp(0)p Fh(.)470 4641 y Fq(W)-8 b(e)42 b(pro)m(v)m(e)h(that)e(in)g (considering)g(Corrolary)f(2)i(,)h(and)f(on)f(accoun)m(t)i(of)e(the)h (fact)324 4761 y(that)32 b Fo(g)k Fq(is)c(not)g(a)h(o)s(dd)f(function.) 324 4882 y(A)m(ccording)f(to)f(this)h(result,)g(to)g(establish)f(the)h (existence)i(of)d(iso)s(c)m(hronous)h(cen)m(ters)i(for)324 5002 y(the)k(Lienard)f(sytem,)j(it)c(is)i(necessary)i(\034rst)e(to)g (mak)m(e)g(sure)g(that)g(the)g(p)s(erio)s(d)f(of)g(the)1894 5251 y(19)p eop %%Page: 20 20 20 19 bop 324 548 a Fq(conserv)-5 b(ativ)m(e)35 b(asso)s(ciated)e (system)i(is)f(decreasing)g(at)g(least)f(in)g(a)h(neigh)m(b)s(orho)s(o) s(d)e(of)i Fp(0)324 668 y Fq(.)42 b(One)30 b(will)d(b)s(e)i(able)g(to)g (use)h(for)f(that)g(the)h(di\033eren)m(t)f(criteria)f(of)h(monotonicit) m(y)e(of)i(the)324 789 y(p)s(erio)s(d)i(function.)470 1129 y(F)-8 b(rom)31 b(Theorem)i(1)f(w)m(e)i(ma)m(y)e(deduce)i(other)f (consequences.)470 1469 y Fi(Corollary)c(4)222 b Fh(L)-5 b(et)30 b(a)g(function)f Fo(f)38 b Fj(2)29 b Fo(C)2118 1433 y Fu(1)2157 1469 y Fp(\([)p Fo(a;)17 b(b)p Fp(]\))30 b Fh(,)h(and)e Fo(g)k Fh(a)c(smo)-5 b(oth)29 b(function,)324 1590 y(verifying)38 b Fo(f)11 b Fp(\(0\))36 b(=)g Fo(g)t Fp(\(0\))p Fo(;)55 b(g)1371 1554 y Fy(0)1394 1590 y Fp(\(0\))35 b Fo(>)h Fp(0)k Fh(and)78 b Fo(g)2039 1554 y Fy(00)2081 1590 y Fp(\(0\))35 b Fj(6)p Fp(=)h(0)p Fo(;)79 b Fh(and)39 b(let)g(the)h(origin)f Fp(0)g Fh(b)-5 b(e)39 b(a)324 1710 y(c)-5 b(enter)25 b(of)f(\(3.1\))h(.)41 b(Then)24 b(ne)-5 b(c)g(essary)25 b(c)-5 b(onditions)23 b(on)i(the)g(function)50 b Fo(f)36 b Fh(so)25 b(that)g(e)-5 b(quation)324 1831 y(\(3.1\))34 b(has)g Fo(T)49 b Fh(c)-5 b(onstant)34 b(in)h(a)f(neighb) -5 b(orho)g(o)g(d)33 b(of)i Fp(0)f Fh(ar)-5 b(e)324 2119 y Fo(f)383 2078 y Fy(0)406 2119 y Fp(\(0\))27 b(=)h Fj(\006)739 1970 y Fk(s)p 822 1970 911 4 v 149 x Fp(3)p Fo(g)922 2090 y Fu(\(3\))1016 2119 y Fp(\(0\))21 b Fj(\000)1347 2052 y Fp(5)p 1272 2096 199 4 v 1272 2188 a Fo(g)1323 2159 y Fy(0)1346 2188 y Fp(\(0\))1480 2119 y Fo(g)1531 2090 y Fy(00)p Fu(2)1608 2119 y Fp(\(0\))p Fo(;)216 b(f)2035 2078 y Fy(00)2077 2119 y Fp(\(0\))27 b(=)h Fj(\006)2435 2052 y Fo(g)2486 2016 y Fy(00)2528 2052 y Fp(\(0\))p 2420 2096 247 4 v 2420 2188 a(2)p Fo(g)2520 2159 y Fy(0)2542 2188 y Fp(\(0\))2677 1970 y Fk(s)p 2760 1970 911 4 v 149 x Fp(3)p Fo(g)2860 2090 y Fu(\(3\))2954 2119 y Fp(\(0\))21 b Fj(\000)3285 2052 y Fp(5)p 3210 2096 199 4 v 3210 2188 a Fo(g)3261 2159 y Fy(0)3283 2188 y Fp(\(0\))3418 2119 y Fo(g)3469 2090 y Fy(00)p Fu(2)3546 2119 y Fp(\(0\))470 2559 y Fq(Indeed,)46 b(it)41 b(su\036ces)k(to)d(remark)g(b)m(y)h (Corollary)d(3)i(the)h(p)s(erio)s(d)e(function)g(of)h(the)324 