%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: mparch.dvi %%Pages: 14 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o mparch.ps mparch.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2000.05.18:1810 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] 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b(transformation)f(on)g(the)h(t)n(w)n (o)f(main)h(F)-7 b(ourier)23 b(mo)r(des)h(of)f(the)i(p)r(erturbation)e (and)h(w)n(e)f(neglect)199 4860 y(higher)k(orders)g(in)h Fj(")f Fl(\(the)i(amplitude)f(of)g(the)g(p)r(erturbation\).)37 b(These)28 b(appro)n(ximations)d(reduce)199 4960 y(the)39 b(renormalization)c(transformation)i Ft(R)h Fl(to)g(a)f(4)h (dimensional)f(map)h(from)g(whic)n(h)g(w)n(e)f(can)199 5059 y(deriv)n(e)27 b(analytical)f(results.)p eop %%Page: 3 3 3 2 bop 199 100 a Fm(Sc)l(aling)31 b(law)f(for)h(the)f(critic)l(al)h (function)e(of)i(an)f(appr)l(oximate)h(r)l(enormalization)484 b Fl(3)199 282 y(F)-7 b(ollo)n(wing)27 b(Ref)i([8],)f(w)n(e)g(deriv)n (e)g(analytically)f(the)h(equation)g(of)g(the)h(critical)e(surface)h Ft(S)35 b Fl(for)27 b(the)199 382 y(appro)n(ximate)c(renormalization)e (transformation,)j(and)f(w)n(e)h(analyze)f(the)h(precise)g(shap)r(e)f (of)h(this)199 482 y(surface)j(near)f(a)h(resonance)f Fj(P)7 b(=Q)p Fl(,)27 b(i.e.,)h(when)f(the)h(frequency)f Fj(!)j 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Fg(P)r(;Q)2890 1133 y Fl(=)44 b(1)p Fj(=Q)39 b Fl(for)199 1232 y(0)23 b Fj(<)f(P)35 b(<)23 b(Q)p Fl(.)35 b(Ho)n(w)n(ev)n(er,)23 b(this)h(exp)r(onen)n(t)g Fj(\013)1571 1244 y Fg(P)r(;Q)1715 1232 y Fl(is)f(not)h(the)g(same)f(for)h(all)f(one-parameter)f(family) 199 1332 y(critical)30 b(function)g(since)g(w)n(e)f(will)i(exhibit)f (examples)f(of)h(Hamiltonians)g(with)g(di\013eren)n(t)g(v)-5 b(alues)199 1432 y(\(see)28 b(Eq.)f(\(25\))g(b)r(elo)n(w\).)365 1631 y(In)39 b(Sec.)g(2,)i(w)n(e)d(deriv)n(e)g(the)h(equiv)-5 b(alence)39 b(b)r(et)n(w)n(een)f(the)h(critical)f(function)i(of)e(a)h (simple)199 1731 y(Hamiltonian)22 b(mo)r(del)g(and)f(the)i(critical)e (function)h(of)g(the)g(follo)n(wing)f(forced)g(p)r(endulum)i(mo)r(dels) f(:)853 1923 y Fj(H)7 b Fl(\()p Fj(p;)14 b(x;)g(t)p Fl(\))23 b(=)1307 1867 y(1)p 1307 1904 42 4 v 1307 1980 a(2)1358 1923 y Fj(p)1400 1889 y Fi(2)1456 1923 y Ft(\000)18 b Fj(")p Fl(\(cos)o(\()p Fj(ax)h Ft(\000)f Fj(bt)p Fl(\))g(+)g(cos)o(\()p Fj(cx)h Ft(\000)f Fj(dt)p 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(of)g(the)g(scaling)g(la)n(ws)f(of)h(the)g(critical)g(function.)45 b(In)30 b(Sec)h(5,)f(w)n(e)199 2712 y(analyze)c(the)h(scaling)f(la)n(w) h(for)f(the)h(forced)g(p)r(endulum)h(mo)r(dels)f(in)g(order)f(to)g(see) h(the)g(generalit)n(y)199 2812 y(of)e(the)g(scaling)f(la)n(w)h(and)f (its)h(c)n(haracteristic)e(exp)r(onen)n(t.)36 b(W)-7 b(e)25 b(also)f(discuss)h(a)f(conjecture)h(on)f(the)199 2911 y(link)k(b)r(et)n(w)n(een)g(the)g(critical)f(function)h(and)f(the) h(Brjuno)f(function.)199 3132 y Fu(2.)42 b(Critical)32 b(function)g(of)f(v)-5 b(arious)32 