%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: pap2.dvi %%Pages: 19 -1 %%PageOrder: Descend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: dvips -pp1-19 pap2.dvi -o pap2.ps %DVIPSParameters: dpi=300, comments removed %DVIPSSource: TeX output 1999.03.11:1639 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}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{dup dup 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]N /FBB[0 0 0 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Fk(\026)e Fn(=)g Fi(\000)p Fn(4,)i Fk(\027)h Fn(=)c(2)p Fk(:)p Fn(1667.)47 b(.)24 b(.)g(.)h(.)54 b(24)103 1233 y(6)88 b(A)25 b(c)o(haotic)g(solution)h(with)f(spatial)h(p)q(erio)q(d)g Fk(\031)h Fn(and)f(homogeneous)f(Neumann)215 1306 y(b)q(oundary)d (conditions)e(that)h(is)f(unstable)h(to)g(p)q(erturbations)g(of)g(p)q (erio)q(d)g(2)p Fk(\031)r Fn(.)33 b(A)215 1378 y(small)15 b(p)q(erturbation)j(with)e(spatial)h(p)q(erio)q(d)g(2)p Fk(\031)h Fn(is)f(added)g(at)g Fk(t)d Fn(=)g(10.)23 b(P)o(arameter)215 1450 y(v)m(alues:)28 b Fk(R)19 b Fn(=)g(4)p Fk(:)p Fn(2,)h Fk(\026)f Fn(=)g Fi(\000)p Fn(4,)h Fk(\027)i Fn(=)d(2)p Fk(:)p Fn(8.)30 b(Note)19 b(that)h(only)f(the)g(real)g(part)g(of)h(the) 215 1522 y(solution)d(is)f(sho)o(wn.)25 b(.)g(.)f(.)h(.)f(.)h(.)f(.)g (.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.) h(.)f(.)h(.)f(.)g(.)h(.)54 b(25)103 1595 y(7)88 b(Dominan)o(t)15 b(Ly)o(apuno)o(v)h(exp)q(onen)o(ts)f(for)h(a)g(solution)g(with)f (spatial)h(p)q(erio)q(d)g Fk(\031)r Fn(,)f(ev)o(en)215 1667 y(ab)q(out)20 b(the)f(origin)g(and)g(o)q(dd)h(ab)q(out)g Fk(\031)r(=)p Fn(4)f(\(solid\),)g(and)g(for)g(p)q(erturbations)h(with) 215 1739 y(spatial)15 b(p)q(erio)q(d)g(2)p Fk(\031)h Fn(\(dashed\).)21 b(Note)14 b(that)h(the)g(Ly)o(apuno)o(v)f(exp)q(onen) o(t)h(for)f(p)q(ertur-)215 1811 y(bations)j(has)g(m)o(ultiplici)o(t)o (y)c(2.)21 b(P)o(arameter)15 b(v)m(alues:)21 b Fk(\026)15 b Fn(=)e Fi(\000)p Fn(4,)j Fk(\027)h Fn(=)d(2.)27 b(.)e(.)f(.)h(.)f(.)g (.)h(.)54 b(26)103 1884 y(8)88 b(A)22 b(c)o(haotic)f(solution)i(with)e (spatial)i(p)q(erio)q(d)f Fk(\031)i Fn(that)e(is)g(ev)o(en)f(ab)q(out)i (the)f(origin)215 1956 y(and)e(o)q(dd)g(ab)q(out)g Fk(x)e Fn(=)g Fk(\031)r(=)p Fn(4)i(and)f(unstable)g(to)g(p)q(erturbations)h (of)g(p)q(erio)q(d)f(2)p Fk(\031)r Fn(.)29 b(A)215 2028 y(small)14 b(p)q(erturbation)i(with)f(spatial)h(p)q(erio)q(d)g(2)p Fk(\031)h Fn(is)e(added)h(at)g Fk(t)d