2680 y(equation)32 b Fo(x)777 2643 y Fy(00)842 2680 y Fp(+)23 b Fo(g)t Fp(\()p Fo(x)p Fp(\))k(=)h(0)66 b Fq(has)33 b(to)f(b)s(e)h(decreasing.)45 b(Then,)34 b(according)e(to)h(criteria)e (of)324 2800 y(Sc)m(haaf)g(it)e(is)i(necessary)i(that)61 b Fp(3)p Fo(g)1593 2764 y Fu(\(3\))1686 2800 y Fp(\()p Fo(x)p Fp(\))19 b Fj(\000)2001 2761 y Fu(5)p 1942 2777 153 4 v 1942 2834 a Fw(g)1978 2815 y Ff(0)2000 2834 y Fu(\()p Fw(x)p Fu(\))2105 2800 y Fo(g)2156 2764 y Fy(00)p Fu(2)2233 2800 y Fp(\()p Fo(x)p Fp(\))28 b Fo(>)f Fp(0)k Fq(in)f(a)g(neigh)m(b)s(orho)s(o)s(d)g(of)g Fp(0)p Fq(.)324 2934 y(This)24 b(result)g(is)g(in)m(teresting)g(since)g(w)m(e)h(do)f (not)h(need)g(to)f(supp)s(ose)49 b Fo(g)t Fp(\()p Fo(x)p Fp(\))24 b Fq(o)s(dd.)40 b(In)25 b(partic-)324 3074 y(ular,)f(if)e Fo(g)675 3038 y Fy(00)717 3074 y Fp(\(0\))27 b(=)h(0)22 b Fq(w)m(e)j(ha)m(v)m(e)1447 3035 y Fw(d)1483 3012 y Fd(2)p 1427 3051 111 4 v 1427 3109 a Fw(dx)1503 3090 y Fd(2)1548 3074 y Fo(C)7 b Fp(\()p Fo(x)p Fp(\))28 b Fj(\021)g Fp(0)23 b Fq(implies)44 b Fo(C)2383 3038 y Fy(000)2444 3074 y Fp(\(0\))28 b(=)2710 3028 y Fw(g)2746 3005 y Fd(\(3\))2829 3028 y Fu(\(0\))p 2710 3051 209 4 v 2740 3109 a Fw(g)2776 3090 y Ff(0)2799 3109 y Fu(\(0\))2932 3074 y Fj(\000)3097 3035 y Fu(2)p 3022 3051 184 4 v 3022 3109 a(3)p Fw(g)3093 3090 y Ff(0)3116 3109 y Fu(\(0\))3216 3074 y Fo(f)3275 3038 y Fy(0)p Fu(2)3333 3074 y Fp(\(0\))g(=)324 3195 y(0)p Fq(.)470 3535 y(On)j(the)h(other)g(hand,)g(in)e(considering) h(the)h(follo)m(wing)c(function)j(dep)s(ending)h(on)f Fo(f)324 3656 y Fq(and)h Fo(g)349 3914 y(\033)t Fp(\()p Fo(x)p Fp(\))c(=)g(2)p Fo(x)775 3873 y Fu(2)899 3847 y Fp(1)p 824 3891 199 4 v 824 3983 a Fo(g)875 3954 y Fy(0)898 3983 y Fp(\(0\))1032 3914 y Fo(f)11 b Fp(\()p Fo(x)p Fp(\))1239 3797 y Fk(Z)1322 3824 y Fw(x)1285 3986 y Fu(0)1383 3914 y Fo(sf)g Fp(\()p Fo(s)p Fp(\))p Fo(ds)21 b Fj(\000)h Fp(4)1961 3847 y(1)p 1886 3891 V 1886 3983 a Fo(g)1937 3954 y Fy(0)1960 3983 y Fp(\(0\))2094 3914 y([)2121 3797 y Fk(Z)2205 3824 y Fw(x)2168 3986 y Fu(0)2265 3914 y Fo(sf)11 b Fp(\()p Fo(s)p Fp(\))p Fo(ds)p Fp(])2616 3873 y Fu(2)2677 3914 y Fp(+)22 b Fo(x)2830 3873 y Fu(3)2870 3914 y Fo(g)2917 3929 y Fw(n)2963 3914 y Fp(\()p Fo(x)p Fp(\))h Fj(\000)f Fo(x)3271 3873 y Fu(4)3311 3914 y Fo(g)3362 3873 y Fy(0)3358 3939 y Fw(n)3405 3914 y Fp(\()p Fo(x)p Fp(\))324 4198 y Fq(where)32 b Fo(g)651 4213 y Fw(n)698 4198 y Fp(\()p Fo(x)p Fp(\))c(=)1027 4158 y Fu(1)p 970 4174 149 4 v 970 4232 a Fw(g)1006 4213 y Ff(0)1029 4232 y Fu(\(0\))1129 4198 y Fo(g)t Fp(\()p Fo(x)p Fp(\))19 b Fj(\000)h Fo(x)p Fq(.)32 b(Then,)g(for)f(this)g Fo(\033)t Fp(\()p