b(t)m(yp)s(es)g(of)f(forced)i(p)s (endulum)c(mo)s(dels)199 3314 y Fl(W)-7 b(e)28 b(consider)f(the)h (follo)n(wing)e(families)i(of)f(Hamiltonian)h(systems)f(:)797 3475 y Fj(H)j Fl(=)993 3418 y(1)p 993 3455 V 993 3531 a(2)1045 3475 y Fj(p)1087 3440 y Fi(2)1142 3475 y Ft(\000)18 b Fj(")1264 3487 y Fi(1)1315 3475 y Fl(cos)o(\()p Fj(ax)h Ft(\000)f Fj(bt)p Fl(\))h Ft(\000)f Fj(")1890 3487 y Fi(2)1941 3475 y Fl(cos)o(\()p Fj(cx)h Ft(\000)f Fj(dt)p 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Fj(H)j Fl(=)1041 4169 y(1)p 1041 4206 V 1041 4282 a(2)1092 4225 y Fj(p)1134 4191 y Fi(2)1190 4225 y Fl(+)18 b Fj(!)s(p)g Ft(\000)g Fj(")1510 4237 y Fi(1)1560 4225 y Fl(cos\()p Fj(ax)h Ft(\000)f Fj(bt)p Fl(\))g Ft(\000)g Fj(")2135 4237 y Fi(2)2186 4225 y Fl(cos)o(\()p Fj(cx)h Ft(\000)f Fj(dt)p Fl(\))p Fj(:)199 4406 y Fl(This)39 b(Hamiltonian)g(is)g(also)f(mapp)r(ed)i(in)n(to)f(a)g(time-indep)r (enden)n(t)h(t)n(w)n(o)e(degrees)g(of)h(freedom)199 4506 y(Hamiltonian)28 b(:)797 4651 y Fj(H)7 b Fl(\()p Fk(A)p Fj(;)14 b Fk(')p Fl(\))24 b(=)e Fk(!)f Ft(\001)e Fk(A)f Fl(+)1525 4595 y(1)p 1525 4632 V 1525 4708 a(2)1577 4651 y Fj(A)1639 4616 y Fi(2)1639 4671 y(1)1695 4651 y Ft(\000)g Fj(")1817 4663 y Fi(1)1868 4651 y Fl(cos)o(\()p Fj(a')2109 4663 y Fi(1)2165 4651 y Fl(+)g Fj(b')2338 4663 y Fi(2)2375 4651 y Fl(\))h Ft(\000)f Fj(")2548 4663 y Fi(2)2599 4651 y Fl(cos)o(\()p Fj(c')2832 4663 y Fi(1)2888 4651 y Fl(+)g Fj(d')3068 4663 y Fi(2)3106 4651 y Fl(\))p Fj(;)j Fl(\(6\))199 4805 y(where)27 b Fk(!)f Fl(=)d(\()p Fj(!)s(;)14 b Ft(\000)p Fl(1\).)36 b(This)28 b(Hamiltonian)g(b)r(elongs)f(to)g(the)h(family)g (\(1\),)g(and)f(it)i(is)e(canonically)199 4905 y(conjugated)g(with)h (the)g(follo)n(wing)f(Hamiltonian)g(:)819 5045 y(^)797 5066 y Fj(H)7 b Fl(\()p Fk(A)p Fj(;)14 b Fk(')p Fl(\))24 b(=)e Fk(!)f Ft(\001)e Fk(A)f Fl(+)1525 5009 y(1)p 1525 5046 V 1525 5122 a(2)1577 5066 y(\()p Fu(\012)h Ft(\001)f Fk(A)p Fl(\))1842 5031 y Fi(2)1898 5066 y Ft(\000)g Fj(f)2022 5078 y Fi(1)2073 5066 y Fl(cos)13 b Fj(')2252 5078 y Fi(1)2308 5066 y Ft(\000)18 b Fj(f)2432 5078 y Fi(2)2483 5066 y Fl(cos)13 b Fj(')2662 5078 y Fi(2)2699 5066 y Fj(;)460 b Fl(\(7\))p eop %%Page: 4 4 4 3 bop 199 100 a Fm(Sc)l(aling)31 b(law)f(for)h(the)f(critic)l(al)h (function)e(of)i(an)f(appr)l(oximate)h(r)l(enormalization)484 b Fl(4)199 282 y(where)31 b Fk(!)h Fl(=)c(\()p Fj(!)s(;)14 b Ft(\000)p Fl(1\))31 b(and)g Fu(\012)e Fl(=)f(\(1)p Fj(;)14 b(k)s Fl(\))31 b(for)g(some)g Fj(k)g(>)e Fl(0.)47 b(The)32 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Ft(7!)h Fl(\()p Fj(f)1795 4910 y Ff(0)1786 4965 y Fi(1)1824 4944 y Fj(;)14 b(f)1911 4910 y Ff(0)1902 4965 y Fi(2)1938 4944 y Fj(;)g Fl(1)p Fj(=!)s(;)g Fl(1)p Fj(=k)s Fl(\))p Fj(:)p eop %%Page: 6 6 6 5 bop 199 100 a Fm(Sc)l(aling)31 b(law)f(for)h(the)f(critic)l(al)h (function)e(of)i(an)f(appr)l(oximate)h(r)l(enormalization)484 b Fl(6)199 282 y(In)28 b(order)e(to)h(simplify)h(the)g(computations,)e (w)n(e)h(denote)2025 260 y(~)2007 282 y Fj(f)2048 