Fn(=)h(0)p Fk(:)p Fn(3.)21 b(P)o(arameter)215 2100 y(v)m(alues:)h Fk(R)14 b Fn(=)g(62,)i Fk(\026)e Fn(=)g Fi(\000)p Fn(4,)i Fk(\027)h Fn(=)d(2.)22 b(Compare)15 b(with)h(Figure)g(7.)e(.)24 b(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)54 b(27)103 2172 y(9)88 b(Dominan)o(t)21 b(Ly)o(apuno)o(v)g(exp)q(onen)o(ts)g(for)h(a)f (solution)h(with)f(spatial)g(p)q(erio)q(d)h(2)p Fk(\031)r(=)p Fn(3)215 2245 y(\(lo)o(w)o(er\))h(and)i(p)q(erturbations)g(with)e (spatial)h(p)q(erio)q(d)h(2)p Fk(\031)h Fn(\(upp)q(er\).)44 b(P)o(arameter)215 2317 y(v)m(alues:)22 b Fk(R)14 b Fn(=)g(45,)i Fk(\027)h Fn(=)d(2)p Fk(:)p Fn(6.)43 b(.)25 b(.)f(.)g(.)h(.)f(.)h(.)f (.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.) g(.)h(.)54 b(28)103 2389 y(10)64 b(A)16 b(c)o(haotic)f(solution)h(with) g(spatial)g(p)q(erio)q(d)g(2)p Fk(\031)r(=)p Fn(3)h(that)f(is)g (unstable)g(to)g(p)q(erturba-)215 2461 y(tions)h(of)g(p)q(erio)q(d)f(2) p Fk(\031)r Fn(.)22 b(A)16 b(small)e(p)q(erturbation)j(with)g(spatial)f (p)q(erio)q(d)h(2)p Fk(\031)h Fn(is)e(added)215 2534 y(at)i Fk(t)e Fn(=)f(1.)26 b(P)o(arameter)16 b(v)m(alues:)23 b Fk(R)17 b Fn(=)e(45,)j Fk(\026)f Fn(=)f Fi(\000)p Fn(1)p Fk(:)p Fn(8,)h Fk(\027)i Fn(=)d(2)p Fk(:)p Fn(6.)25 b(Note)17 b(that)h(only)215 2606 y(the)e(real)g(part)h(of)f Fk(A)g Fn(is)g(plotted.)27 b(.)d(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g (.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)54 b(29)936 2893 y(18)p eop %%Page: 17 3 17 2 bop 30 28 a Fn([14])24 b(J.)16 b(T.)f(Stuart)i(and)f(R.)g(C.)f (DiPrima.)f(The)i(Ec)o(khaus)g(and)h(Benjamin-F)l(eir)c(resonance)j (mec)o(h-)130 101 y(anisms.)f Fl(Pr)n(o)n(c.)i(R)n(oy.)f(So)n(c.)h(L)n (ond.)g(A)f Fe(362)p Fn(,)g(27-41,)i(1978.)30 214 y([15])24 b(B.)19 b(W)l(erner.)g(Eigen)o(v)m(alue)h(problems)e(with)i(the)g (symmetry)c(of)21 b(a)f(group)h(and)g(bifurcations.)130 287 y(In)15 b Fl(Continuation)j(and)f(Bifur)n(c)n(ations:)22 b(Numeric)n(al)17 b(T)l(e)n(chniques)h(and)f(Applic)n(ations)p Fn(,)f(eds.)f(D.)130 359 y(Ro)q(ose,)i(B.)e(De)i(Dier)e(and)i(A.)e(Sp)q (ence,)h(NA)l(TO)f(ASI)h(Series)f(C,)h Fe(313)p Fn(,)g(71-88,)i(1990.) 