Fo(x)p Fp(\))h Fq(w)m(e)g(ha)m(v)m(e)h(the)e(follo)m(wing)e (result)324 4336 y(whic)m(h)i(has)g(b)s(een)g(pro)m(v)m(ed)h(b)m(y)f ([S])g(in)e(the)i(case)h(where)f Fo(g)j Fq(and)c Fo(f)42 b Fq(are)30 b Fo(C)2944 4300 y Fu(1)3014 4336 y Fq(\(Theorem)g(A,)324 4456 y(Theorem)j(2)f(and)g(Corollary)f(1\))1894 5251 y(20)p eop %%Page: 21 21 21 20 bop 470 548 a Fi(Prop)s(osition)48 b(2)238 b Fh(L)-5 b(et)45 b Fo(f)5 b(;)17 b(g)49 b Fj(2)d Fo(C)1942 512 y Fu(3)1981 548 y Fp(\()p Fo(a;)17 b(b)p Fp(\))p Fo(;)g(f)11 b Fp(\(0\))45 b(=)h Fo(g)t Fp(\(0\))f(=)g(0)g Fh(and)f Fo(g)3274 512 y Fy(00)3315 548 y Fp(\(0\))i(=)324 668 y Fo(f)383 632 y Fy(00)425 668 y Fp(\(0\))j(=)g(0)p Fh(.)80 b(The)46 b(origin)g Fp(0)g Fh(b)-5 b(eing)46 b(a)g(c)-5 b(enter)47 b(of)f(\(3.1\))g(.)80 b(If)92 b Fo(xC)7 b Fp(\()p Fo(x)p Fp(\))50 b Fo(>)g Fp(0)93 b Fh(in)46 b(a)324 789 y(punctur)-5 b(e)g(d)35 b(neighb)-5 b(orho)g(o)g(d)33 b Fo(J)44 b Fh(of)34 b Fp(0)p Fh(,)324 909 y(then)g(we)h(have)470 1029 y(\(1\))134 b(if)34 b Fo(\033)t Fp(\()p Fo(x)p Fp(\))28 b Fj(\024)h Fp(0)34 b Fh(for)g Fo(x)29 b Fj(2)f Fo(J)9 b Fh(,)34 b(then)h Fo(T)48 b Fh(is)35 b(de)-5 b(cr)g(e)g(asing)33 b(in)h(a)h(neighb)-5 b(orho)g(o)g(d)33 b(of)h Fp(0)p Fh(;)470 1150 y(\(2\))134 b(if)35 b Fo(\033)t Fp(\()p Fo(x)p Fp(\))28 b Fj(\025)g Fp(0)34 b Fh(for)h Fo(x)28 b Fj(2)g Fo(J)9 b Fh(,)35 b(then)g Fo(T)48 b Fh(is)35 b(incr)-5 b(e)g(asing)33 b(in)i(a)g(neighb)-5 b(orho)g(o)g(d)33 b(of)h Fp(0)p Fh(;)470 1270 y(\(3\))134 b(if)35 b Fo(\033)t Fp(\()p Fo(x)p Fp(\))28 b Fj(\021)g Fp(0)34 b Fh(for)h Fo(x)28 b Fj(2)g Fo(J)9 b Fh(,)35 b(then)g Fo(T)48 b Fh(is)35 b(c)-5 b(onstant)34 b(in)h(a)g(neighb)-5 b(orho)g(o)g(d)33 b(of)h Fp(0)p Fh(.)470 1602 y Fq(Notice)23 b(that)h(while)f (considering)g(the)h(assumption)f Fo(xC)7 b Fp(\()p Fo(x)p Fp(\))28 b Fo(>)g Fp(0)c Fq(in)f(the)h(case)g(where)324 1723 y Fo(g)38 b Fq(is)c Fo(C)586 1687 y Fu(3)660 1723 y Fq(and)g Fo(f)45 b Fq(is)35 b Fo(C)1122 1687 y Fu(1)1161 1723 y Fq(,)g(w)m(e)h(ha)m(v)m(e)f Fo(g)1646 1687 y Fy(0)1669 1723 y Fp(\(0\))c Fo(>)g Fp(0)j Fq(and)h Fo(C)7 b Fp(\()p Fo(x)p Fp(\))34 b Fq(is)h Fo(C)2627 1687 y Fu(3)2700 1723 y Fq(.)50 b(F)-8 b(urthermore,)34 b Fo(\033)t Fp(\()p Fo(x)p Fp(\))324 1843 y Fq(and)e Fo(C)7 b Fp(\()p Fo(x)p Fp(\))33 b Fq(are)g(related)1486 2019 y Fo(\033)t Fp(\()p Fo(x)p Fp(\))28 b(=)g Fj(\000)p Fo(x)1940 1977 y Fu(5)2018 1951 y Fo(d)p 1990 1995 107 4 v 1990 2087 a(dx)2106 2019 y Fp(\()2154 1951 y Fo(C)7 b Fp(\()p Fo(x)p Fp(\))p 2154 1995 209 4 v 2230 2087 a Fo(x)2372 2019 y(:)324 2212 y Fq(This)32 b(function)g(ma)m(y)h(b)s(e)g(written)f(in)g(a)g(neigh)m (b)s(orho)s(o)s(d)f(of)h Fp(0)p