294 y Fi(2)2108 282 y Fl(=)22 b Fj(k)2241 252 y Fi(2)2278 282 y Fj(f)2319 294 y Fi(2)2356 282 y Fj(=!)2453 252 y Fi(2)2490 282 y Fl(.)36 b(The)28 b(renormalization)199 382 y(op)r(erators)e(is)h(giv)n(en)g(b)n(y)g(:)797 519 y Fj(f)847 485 y Ff(0)838 539 y Fi(1)898 519 y Fl(=)1004 497 y(~)986 519 y Fj(f)1027 531 y Fi(2)1063 519 y Fj(;)815 628 y Fl(~)797 653 y Fj(f)847 625 y Ff(0)838 675 y Fi(2)898 653 y Fl(=)c Fj(f)1027 665 y Fi(1)1063 653 y Fj(:)199 790 y Fl(W)-7 b(e)28 b(notice)g(that)g(the)g(new)f(frequency)g Fj(!)1511 760 y Ff(0)1557 790 y Fl(=)c(1)p Fj(=!)29 b Fl(is)f(smaller)e(than)i(one.)199 1005 y Fm(3.2.)40 b(De\014nition)30 b(of)g Ft(R)932 1017 y Fg(S)199 1158 y Fl(The)f(renormalization)e(op)r (erator)g Ft(R)1374 1170 y Fg(S)1452 1158 y Fl(acts)h(on)h(the)g (family)h(of)e(Hamiltonians)h(\(11\))g(for)f Fj(!)g(<)d Fl(1.)199 1258 y(It)40 b(con)n(tains)f(an)g(elimination)g(of)h(the)g (mo)r(de)f(\(0)p Fj(;)14 b Fl(1\))39 b(of)h(the)g(p)r(erturbation,)i (and)d(a)g(rescaling)199 1357 y(pro)r(cedure)26 b(\(shift)i(of)f(the)g (resonances,)e(rescaling)g(of)i(time)h(and)e(of)h(the)g(actions\))g (suc)n(h)f(that)h(the)199 1457 y(image)33 b(of)g(a)g(Hamiltonian)g Fj(H)40 b Fl(giv)n(en)32 b(b)n(y)h(Eq.)f(\(11\))h(is)g(of)h(the)f(same) g(general)f(form)g(as)h Fj(H)40 b Fl(and)199 1556 y(describ)r(es)27 b(the)h(system)g(on)f(a)g(smaller)g(scale)g(in)g(phase)g(space)g(and)h (on)f(a)g(longer)g(time)h(scale.)199 1656 y(W)-7 b(e)50 b(eliminate)g(the)g(mo)r(de)f(\(0)p Fj(;)14 b Fl(1\))49 b(of)h(the)g(p)r(erturbation,)k(b)n(y)c(a)f(near-iden)n(tit)n(y)f (canonical)199 1756 y(transformation.)f(W)-7 b(e)32 b(p)r(erform)f(a)g (Lie)h(transformation)d(generated)i(b)n(y)g(a)g(function)h Fj(F)12 b Fl(\()p Fk(A)p Fj(;)i Fk(')p Fl(\).)199 1855 y(The)28 b(image)f(of)g(a)g(Hamiltonian)h Fj(H)34 b Fl(is)28 b(giv)n(en)f(b)n(y)g(:)797 2027 y Fj(H)873 1992 y Ff(0)919 2027 y Fl(=)c(exp\()1185 2006 y(^)1166 2027 y Fj(F)12 b Fl(\))p Fj(H)30 b Fl(=)23 b Fj(H)i Fl(+)18 b Ft(f)p Fj(F)r(;)c(H)7 b Ft(g)18 b Fl(+)1990 1971 y(1)p 1990 2008 42 4 v 1990 2084 a(2)2041 2027 y Ft(f)p Fj(F)r(;)c Ft(f)p Fj(F)r(;)g(H)7 b Ft(gg)17 b Fl(+)i Ft(\001)14 b(\001)g(\001)f Fj(;)437 b Fl(\(12\))199 2191 y(where)27 b Ft(f)14 b Fj(;)27 b Ft(g)h Fl(denotes)f(the)h(P)n(oisson)d(brac)n(k)n (et)i(:)1191 2410 y Ft(f)p Fj(f)t(;)14 b(g)s Ft(g)21 b Fl(=)1525 2354 y Fj(@)5 b(f)p 1519 2391 111 4 v 1519 2467 a(@)g Fk(')1658 2410 y Ft(\001)1724 2354 y Fj(@)g(g)p 1709 2391 121 4 v 1709 2467 a(@)g Fk(A)1859 2410 y Ft(\000)1961 2354 y Fj(@)g(g)p 1952 2391 111 4 v 1952 2467 a(@)g Fk(')2091 2410 y Ft(\001)2154 2354 y Fj(@)g(f)p 2142 2391 121 4 v 2142 2467 a(@)g Fk(A)2273 2410 y Fj(;)199 2646 y Fl(and)29 b(the)h(op)r(erator)862 2625 y(^)843 2646 y Fj(F)41 b Fl(acts)29 b(on)g Fj(H)36 b Fl(lik)n(e)1504 2625 y(^)1486 2646 y Fj(F)11 b(H)33 b Fl(=)25 b Ft(f)p Fj(F)r(;)14 b(H)7 b Ft(g)p Fl(.)41 b(The)29 b(generating)f(function)i(is)f(c)n (hosen)199 2746 y(linear)e(in)h(the)g(actions)f(and)g(of)h(the)g(form)f (:)797 2883 y Fj(F)12 b Fl(\()p Fk(A)p Fj(;)i Fk(')p Fl(\))24 b(=)e Ft(\000)p Fj(f)1314 2895 y Fi(2)1351 2883 y Fl(\(1)c(+)g Fj(k)s Fu(\012)g Ft(\001)h Fk(A)p Fl(\))14 b(sin)g Fj(')1989 2895 y Fi(2)2026 2883 y Fj(:)1091 b Fl(\(13\))199 3020 y(The)28 b(generating)e(function)i Fj(F)40 b Fl(satis\014es)27 b(:)1192 3195 y Ft(f)p Fj(F)r(;)14 b(H)1395 3207 y Fi(0)1432 3195 y Ft(g)23 b Fl(=)f Fj(f)1625 3207 y Fi(2)1676 3195 y Fl(cos)13 b Fj(')1855 3207 y Fi(2)1911 3195 y Fl(+)18 b Fj(O)r Fl(\()p Fj(")p Fk(A)2203 3159 y 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3715 y(1)p 2261 3752 42 4 v 2261 3828 a(2)2313 3771 y Ft(f)p Fj(F)r(;)c Ft(f)p Fj(F)r(;)g(H)2650 3783 y Fi(0)2687 3771 y Ft(gg)k Fl(+)g Fj(O)r Fl(\()p Fj(")3008 3737 y Fi(3)3046 3771 y Fl(\))p Fj(:)199 3948 y Fl(W)-7 b(e)22 b(neglect)g(the)g(order)e Fj(O)r Fl(\()p Fj(")p Fk(A)1169 3911 y Fi(2)1206 3948 y Fl(\))i(pro)r(duced)g(b)n(y)f(the)h(transformation)e(together)h(with) h(the)g(orders)199 4047 y Fj(O)r Fl(\()p Fj(")335 4017 y Fi(3)373 4047 y Fl(\).)41 b(These)29 b(appro)n(ximations)d(are)i (consisten)n(t)h(since)f(for)h(small)f Fj(")p Fl(,)h Fk(A)p Fl(\()p Fj(t)p Fl(\))h(is)f(of)f(order)g Fj(")h Fl(near)199 4147 y(the)21 b(torus)f(with)h(frequency)f Fj(!)s Fl(.)35 b(The)20 b(term)h Ft(\000)p Fj(f)1666 4159 y Fi(2)1703 4147 y Ft(f)p Fj(F)r(;)14 b Fl(cos)f Fj(')2016 4159 y Fi(2)2053 4147 y Ft(g)20 b Fl(is)h(indep)r(enden)n(t)g (of)g(the)g(actions)f(and)199 4247 y(con)n(tains)29 b(the)h(mo)r(des)g Fu(0)f Fl(and)h(\(0)p Fj(;)14 b Fl(2\))29 b(that)h(w)n(e)f(neglect.)43 b(The)30 b(term)g Ft(\000)p Fj(f)2492 4259 y Fi(1)2528 4247 y Ft(f)p Fj(F)r(;)14 b Fl(cos)f Fj(')2841 4259 y Fi(1)2879 4247 y Ft(g)29 b Fl(generates)199 4346 y(the)34 b(mo)r(des)e(\(1)p Fj(;)14 b Fl(1\))33 b(and)g(\(1,-1\).)52 b(W)-7 b(e)34 b(neglect)e(the)i(last)e(one)h(and)g(w)n(e)f(k)n(eep)h (the)g(next)g(relev)-5 b(an)n(t)199 4446 y(F)e(ourier)32 b(mo)r(de)g Ft(\000)795 4413 y Fi(1)p 795 4427 34 4 v 795 4474 a(2)838 4446 y Fj(k)s(f)925 4458 y Fi(1)962 4446 y Fj(f)1003 4458 y Fi(2)1053 4446 y Fl(cos\()p Fj(')1251 4458 y Fi(1)1310 4446 y Fl(+)21 b Fj(')1450 4458 y Fi(2)1488 4446 y Fl(\).)52 b(Then)33 b(after)f(the)h(elimination)f(of)h(the)g(mo) r(de)f(\(0)p Fj(;)14 b Fl(1\),)199 4545 y(the)28 b(new)g(Hamiltonian)f (is)h(equal)f(to)g(:)797 4717 y Fj(H)873 4683 y Ff(0)919 4717 y Fl(=)c Fk(!)e Ft(\001)d Fk(A)h Fl(+)f(\()p Fu(\012)h Ft(\001)f Fk(A)p Fl(\))1568 4683 y Fi(2)1606 4717 y Fj(=)p Fl(2)f Ft(\000)h Fj(f)1831 4729 y Fi(1)1882 4717 y Fl(cos)13 b Fj(')2061 4729 y Fi(1)2117 4717 y Ft(\000)2210 4661 y Fl(1)p 2210 4698 42 4 v 2210 4774 a(2)2262 4717 y Fj(k)s(f)2349 4729 y Fi(1)2385 4717 y Fj(f)2426 4729 y Fi(2)2477 4717 y Fl(cos)o(\()p Fj(')2674 4729 y Fi(1)2731 4717 y Fl(+)18 b Fj(')2868 4729 y Fi(2)2905 4717 y Fl(\))p Fj(:)180 b Fl(\(14\))199 4882 y(In)27 b(this)h(elimination)f(pro)r(cedure,)f(w)n (e)g(ha)n(v)n(e)g(neglected)h(the)g(F)-7 b(ourier)26 b(mo)r(des)h(\(1)p Fj(;)14 b