936 2893 y(17)p eop %%Page: 16 4 16 3 bop 30 28 a Fm(References)54 150 y Fn([1])24 b(P)l(.)15 b(Ash)o(win,)f(J.)g(Buescu)g(and)h(I.)f(Stew)o(art.)h(Bubbling)f(of)h (attractors)g(and)h(sync)o(hronisation)f(of)130 222 y(oscillators.)h Fl(Phys.)h(L)n(ett.)g(A.)g Fe(193)p Fn(.)e(p.)h(126.)h(1994.)54 336 y([2])24 b(P)l(.)16 b(J.)f(Aston.)g(Analysis)h(and)g(computation)f (of)h(symmetry)o(-breaking)d(bifurcation)j(and)g(scal-)130 408 y(ing)h(la)o(ws)f(using)g(group)i(theoretic)d(metho)q(ds.)g Fl(SIAM)j(J.)f(Math.)g(A)o(nal.)g Fe(22)p Fn(,)e(181-212,)k(1991.)54 522 y([3])24 b(P)l(.)16 b(J.)f(Aston)h(and)g(E.)f(L.)h(Allgo)o(w)o(er.) e(Symmetry)e(reductions)j(for)h(the)g(n)o(umerical)d(solution)j(of)130 594 y(b)q(oundary)h(v)m(alue)d(problems.)g(In)h Fl(The)h(Mathematics)h (of)f(Numeric)n(al)h(A)o(nalysis)p Fn(,)e(Pro)q(c.)g(AMS-)130 666 y(SIAM)20 b(Summer)d(Seminar)i(in)i(Appl.)e(Math.,)i(Utah,)g(eds.)f (J.)h(Renegar,)g(M.)f(Sh)o(ub)g(and)h(S.)130 739 y(Smale,)14 b(Lectures)i(in)g(Appl.)f(Math.,)h(V)l(ol)g(32,)g(29-47,)i(1996.)54 852 y([4])24 b(P)l(.)15 b(J.)g(Aston)h(and)g(M.)e(Dellnitz.)g(Symmet)o (ry)f(breaking)i(bifurcations)g(of)h(c)o(haotic)f(attractors.)130 925 y Fl(Int.)j(J.)f(Bifn.)h(and)g(Chaos.)d Fe(5)p Fn(.)h(No.)g(6.)g (pp.)g(1643-1676.)j(1995.)54 1038 y([5])24 b(P)l(.)f(J.)f(Aston)h(and)g (C.)f(R.)g(Laing.)i(Symme)o(try)c(and)j(Chaos)h(in)e(the)h(Complex)d (Ginzburg{)130 1111 y(Landau)14 b(Equation.)f(I:)f(Re\015ectional)f (Symmetri)o(es.)e(Submitted)i(to)i(Dynamics)e(and)i(Stabilit)o(y)130 1183 y(of)k(Systems,)d(1998.)54 1297 y([6])24 b(T.)c(B.)f(Benjamin)f (and)i(J.)g(E.)g(F)l(eir.)e(The)i(disin)o(tegration)f(of)i(w)o(a)o(v)o (e)e(trains)h(on)g(deep)g(w)o(ater.)130 1369 y(P)o(art)d(I.)e(Theory)l (.)h Fl(J.)h(Fluid)h(Me)n(ch.)e Fe(27,)g Fn(pt.)f(3.)i(pp.)f(417-430.)i (1967.)54 1483 y([7])24 b(E.)19 b(Co)o(v)m(as,)h(P)l(.)f(Ash)o(win)f (and)i(R.)e(T)l(a)o(v)m(ak)o(ol.)h(Non{normal)f(parameter)g(blo)o(w)o (out)h(bifurcation)130 1555 y(in)c(a)h(truncated)g(mean)e(\014eld)h (dynamo.)f(T)l(ec)o(hnical)g(Rep)q(ort)i(97/15.)h(Departmen)o(t)d(of)i (Mathe-)130 1627 y(matics)f(and)i(Statistics,)e(Univ)o(ersit)o(y)f(of)i (Surrey)l(,)g(1997.)54 1741 y([8])24 b(J.D.)g(Cra)o(wford,)i(M.)d (Golubitsky)l(,)i(M.G.M.)d(Gomes,)i(E.)g(Knoblo)q(c)o(h)g(and)g(I.N.)f (Stew)o(art.)130 1813 y(Boundary)d(conditions)f(as)h(symmetry)c (constrain)o(ts.)j(In)g Fl(Singularity)j(The)n(ory)d(and)h(Its)g(Ap-) 130 1885 y(plic)n(ations,)e(Part)f(II)p Fn(,)f(eds.)f(M.)h(Rob)q(erts)h (and)g(I.)e(Stew)o(art,)h(LNM)g(1463,)h(Springer,)f(1991.)54 1999 y([9])24 b(W.)16 b(Ec)o(khaus.)g Fl(Studies)i(in)g(Non{Line)n(ar)f (Stability)i(The)n(ory)p Fn(,)14 b(Springer-V)l(erlag,)i(1965.)30 2113 