Fq(,)1159 2456 y Fo(\033)t Fp(\()p Fo(x)p Fp(\))c(=)g Fj(\000)p Fp(2)p Fo(x)1662 2415 y Fu(6)1702 2456 y Fp([)p Fo(g)1780 2415 y Fu(\(3\))1873 2456 y Fp(\()p Fo(x)p Fp(\))23 b Fj(\000)2136 2389 y Fp(2)p 2136 2433 49 4 v 2136 2525 a(3)2195 2456 y Fo(f)2254 2415 y Fy(0)p Fu(2)2312 2456 y Fp(\()p Fo(x)p Fp(\)])g(+)f Fo(:::::)324 2693 y Fq(That)33 b(imp)s(oses)f(an)h(additional)d (condition)65 b Fo(g)2045 2657 y Fy(00)2087 2693 y Fp(\(0\))28 b(=)g(0)33 b Fq(whic)m(h)h(turns)f(out)g(to)g(b)s(e)g(nec-)324 2813 y(essary)41 b(since)g(w)m(e)g(ha)m(v)m(e)g(seen)h Fo(C)1549 2777 y Fy(00)1591 2813 y Fp(\(0\))f(=)f Fo(g)1924 2777 y Fy(00)1966 2813 y Fp(\(0\))p Fq(.)66 b(Moreo)m(v)m(er,)43 b(it)c(implies)f(necessarely)324 2934 y Fo(f)383 2898 y Fy(00)425 2934 y Fp(\(0\))27 b(=)h(0)k Fq(b)m(y)i(Lemma)d(2.)324 3341 y Fe(3.1)130 b(Other)44 b(class)g(of)g(equations)324 3526 y Fq(In)33 b([C2],)g(Chicone)f(consider)h(the)g(class)g(of)f (di\033eren)m(tial)f(equations)h(of)g(the)h(form)1464 3731 y Fo(x)1519 3690 y Fy(00)1584 3731 y Fp(+)22 b Fo(F)14 b Fp(\()p Fo(x)1852 3690 y Fy(0)1875 3731 y Fp(\))22 b(+)g Fo(G)p Fp(\()p Fo(x)p Fp(\))28 b(=)g(0)324 3937 y Fq(with)k(Diric)m(hlet)g(or)h(Neumann)g(b)s(oundary)h(v)-5 b(alues.)45 b(These)34 b(functions)f(v)m(erify)h Fo(F)14 b Fp(\(0\))28 b(=)324 4057 y Fo(G)p Fp(\(0\))f(=)h(0)k Fq(.)43 b(This)33 b(is)f(equiv)-5 b(alen)m(t)32 b(to)h(the)f(system) 1277 4284 y Fk(\032)1356 4345 y Fo(x)1411 4308 y Fy(0)1463 4345 y Fp(=)27 b Fj(\000)p Fo(y)1356 4465 y(y)1408 4429 y Fy(0)1458 4465 y Fp(=)h Fo(x)22 b Fj(\000)h Fo(x)s Fp(~)-52 b Fo(g)t Fp(\()p Fo(x)p 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b Fp(~)-53 b Fo(g)1145 727 y Fy(0)1168 768 y Fp(\()p Fo(s)p Fp(\))28 b Fj(\025)g Fp(0)p Fo(;)1634 742 y Fp(~)1613 768 y Fo(f)1672 727 y Fy(00)1714 768 y Fp(\()p Fo(s)p Fp(\))f Fj(\025)i Fp(0)p Fo(;)117 b Fp(~)-52 b Fo(g)2210 727 y Fy(00)2251 768 y Fp(\()p Fo(s)p Fp(\))28 b Fj(\025)g Fp(0)p Fo(;)114 b Fg(for)108 b Fo(s)28 b Fj(2)g Fp([0)p Fo(;)17 b(a)p Fp(])p Fo(:)470 988 y Fq(The)33 b(tra)5 b(jectories)33 b(of)e(this)h(system)h(are)g (symmetric)e(with)h(resp)s(ect)h(to)f(the)h(x-axis.)324 1108 y(So,)28 b(it)e(has)i(ob)m(viously)f(a)g(cen)m(ter)i(at)e(the)g (origin)f(of)g(the)i(phase)g(plane.)42 b(Chicone)27 b(pro)m(v)m(ed)324 1229 y(that)h(under)h(the)g(ab)s(o)m(v)m(e)h(conditions,)e(the)h(p)s (erio)s(d)f(function)g(is)g(monotone)f(increasing.)324 1349 y(The)j(follo)m(wing)c(class)j(of)f(Raleigh)f(equations)i(with)f (linear)f(restoring)h(term)h(ma)m(y)f(ha)m(v)m(e)324 1469 y(a)k(monotone)g(increasing)f(p)s(erio)s(d)h(in)g(a)g(neigh)m(b)s (orho)s(o)s(d)f(of)h(the)h(cen)m(ter)h