Ft(\000)p Fl(1\))26 b(and)h(\(0)p Fj(;)14 b Fl(2\))199 4981 y(whic)n(h)19 b(are)f(of)h(the)h(same)e (order)g Fj(")1226 4951 y Fi(2)1282 4981 y Fl(as)g(the)h(mo)r(de)g(\(1) p Fj(;)14 b Fl(1\))19 b(that)g(w)n(e)g(k)n(ept.)34 b(The)19 b(justi\014cation)g(for)f(this)199 5081 y(c)n(hoice)27 b(is)g(that)g(the)h(mo)r(de)f(\(1)p Fj(;)14 b Fl(1\))27 b(is)g(asso)r(ciated)f(with)i(the)f(smallest)g(small)g(denominator)f Fk(!)20 b Ft(\001)e Fk(k)p eop %%Page: 7 7 7 6 bop 199 100 a Fm(Sc)l(aling)31 b(law)f(for)h(the)f(critic)l(al)h (function)e(of)i(an)f(appr)l(oximate)h(r)l(enormalization)484 b Fl(7)199 282 y(among)24 b(these)i(three)f(mo)r(des)g(:)36 b Ft(j)p Fk(!)16 b Ft(\001)e Fl(\(1)p Fj(;)g Fl(1\))p Ft(j)22 b Fl(=)h(1)14 b Ft(\000)g Fj(!)24 b(<)f Ft(j)p Fk(!)16 b Ft(\001)e Fl(\(1)p Fj(;)g Ft(\000)p Fl(1\))p Ft(j)22 b Fl(=)h(1)14 b(+)g Fj(!)24 b(<)f Ft(j)p Fk(!)16 b Ft(\001)e Fl(\(0)p Fj(;)g Fl(2\))p Ft(j)22 b Fl(=)h(2.)199 382 y(Th)n(us)38 b(w)n(e)f(exp)r(ect)h(that)g(the)g(dynamics)g(is)f (mainly)h(in\015uenced)g(b)n(y)f(the)i(mo)r(de)e(\(1)p Fj(;)14 b Fl(1\))38 b(at)f(the)199 482 y(order)27 b Fj(")456 451 y Fi(2)521 482 y Fl(of)h(the)h(p)r(erturbation.)38 b(In)28 b(phase)g(space,)f(this)i(means)f(that)g(the)g(p)r(erio)r(dic)h (orbit)e(with)199 581 y(frequency)20 b(v)n(ector)g(\(1)p Fj(;)14 b Fl(1\))20 b(is)h(closer)e(to)i(the)g(torus)f(\(with)h (frequency)f Fj(!)s Fl(\))h(than)g(the)g(p)r(erio)r(dic)f(orbits)199 681 y(with)28 b(frequency)f(v)n(ectors)g(\(1)p Fj(;)14 b Ft(\000)p Fl(1\))26 b(and)i(\(0)p Fj(;)14 b Fl(2\).)199 780 y(W)-7 b(e)21 b(shift)h(the)f(F)-7 b(ourier)20 b(mo)r(des)g (according)f(to)i(the)g(follo)n(wing)f(linear)g(canonical)f (transformation)h(:)1135 963 y(\()p Fk(A)p Fj(;)14 b Fk(')p Fl(\))24 b Ft(7!)f Fl(\()p Fk(A)1604 927 y Ff(0)1628 963 y Fj(;)14 b Fk(')1726 929 y Ff(0)1750 963 y Fl(\))23 b(=)g(\()p Fj(S)1981 929 y Ff(\000)p Fi(1)2070 963 y Fk(A)p Fj(;)14 b(S)5 b Fk(')p Fl(\))p Fj(;)199 1146 y Fl(where)1456 1274 y Fj(S)28 b Fl(=)1623 1157 y Fe(\022)1725 1224 y Fl(1)83 b(1)1725 1323 y(1)g(0)1933 1157 y Fe(\023)2008 1274 y Fj(;)199 1474 y Fl(suc)n(h)21 b(that)h(cos)13 b Fj(')733 1486 y Fi(1)793 1474 y Fl(=)23 b(cos)13 b Fj(')1060 1444 y Ff(0)1060 1494 y Fi(2)1119 1474 y Fl(and)21 b(cos)o(\()p Fj(')1471 1486 y Fi(1)1514 1474 y Fl(+)6 b Fj(')1639 1486 y Fi(2)1676 1474 y Fl(\))23 b(=)g(cos)13 b Fj(')1998 1444 y Ff(0)1998 1494 y Fi(1)2036 1474 y Fl(.)34 b(The)22 b(v)n(ectors)d Fk(!)24 b Fl(and)d Fu(\012)h Fl(are)e(c)n(hanged)199 1573 y(in)n(to)k Fj(S)5 b Fk(!)26 b Fl(=)c(\()p Fj(!)15 b Ft(\000)d Fl(1)p Fj(;)i(!)s Fl(\))23 b(and)h Fj(S)5 b Fu(\012)22 b Fl(=)h(\()p Fj(k)15 b Fl(+)d(1)p Fj(;)i Fl(1\).)34 b(Th)n(us)24 b(the)g(normalization)f(conditions)h(on) g(these)199 1673 y(v)n(ectors)i(giv)n(e)h(the)h(image)f(of)g Fj(!)k Fl(and)c Fj(k)k Fl(:)797 1814 y Fj(!)852 1780 y Ff(0)898 1814 y Fl(=)23 b(\(1)p Fj(=!)s Fl(\))17 b Ft(\000)h Fl(1)p Fj(;)797 1939 y(k)843 1904 y Ff(0)889 1939 y Fl(=)23 b(1)p Fj(=)p Fl(\()p Fj(k)d