y([10])24 b(H.)14 b(F)l(ujisak)m(a,)h(K.)f(Ouc)o(hi,)f(H.)h(Hata,) h(B.)e(Masaok)m(a)j(and)f(S.)g(Miy)o(azaki.)d(On{o\013)k(in)o(termitte) o(ncy)130 2185 y(in)g(oscillatory)g(media.)e Fl(Physic)n(a)j(D)f Fe(114)p Fn(.)g(pp.)g(237-250.)i(1998.)30 2299 y([11])24 b(M.)15 b(Golubitsky)l(,)f(I.)h(Stew)o(art)g(and)g(D.G.)g(Sc)o (hae\013er.)g Fl(Singularities)j(and)e(Gr)n(oups)g(in)g(Bifur)n(c)n(a-) 130 2371 y(tion)i(The)n(ory.)e(V)l(ol.)j(II.)c Fn(Applied)g(Math.)h (Sciences)f Fe(69)p Fn(,)h(Springer{V)l(erlag,)f(1988.)30 2485 y([12])24 b(J.)12 b(M.)f(Hyman,)g(B.)g(Nicolaenk)o(o)g(and)i(S.)f (Zaleski.)e(Order)i(and)h(complexit)o(y)8 b(in)k(the)g(Kuramoto-)130 2557 y(Siv)m(ashinsky)k(mo)q(del)f(of)i(w)o(eakly)e(turbulen)o(t)g(in)o (terfaces.)g Fl(Physic)n(a)i(D)f Fe(23)p Fn(,)g(265-292,)i(1986.)30 2671 y([13])24 b(E.)d(Ott)g(and)g(J.)f(Sommerer.)e(Blo)o(w)o(out)i (bifurcations:)31 b(the)20 b(o)q(ccurrence)g(of)h(riddled)f(basins)130 2743 y(and)d(on{o\013)h(in)o(termittency)-5 b(.)13 b Fl(Phys.)k(L)n(ett.)g(A.)g Fe(188)p Fn(,)e(39.)i(1994.)936 2893 y(16)p eop %%Page: 15 5 15 4 bop 103 28 a Fn(W)l(e)14 b(ha)o(v)o(e)g(also)h(found)g(that)g (this)f(solution,)g(whether)g(regarded)h(as)g(a)g(solution)f(of)h(the)f (p)q(erio)q(dic)30 101 y(problem)j(or)j(the)e(Neumann)g(problem,)f (undergo)q(es)j(a)f(sup)q(ercritical)f(blo)o(w)o(out)h(bifurcation.)29 b(F)l(or)30 173 y(the)11 b(Neumann)f(problem,)g(this)h(in)o(v)o(olv)o (es)f(breaking)h(only)g(a)h(re\015ectional)e(symmetry)e(of)k(the)f (solution)30 245 y(while)k(for)h(the)f(p)q(erio)q(dic)h(problem,)d (this)j(is)f(equiv)m(alen)o(t)f(to)i(a)h(p)q(erio)q(d-increasing)f (bifurcation.)k(W)l(e)30 317 y(b)q(eliev)o(e)15 b(this)i(to)h(b)q(e)g (the)f(\014rst)g(observ)m(ation)h(of)g(a)g(blo)o(w)o(out)f(bifurcation) g(from)f(a)i(c)o(haotic)f(solution)30 390 y(of)g(a)f(PDE)h(asso)q (ciated)g(with)f(a)h(side{band)g(instabilit)o(y)l(.)103 462 y(F)l(or)h(solutions)g(with)f(spatial)h(p)q(erio)q(d)g(2)p Fk(\031)r(=)p Fn(3)g(and)g(v)m(arying)g(amoun)o(ts)f(of)h(symmetry)c(w) o(e)j(found)30 534 y(that)g(c)o(haotic)e(solutions)i(are)f(alw)o(a)o (ys)g(unstable)h(with)f(resp)q(ect)g(to)g(p)q(erturbations)i(of)e(p)q (erio)q(d)h(2)p Fk(\031)r Fn(.)30 660 y Fe(Ac)n(kno)n(wledgemen)n(ts)30 786 y Fn(This)f(w)o(ork)h(w)o(as)f(supp)q(orted)i(b)o(y)d(the)h(EPSR)o (C)h(Applied)e(Nonlinear)g(Mathematics)g(Initiativ)o(e.)936 2893 y(15)p eop %%Page: 14 6 14 5 bop 