Fp(0)p Fq(.)1354 1689 y Fp(\()p Fo(R)q Fp(\))195 b Fo(x)1755 1648 y Fy(00)1820 1689 y Fp(+)22 b Fo(F)14 b Fp(\()p Fo(x)2088 1648 y Fy(0)2111 1689 y Fp(\))22 b(+)g Fo(x)28 b Fp(=)g(0)p Fo(:)470 1910 y Fq(Without)k(supp)s(osing)h(h)m(yp)s(otheses)i(ab)s(o)m(v)m(e,)f(w)m (e)g(ma)m(y)e(pro)m(v)m(e)i(an)f(analogous)f(result)324 2030 y(whic)m(h)44 b(also)e(impro)m(v)m(es)h(Corollary)f(9)h(of)f([S].) i(W)-8 b(e)44 b(only)e(need)j(for)d(that)h(to)g(supp)s(ose)324 2150 y Fo(F)14 b Fp(\()p Fo(x)p Fp(\))32 b Fq(is)g(an)h(ev)m(en)h (function)e(with)65 b Fo(F)1734 2114 y Fy(00)1776 2150 y Fp(\(0\))27 b Fj(6)p Fp(=)h(0)p Fo(;)k Fq(instead)g(of)h(conditions)e Fp(\()p Fb(C)p Fp(\))p Fq(.)470 2491 y Fi(Corollary)g(5)223 b Fh(L)-5 b(et)32 b Fo(h)f Fh(b)-5 b(e)30 b(an)h(analytic)f(even)h (function)f(such)h(that)62 b Fo(F)14 b Fp(\(0\))27 b(=)h(0)p Fo(;)324 2611 y Fh(and)76 b Fo(F)632 2575 y Fy(00)674 2611 y Fp(\(0\))33 b Fj(6)p Fp(=)h(0)p Fh(.)k(The)f(origin)h Fp(0)g Fh(b)-5 b(e)38 b(a)g(c)-5 b(enter)38 b(of)f 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b(monotonicit)m(y)d(condition)g(as)i(w)m(ell)g(as)g(the)g(iso) s(c)m(hronic-)324 973 y(it)m(y)34 b(case)i(can)g(v)m(erify)f(itself)e (more)i(easily)-8 b(.)49 b(So)35 b(m)m(uc)m(h)h(as)f(the)g(function)g Fo(C)7 b Fp(\()p Fo(x)p Fp(\))35 b Fq(reduces)324 1094 y(itself)24 b(to)g(the)i(function)e Fo(g)t Fp(\()p Fo(x)p Fp(\))h Fq(when)h Fo(f)11 b Fp(\()p Fo(x)p Fp(\))28 b Fj(\021)g Fp(0)d Fq(in)f(a)h(neigh)m(b)s(orho)s(o)s(d)f(of)h Fp(0)p Fq(.)40 b(This)26 b(puts)f(in)324 1214 y(evidence)i(the)f (parallel)c(b)s(et)m(w)m(een)28 b(our)d(result)h(and)f(the)h(one)g(of)f (Opial.)39 b(More)26 b(precisely)-8 b(,)324 1335 y(a)32 b(calculation)e(giv)m(es)629 1570 y Fo(C)706 1529 y Fy(00)748 1570 y Fp(\(0\))e(=)f Fo(g)1055 1529 y Fy(00)1097 1570 y Fp(\(0\))22 b Fj(\000)1353 1503 y Fp(2)p 1353 1547 49 4 v 1353 1639 a(3)1412 1570 y Fo(f)11 b Fp(\(0\))p Fo(f)1655 1529 y Fy(0)1677 1570 y Fp(\(0\))98 b Fg(and)107 b Fo(C)2239 1529 y Fy(000)2300 1570 y Fp(\(0\))28 b(=)f Fo(g)2607 1529 y Fy(000)2668 1570 y Fp(\(0\))22 b Fj(\000)2925 1503 y Fp(2)p 2925 1547 V 2925 1639 a(3)2983 1570 y Fo(f)3042 1529 y Fy(0)3065 1570 y Fp(\(0\))3190 1529 y Fu(2)3229 1570 y Fo(:)324 1798 y Fq(Then)33 b(w)m(e)h(ha)m(v)m(e)g(necessarely)g Fo(C)1526 1762 y Fy(00)1569 1798 y Fp(\(0\))27 b(=)g(0)p Fq(.)324 1918 y(W)-8 b(e)39 b(ma)m(y)g(remark)g(that)g(condition)f Fo(C)1786 1882 y Fy(000)1847 1918 y Fp(\()p Fo(x)p Fp(\))i Fo(<)e Fp(0)h Fq(for)g Fo(x)g Fj(2)h Fp(\()p Fo(a;)17 b(b)p Fp(\))39 b Fq(implies)e(\(2.7\))i(and)324 2039 y Fo(C)401 2003 y Fy(000)462 2039 y Fp(\()p Fo(x)p Fp(\))33 b Fo(>)f