Fl(+)e(1\))p Fj(:)199 2080 y Fl(W)-7 b(e)33 b(rescale)e(time)i(in)g(suc)n(h)f(a)g(w) n(a)n(y)f(that)i(the)g(linear)f(term)g(in)h(the)g(actions)e(of)i Fj(H)2832 2092 y Fi(0)2902 2080 y Fl(is)f(equal)g(to)199 2179 y Fk(!)262 2149 y Ff(0)308 2179 y Ft(\001)22 b Fk(A)34 b Fl(with)f Fk(!)716 2149 y Ff(0)772 2179 y Fl(=)f(\()p Fj(!)956 2149 y Ff(0)980 2179 y Fj(;)14 b Ft(\000)p Fl(1\))32 b(:)49 b(w)n(e)33 b(m)n(ultiply)h(the)f(Hamiltonian)h Fj(H)2434 2149 y Ff(0)2490 2179 y Fl(b)n(y)f(a)g(factor)g(1)p Fj(=!)s Fl(.)53 b(The)199 2279 y(quadratic)24 b(term)i(of)f(the)h(new)f (Hamiltonian)g(is)g(equal)g(to)g(\()p Fj(k)17 b 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y(f)1337 4485 y Fi(1)1392 4451 y Fl(~)1374 4473 y Fj(f)1415 4485 y Fi(2)1452 4473 y Fj(;)815 4648 y Fl(~)797 4673 y Fj(f)847 4644 y Ff(0)838 4695 y Fi(2)898 4673 y Fl(=)1125 4617 y(1)p 996 4654 300 4 v 996 4730 a(\(1)g Ft(\000)g Fj(!)s Fl(\))1258 4706 y Fi(2)1305 4673 y Fj(f)1346 4685 y Fi(1)1383 4673 y Fj(:)p eop %%Page: 8 8 8 7 bop 199 100 a Fm(Sc)l(aling)31 b(law)f(for)h(the)f(critic)l(al)h (function)e(of)i(an)f(appr)l(oximate)h(r)l(enormalization)484 b Fl(8)199 282 y Fm(3.3.)40 b(R)l(enormalization)31 b(op)l(er)l(ator)g Ft(R)199 436 y Fl(W)-7 b(e)31 b(de\014ne)f(the)h(renormalization)d(op)r (erator)g Ft(R)j Fl(according)d(to)i(the)h(v)-5 b(alue)30 b(of)g(the)g(frequency)g Fj(!)199 535 y Fl(in)e(the)g(follo)n(wing)f(w) n(a)n(y)f(:)797 676 y Ft(R)d Fl(=)g Ft(R)1048 688 y Fg(C)1215 676 y Fl(if)28 b Fj(!)e(>)c Fl(1)p Fj(;)797 801 y Ft(R)h Fl(=)g Ft(R)1048 813 y Fg(S)1207 801 y Fl(if)28 b Fj(!)e(<)d Fl(1)p Fj(:)199 942 y Fl(In)36 b(order)e(to)h(kno)n(w)g(the)g(sequence) 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b(action)g(of)g(the)199 4773 y(renormalization)e(op)r(erator)g Ft(R)1200 4785 y Fg(a)1268 4773 y Fl(b)r(ecomes)h(:)797 4914 y Fk(x)852 4880 y Ff(0)898 4914 y Fl(=)c Fj(N)1053 4926 y Fg(a)1093 4914 y Fk(x)18 b Fl(+)g Fk(\015)5 b Fl(\()p Fj(!)s(;)14 b(k)s Fl(\))p Fj(;)1612 b Fl(\(17\))p eop %%Page: 9 9 9 8 bop 199 100 a Fm(Sc)l(aling)31 b(law)f(for)h(the)f(critic)l(al)h (function)e(of)i(an)f(appr)l(oximate)h(r)l(enormalization)484 b Fl(9)199 282 y(where)1429 411 y Fj(N)1496 423 y Fg(a)1559 411 y Fl(=)1647 294 y Fe(\022)1750 360 y Fj(a)82 b Fl(1)1751 460 y(1)h(0)1960 294 y Fe(\023)2034 411 y Fj(:)199 606 y Fl(By)28 b(iteration,)f(w)n(e)g(obtain)g(the)h(general)e(equation)h (for)g(a)h(renormalization)d(tra)5 b(jectory)26 b(:)797 821 y Fk(x)852 833 y Fg(n)920 821 y Fl(=)d Fj(N)1075 833 y Fg(a)1111 841 y Fc(n)p Fb(\000)p Fd(1)1242 821 y Ft(\001)14 b(\001)g(\001)g Fj(N)1420 833 y Fg(a)1456 841 y Fd(0)1492 821 y Fk(x)1547 833 y Fi(0)1584 821 y Fl(+)p Fk(\015)5 b Fl(\()p Fj(!)1787 833 y Fg(n)p 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y Fg(l)p Fi(+1)1962 2019 y Fl(:)450 2262 y Fj(N)517 2274 y Fg(a)553 2282 y Fc(n)p Fb(\000)p Fd(1)685 2262 y Ft(\001)14 b(\001)g(\001)g Fj(N)863 2274 y Fg(a)899 2283 y Fc(l)p Fd(+1)1021 2262 y Fl(=)22 b(\()p Ft(\000)p Fl(1\))1279 2228 y Fg(l)p Fi(+1)1402 2145 y Fe(\022)1590 2212 y Fj(q)1627 2224 y Fg(n)1672 2212 y Fj(p)1714 2224 y Fg(l)1758 2212 y Ft(\000)c Fj(p)1883 2224 y Fg(n)1928 2212 y Fj(q)1965 2224 y Fg(l)2243 2212 y Fj(p)2285 2224 y Fg(n)2330 2212 y Fj(q)2367 2224 y Fg(l)p Fi(+1)2495 2212 y Ft(\000)g Fj(q)2615 2224 y Fg(n)2661 2212 y Fj(p)2703 2224 y Fg(l)p Fi(+1)1505 2311 y Fj(q)1542 2323 y Fg(n)p Ff(\000)p Fi(1)1672 2311 y Fj(p)1714 2323 y Fg(l)1758 2311 y Ft(\000)g Fj(p)1883 2323 y Fg(n)p Ff(\000)p Fi(1)2013 2311 y Fj(q)2050 2323 y Fg(l)2158 2311 y Fj(p)2200 2323 y Fg(n)p Ff(\000)p Fi(1)2330 2311 y Fj(q)2367 2323 y Fg(l)p Fi(+1)2495 2311 y Ft(\000)g Fj(q)2615 2323 y Fg(n)p Ff(\000)p Fi(1)2746 2311 y Fj(p)2788 2323 y Fg(l)p Fi(+1)2939 2145 y Fe(\023)3013 2262 y Fj(:)199 2523 y Fm(4.2.)40 b(Critic)l(al)31 b(surfac)l(e)199 2677 y Fl(F)-7 b(ollo)n(wing)32 b(Ref.)i([8)o(],)h(w)n(e)e(determine)g(the)g(critical)f(surface)h(that) g(separates)e(p)r(oin)n(ts)i(for)g(whic)n(h)199 2776 y Fk(x)254 2788 y Fg(n)331 2776 y Ft(!)f(\0001)h Fl(lik)n(e)f Ft(\000)p Fj(q)886 2788 y Fg(n)964 2776 y Fl(and)h(from)g(those)f(for)h (whic)n(h)g Fk(x)1985 2788 y Fg(n)2063 2776 y Ft(!)f Fl(+)p Ft(1)g Fl(lik)n(e)h Fj(q)2553 2788 y Fg(n)2598 2776 y Fl(.)53 b(Th)n(us)33 b(the)g(critical)199 2876 y(surface)d(is)h(obtained)f(b)n(y)h(the)g(condition)f Fk(x)1600 2888 y Fg(n)1645 2876 y Fj(=q)1724 2888 y Fg(n)1797 2876 y Ft(!)f Fl(0)h(when)h Fj(n)d Ft(!)g(1)p Fl(.)47 b(The)31 b(\014rst)f(comp)r(onen)n(t)199 2976 y(of)e(Eq.)f(\(18\))g(b)r (ecomes)g(:)209 3181 y Fj(x)256 3193 y Fg(n)p 209 3218 93 4 v 214 3294 a Fj(q)251 3306 y Fg(n)335 3237 y Fl(=)c Fj(x)p Fl(+)545 3181 y Fj(p)587 3193 y Fg(n)p 544 3218 87 4 v 546 3294 a Fj(q)583 3306 y Fg(n)641 3237 y Fj(y)s Fl(+)750 3133 y Fg(n)p Ff(\000)p 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4 v 3327 3294 a Fj(q)3364 3306 y Fg(n)3646 3237 y Fj(:)199 3499 y Fl(W)-7 b(e)28 b(use)g(the)g(fact)f(that)h Fj(p)1014 3511 y Fg(n)1059 3499 y Fj(=q)1138 3511 y Fg(n)1206 3499 y Ft(!)23 b Fj(!)s Fl(.)37 b(Then)28 b(w)n(e)f(ha)n(v)n(e)f(:)797 3707 y Fj(x)19 b Fl(+)f Fj(!)s(y)j Fl(+)1173 3603 y Ff(1)1146 3628 y Fe(X)1153 3806 y Fg(l)p Fi(=0)1279 3707 y Fl([)q Fj(\014)1350 3719 y Fg(l)1375 3707 y Fj(\015)1418 3719 y Fi(1)1455 3707 y Fl(\()p Fj(!)1539 3719 y Fg(l)1565 3707 y Fj(;)14 b(k)1645 3719 y Fg(l)1670 3707 y Fl(\))19 b(+)f Fj(\014)1851 3719 y Fg(l)p Fi(+1)1961 3707 y Fj(\015)2004 3719 y Fi(2)2041 3707 y Fl(\()p Fj(!)2125 3719 y Fg(l)2150 3707 y Fl(\)])24 b(=)e(0)p Fj(;)759 b Fl(\(19\))199 3935 y(where)26 b Fj(\014)485 3947 y Fg(l)533 3935 y Fl(=)d(\()p Ft(\000)p Fl(1\))792 3905 y Fg(l)p Fi(+1)901 3935 y Fl(\()p Fj(p)975 3947 y Fg(l)1015 3935 y Ft(\000)15 b Fj(!)s(q)1187 3947 y Fg(l)1212 3935 y Fl(\).)37 b(W)-7 b(e)26 b(notice)g(that)g(the)g (second)g(comp)r(onen)n(t)f(of)h(Eq.)f(\(18\))h(giv)n(es)199 4035 y(the)i(same)f(equation)g(\(19\))g(for)g(the)h(critical)f (surface.)36 b(F)-7 b(ollo)n(wing)26 