Fp(0)j Fq(implies)e(\(2.8\))i(\(this)f(requires)i(necessarily) g Fo(C)2553 2003 y Fy(000)2615 2039 y Fp(\(0\))c Fo(<)g Fp(0\))j Fq(when)h Fo(g)i Fq(is)d Fo(C)3522 2003 y Fu(3)324 2159 y Fq(and)d Fo(g)564 2123 y Fy(000)625 2159 y Fp(\(0\))c Fo(<)f Fp(0)p Fq(\).)324 2279 y(It)37 b(naturally)f(seems)i(that)e(the) i(function)f Fo(C)7 b Fp(\()p Fo(x)p Fp(\))37 b Fq(pla)m(ys)g(a)g(same) g(role)g(for)f(the)i(system)324 2400 y(\(3.3\))j(as)i(the)f(function)g Fo(g)t Fp(\()p Fo(x)p Fp(\))f Fq(for)h(the)h(conserv)-5 b(ativ)m(e)43 b(system.)73 b(Indeed,)46 b(if)41 b(w)m(e)i(tak)m(e)324 2520 y Fo(f)11 b Fp(\()p Fo(x)p Fp(\))27 b Fj(\021)i Fp(0)p Fq(,)j(conditions)f(\(3.7\))h(and)h(\(3.8\))f(reduce)h(to)f(the) h(conditions)f(\(2.2\))g(and)g(\(2.3\))324 2641 y(resp)s(ectiv)m(ely)e (of)f(the)h(Prop)s(osition)d(1.)42 b(Notice)29 b(that)g(Rothe)h (condition)e(for)g(the)i(mono-)324 2761 y(tonicit)m(y)h(of)h(the)h(p)s (erio)s(d)f(function)324 3011 y Fo(H)405 3026 y Fu(4)444 3011 y Fp(\()p Fo(x)p Fp(\))c(=)g Fo(x)p Fp([3)p Fo(g)889 2970 y Fy(0)911 3011 y Fp(\()p Fo(x)p Fp(\))1042 2970 y Fu(2)1091 3011 y Fj(\000)9 b Fo(g)t Fp(\()p Fo(x)p Fp(\))p Fo(g)1410 2970 y Fy(00)1452 3011 y Fp(\()p Fo(x)p Fp(\))g Fj(\000)g Fp(\(3)1775 2944 y Fo(g)1826 2908 y Fy(0)1848 2944 y Fp(\(0\))1973 2908 y Fu(2)p 1775 2988 238 4 v 1785 3080 a Fo(g)1836 3051 y Fy(00)1878 3080 y Fp(\(0\))2022 3011 y(\))p Fo(g)2111 2970 y Fy(00)2153 3011 y Fp(\()p Fo(x)p Fp(\)])28 b Fj(\025)g Fp(0\()p Fj(\024)g Fp(0\))e Fo(;)43 b Fg(for)205 b Fo(x)28 b Fj(2)g Fp(\()p Fo(a;)17 b(b)p Fp(\))p Fo(;)324 3264 y Fq(is)32 b(more)g(general)g(than)g(\(2.2\))g(and)h(\(2.3\))f(\(see)h(Remark)f(1) h(and)f([R]\).)324 3384 y(W)-8 b(e)37 b(ma)m(y)g(exp)s(ect)h(that)f (the)g(results)g(of)f(the)i(ab)s(o)m(v)m(e)f(Theorem)h(1)e(can)h(b)s(e) g(generalize.)324 3505 y(In)f(the)g(sense)i(where)73 b Fo(C)7 b Fp(\()p Fo(x)p Fp(\))36 b Fq(is)f(replaced)h(b)m(y)h (another)f(more)f(general)g(appropriated)324 3625 y(function)d(let)64 b Fo(D)s Fp(\()p Fo(x)p Fp(\))p Fq(,)33 b(whic)m(h)g(itself)e(can)i(b)s (e)g(reduce)g(to)g(the)g(Rothe)f(function)1169 3881 y Fp(\(3)1266 3814 y Fo(g)1317 3778 y Fy(0)1340 3814 y Fp(\(0\))1465 3778 y Fu(2)p 1266 3858 V 1276 3949 a Fo(g)1327 3921 y Fy(00)1369 3949 y Fp(\(0\))1514 3881 y(\))p Fo(g)1603 3840 y Fy(00)1644 3881 y Fp(\()p Fo(x)p Fp(\))23 b Fj(\000)f Fp(3)p Fo(g)1997 3840 y Fy(0)2020 3881 y Fp(\()p Fo(x)p Fp(\))2151 3840 y Fu(2)2213 3881 y Fp(+)g Fo(g)t Fp(\()p Fo(x)p Fp(\))p Fo(g)2544 3840 y Fy(00)2585 3881 y Fp(\()p Fo(x)p Fp(\))324 4134 y Fq(for)32 b(the)h(conserv)-5 b(ativ)m(e)33 b(case)h Fo(f)11 b Fp(\()p Fo(x)p Fp(\))27 b Fj(\021)i