b(Ref.)i([8],)f(one)h(can)f(pro)n (v)n(e)e(that)199 4134 y(the)j(sum)g(in)g(Eq.)f(\(19\))g(con)n(v)n (erges)e(if)j(and)g(only)f(if)h Fj(!)i Fl(satis\014es)d(the)h(Bruno)f (condition)g(:)1419 4280 y Ff(1)1392 4305 y Fe(X)1399 4483 y Fg(l)p Fi(=0)1536 4327 y Fl(log)14 b Fj(q)1694 4339 y Fg(l)p Fi(+1)p 1536 4364 268 4 v 1638 4440 a Fj(q)1675 4452 y Fg(l)1837 4383 y Fj(<)22 b Fl(+)p Ft(1)p Fj(:)p eop %%Page: 10 10 10 9 bop 199 100 a Fm(Sc)l(aling)31 b(law)f(for)h(the)f(critic)l(al)h (function)e(of)i(an)f(appr)l(oximate)h(r)l(enormalization)443 b Fl(10)199 282 y Fm(4.3.)40 b(Sc)l(aling)31 b(law)f(ne)l(ar)g Fj(!)i Fm(r)l(ational)199 436 y Fl(In)h(order)e(to)i(in)n(v)n(estigate) e(the)i(b)r(eha)n(vior)e(of)i(the)f(critical)g(surface)g(near)g(a)g (rational)f(frequency)199 535 y Fj(P)7 b(=Q)33 b Fl(=)f([)p Fj(a)564 547 y Fi(0)602 535 y 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1655 y Fg(J)423 1643 y Fl(is)f(equal)g(to)g([)p Fj(n;)14 b(a)986 1655 y Fg(J)5 b Fi(+1)1117 1643 y Fj(;)14 b(:)g(:)g(:)o Fl(].)43 b(Then)29 b(w)n(e)g(ha)n(v)n(e)f Fj(!)1926 1655 y Fg(J)1998 1643 y Fl(=)e(1)p Fj(=n)18 b Fl(+)i Fj(O)r Fl(\(1)p Fj(=n)2557 1613 y Fi(2)2594 1643 y Fl(\).)42 b(F)-7 b(urthermore,)29 b Fj(k)3241 1655 y Fg(J)199 1743 y Fl(is)h(indep)r(enden)n(t)h(of)f Fj(n)h Fl(since)e(its)i(con)n(tin)n(ued)f(fraction)f(expansion)g(is)h ([)p Fj(a)2491 1755 y Fg(J)5 b Ff(\000)p Fi(1)2623 1743 y Fj(;)14 b(:)g(:)g(:)f(;)h(a)2851 1755 y Fi(0)2888 1743 y Fj(;)g(b)2961 1755 y Fi(0)2998 1743 y Fj(;)g(b)3071 1755 y Fi(1)3108 1743 y Fj(;)g(:)g(:)g(:)p Fl(].)199 1842 y(W)-7 b(e)28 b(use)g(the)g(follo)n(wing)e(asymptotic)h (expansions)g(:)850 1942 y Fg(n)811 1966 y Fe(X)818 2145 y Fg(l)p Fi(=1)945 2045 y Fl(log\()p Fj(k)1127 2057 y Fg(J)1192 2045 y Fl(+)18 b Fj(l)r Fl(\))23 b(=)g Fj(n)14 b Fl(log)f Fj(n)19 b Ft(\000)f Fj(n)g Fl(+)g Fj(O)r Fl(\(log)d Fj(n)p Fl(\))p Fj(;)815 2336 y 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Ft(\031)1286 3887 y Fl(1)p 1274 3924 66 4 v 1274 4000 a Fj(Q)1363 3943 y Fl(log)14 b Ft(j)p Fj(!)22 b Ft(\000)c Fj(P)7 b(=Q)p Ft(j)18 b Fl(+)g Fj(C)2015 3958 y Fg(P)5 b(=Q)2153 3943 y Fj(;)964 b Fl(\(23\))199 4124 y(where)450 4335 y Fj(C)509 4350 y Fg(P)5 b(=Q)671 4335 y Fl(=)780 4279 y(1)p 768 4316 V 768 4392 a Fj(Q)844 4335 y Fl(\(log)14 b(2)k Ft(\000)g Fl(1)g(+)g(2)c(log)g Fj(Q)p Fl(\))k(+)g(2)1711 4279 y Fj(P)p 1711 4316 V 1711 4392 a(Q)1800 4335 y Fl(log)1931 4279 y Fj(P)p 1931 4316 V 1931 4392 a(Q)2025 4335 y Fl(+)g(2)2164 4218 y Fe(\022)2224 4335 y Fl(1)g Ft(\000)2378 4279 y Fj(P)p 2377 4316 V 2377 4392 a(Q)2453 4218 y Fe(\023)2528 4335 y Fl(log)2649 4218 y Fe(\022)2710 4335 y Fl(1)g Ft(\000)2864 4279 y Fj(P)p 2863 4316 V 2863 4392 a(Q)2939 4218 y Fe(\023)3014 4335 y Fj(:)199 4563 y Fl(W)-7 b(e)28 b(notice)g(that)g Fj(C)826 4578 y Fg(P)5 b(=Q)992 4563 y Fl(is)27 b(symmetric)g(:)37 b Fj(C)1598 4578 y Fg(P)5 b(=Q)1760 4563 y Fl(=)22 b Fj(C)1906 4578 y Fi(1)p Ff(\000)p Fg(P)5 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