Fp(0)p Fo(:)j Fq(This)h(function)f(ha)m(v)m(e)h(to)g(b)s(e) f(suc)m(h)i(that)1054 4375 y Fo(D)1138 4334 y Fu(\(3\))1232 4375 y Fp(\(0\))28 b(=)f Fo(g)1539 4334 y Fu(\(3\))1633 4375 y Fp(\(0\))22 b Fj(\000)1988 4307 y Fp(5)p 1889 4352 247 4 v 1889 4443 a(3)p Fo(g)1989 4414 y Fy(0)2011 4443 y Fp(\(0\))2146 4375 y Fo(g)2197 4334 y Fy(00)p Fu(2)2274 4375 y Fp(\(0\))g Fj(\000)2530 4307 y Fp(2)p 2530 4352 49 4 v 2530 4443 a(3)2589 4375 y Fo(f)2648 4334 y Fy(0)p Fu(2)2707 4375 y Fp(\(0\))324 4628 y Fq(One)33 b(separates)g(th)m(us)h(the)f(necessary)i(condition)c Fo(g)2247 4591 y Fy(00)2288 4628 y Fp(\(0\))d(=)f(0)p Fo(:)324 4748 y Fq(W)-8 b(e)33 b(ma)m(y)f(also)g(exp)s(ect)h(that)g (the)g(sign)f(of)g(the)h(follo)m(wing)d(function)1155 4984 y Fo(g)1206 4942 y Fy(0)1229 4984 y Fp(\()p Fo(x)p Fp(\))p Fo(g)1411 4942 y Fu(\(3\))1505 4984 y Fp(\()p Fo(x)p Fp(\))22 b Fj(\000)1768 4916 y Fp(5)p 1768 4960 V 1768 5052 a(3)1827 4984 y Fo(g)1878 4942 y Fy(00)p Fu(2)1955 4984 y Fp(\()p Fo(x)p Fp(\))g Fj(\000)2218 4916 y Fp(2)p 2218 4960 V 2218 5052 a(3)2276 4984 y Fo(f)2335 4942 y Fy(0)p Fu(2)2394 4984 y Fp(\()p Fo(x)p Fp(\))p Fo(g)2576 4942 y Fy(0)2599 4984 y Fp(\()p Fo(x)p Fp(\))1894 5251 y Fq(23)p eop %%Page: 24 24 24 23 bop 324 548 a Fq(determines)35 b(the)g(global)d(monotonicit)m(y)h (of)i(the)g(p)s(erio)s(d)f(function)g Fo(T)49 b Fq(of)34 b(the)h(Lienard)324 668 y(system.)49 b(In)35 b(the)f(conserv)-5 b(ativ)m(e)35 b(case,)h(the)e(last)g(function)f(reduces)j(to)e Fo(g)3031 632 y Fy(0)3054 668 y Fp(\()p Fo(x)p Fp(\))p Fo(g)3236 632 y Fu(\(3\))3330 668 y Fp(\()p Fo(x)p Fp(\))23 b Fj(\000)334 749 y Fu(5)p 334 765 36 4 v 334 823 a(3)379 789 y Fo(g)430 753 y Fy(00)p Fu(2)507 789 y Fp(\()p Fo(x)p Fp(\))32 b Fq(whic)m(h)g(in)m(terv)m(enes)i(in)d(the)h(Sc)m(haaf)g (condition)f(\()g(see)i(condition)e Fp(\()p Fj(C)3166 804 y Fu(3)3205 789 y Fp(\))h Fq(in)f(Sec-)324 909 y(tion)47 b(2)i(ab)s(o)m(v)m(e\).)92 b(Recall)47 b(that)i(the)g(last)f(one)h(is)f (less)h(inclusiv)m(e)f(than)h(the)g(Rothe)324 1029 y(condition.)324 1150 y(Moreo)m(v)m(er,)32 b(according)e(Corollary)f(4)h(to)g(determine) g(the)h(iso)s(c)m(hronous)g(cen)m(ters)h(at)e(the)324 1270 y(origin)g(for)h(the)i(Lienard)e(system)i(\(other)g(than)f(those)h (determined)f(b)m(y)h([S])f(\))g(w)m(e)h(ha)m(v)m(e)324 1391 y(to)24 b(insure)g(that)g(the)g(asso)s(ciated)g(conserv)-5 b(ativ)m(e)25 b(sytem)g(has)f(a)g(decreasing)h(p)s(erio)s(d)e(func-)324 1511 y(tion.)1894 5251 y(24)p eop %%Page: 25 25 25 24 bop 1479 548 a Fa(REFERENCES)470 876 y Fq([C])229 b(C.Chicone)131 b Fh(The)35 b(monotonicity)g(of)g(the)h(p)-5 b(erio)g(d)34 b(function)h(for)g(planar)324 996 y(Hamiltonian)f(ve)-5 b(ctor)34 b(\034elds)130 b Fq(J.)32 b(Di\033)g(Eq.,)i(v)m(ol)e(69,)g (p.)44 b(310-321,)30 b(\(1987\).)470 1237 y([C2])224 b(C.Chicone)127 b Fh(Ge)-5 b(ometric)31 b(metho)-5 b(ds)30 b(for)h(two-p)-5 b(oint)30 b(nonline)-5 b(ar)30 b(b)-5 b(ound-)324 1357 y(ary)35 b(value)f(pr)-5 b(oblems)129 b Fq(J.)33 b(Di\033)f(Eq.,)h(v)m(ol)f(72,)h(p.)43 b(360-407,)31 b(\(1988\).)470 1598 y([Ch-C])225 b(R.Chouikha)30 b(and)f(F.)h(Cuv)m (elier)127 b Fh(R)-5 b(emarks)31 b(on)h(some)f(monotonic-)324 1719 y(ity)36 b(c)-5 b(onditions)34 b(for)h(the)g(p)-5 b(erio)g(d)35 b(function)97 b Fq(Applic.)44 b(Math.,)34 b(v)m(ol)e(26,)h(3,)g(p.)45 b(243-252,)324 1839 y(\(1999\).)470 2080 y([C-W])226 b(S.N.)30 b(Cho)m(w)h(and)f(D.)h(W)-8 b(ang)127 b Fh(On)32 b(the)h(monotonicity)f(of)g(the)g(p)-5 b(erio)g(d)324 2200 y(function)33 b(of)g(some)g(se)-5 b(c)g(ond)33 b(or)-5 b(der)33 b(e)-5 b(quations)97 b Fq(Casopis)31 b(P)m(est.)45 b(Mat.)e(111,)31 b(p.)43 b(14-25,)324 2320 y(\(1986\).)470 2561 y([C-D])226 b(CJ)31 b(Christopher)f(and)g(J.)h(Devlin)127 b Fh(Iso)-5 b(chr)g(onous)31 b(c)-5 b(enters)32 b(in)g(planar)324 2682 y(p)-5 b(olynomial)33 b(systems)130 b Fq(SIAM)33 b(J.)g(Math.)44 b(Anal.,)32 b(v)m(ol)g(28,)g(p.162-177,)f(\(1997\).)470 2922 y([L])240 b(W.S.)45 b(Loud)142 b Fh(The)46 b(b)-5 b(ehavior)44 b(of)i(the)g(p)-5 b(erio)g(d)45 b(of)h(solutions)f(ofc)-5 b(ertain)324 3043 y(plane)37 b(autonomous)h(systems)g(ne)-5 b(ar)38 b(c)-5 b(enters)135 b Fq(Con)m(tr.)56 b(Di\033eren)m(tial)34 b(Equations,)k(3,)324 3163 y(p.)43 b(21-36,)32 b(\(1964\).)470 3404 y([O])232 b(Z.)37 b(Opial)133 b Fh(Sur)39 b(les)g(p)-5 b(erio)g(des)38 b(des)h(solutions)f(de)h(l'e)-5 b(quation)39 b(di\033er)-5 b(en-)324 3524 y(tiel)5 b(le)69 b Fo(x)655 3488 y Fy(00)720 3524 y Fp(+)22 b Fo(g)t Fp(\()p Fo(x)p Fp(\))27 b(=)h(0)130 b Fq(Ann.)44 b(P)m(olon.)f(Math.,)33 b(10,)f(p.)43 b(49-72,)31 b(\(1961\).)470 3765 y([R])230 b(F.)35 b(Rothe)133 b Fh(R)-5 b(emarks)36 b(on)h(p)-5 b(erio)g(ds)36 b(of)h(planar)g(Hamiltonian)f(systems.)324 3885 y Fq(SIAM)d(J.)g(Math.)44 b(Anal.,)31 b(24,)h(p.129-154,)f (\(1993\).)470 4126 y([S])229 b(M.)33 b(Sabatini)129 b Fh(On)35 b(the)h(p)-5 b(erio)g(d)34 b(function)h(of)h(Lienar)-5 b(d)34 b(systems)131 b Fq(J.)33 b(of)324 4247 y(Di\033.)43 b(Eq.,)33 b(152,)f(p.)44 b(467-487,)30 b(\(1999\).)470 4367 y([Sc])232 b(R.Sc)m(haaf)135 b Fh(A)39 b(class)e(of)i(Hamiltonian) e(systems)i(with)f(incr)-5 b(e)g(asing)37 b(p)-5 b(e-)324 4487 y(rio)g(ds)129 b Fq(J.)33 b(Reine)f(Angew.)44 b(Math.,)33 b(363,)f(p.)44 b(96-109,)31 b(\(1985\).)1894 5251 y(25)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0105110510408--