30 28 a Fn(with)14 b(resp)q(ect)g(to)g(p)q(erturbations)i (with)e(p)q(erio)q(d)g(three)g(times)e(that)j(of)f(the)g(solution,)h(w) o(e)e(w)o(ould)i(not)30 101 y(exp)q(ect)g(to)i(see)f(this)g(solution)h (o)q(ccurring)f(starting)h(with)f(an)h(arbitrary)f(initial)f (condition.)103 173 y(Finally)l(,)i(w)o(e)h(consider)g(solutions)h(in)g (Fix)o(\(\006)938 180 y Fh(4)958 173 y Fn(\))f(and)h(compute)e(the)h (dominan)o(t)g(Ly)o(apuno)o(v)g(ex-)30 245 y(p)q(onen)o(ts)h(asso)q (ciated)h(with)f(p)q(erturbations)g(corresp)q(onding)h(to)f(the)g(t)o (w)o(o)f(t)o(w)o(o{dimensional)g(irre-)30 317 y(ducible)g(represen)o (tations)i(of)f(\006)636 324 y Fh(4)676 317 y Fn(as)h(a)g(function)f (of)h Fk(\026)g Fn(for)g Fk(R)g Fn(=)g(80,)g Fk(\027)j Fn(=)c(2.)32 b(W)l(e)19 b(see)h(for)f(these)30 390 y(parameter)f(v)m (alues)i(that)g(as)g Fk(\026)f 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y(conditions)j(are)g(imp)q(osed)g(on)g (them.)32 b(W)l(e)20 b(w)o(ould)g(not)h(exp)q(ect)e(to)i(see)e(suc)o(h) h(solutions)h(forming)30 1779 y(starting)c(with)f(an)h(arbitrary)l(,)f (non-p)q(erio)q(dic)g(initial)f(condition.)22 b(Th)o(us,)16 b(w)o(e)g(conclude)f(that)i(while)30 1852 y(p)q(erio)q(dic)k(b)q (oundary)h(conditions)f(are)f(often)h(mathematically)c(v)o(ery)j(con)o (v)o(enien)o(t,)f(they)i(are)g(not)30 1924 y(necessarily)15 b(ph)o(ysically)g(relev)m(an)o(t)g(for)h(c)o(haotic)g(solutions.)103 1996 y(Homogeneous)g(Neumann)g(b)q(ounda)o(y)h(conditions)g(are)g (often)g(ph)o(ysically)e(relev)m(an)o(t)h(and)h(while)30 2068 y(the)g(application)f(of)i(these)e(b)q(oundary)i(conditions)f(w)o (ould)g(app)q(ear)h(to)g(restrict)d(the)i(symmetry)d(of)30 2140 y(the)f(problem)f(it)h(is)g(w)o(ell)f(kno)o(wn)i(that)g(this)f (problem)f(can)h(b)q(e)h(em)o(b)q(edded)d(in)i(the)g(p)q(erio)q(dic)h (problem)30 2213 y(whic)o(h)20 b(has)h(m)o(uc)o(h)d(more)h(symmet)o(ry) e([2,)j(8)q(].)32 b(In)20 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Fn(A)23 b(c)o(haotic)h(solution)g(in)f(Fix)o(\(\006) 719 35 y Fh(1)739 28 y Fn(\))h(with)f(spatial)h(p)q(erio)q(d)g Fk(\031)h Fn(and)g(homogeneous)e(Neumann)30 101 y(b)q(oundary)16 b(conditions)f(that)g(is)g(unstable)f(with)h(resp)q(ect)g(to)g(p)q (erturbations)g(of)g(spatial)g(p)q(erio)q(d)h(2)p Fk(\031)r Fn(,)30 173 y(i.e.)g(after)h(the)g(blo)o(w)o(out)h(bifurcation)f(is)g (sho)o(wn)h(in)f(Fig.)g(6.)25 b(A)17 b(small)f(p)q(erturbation)i(with)f (spatial)30 245 y(p)q(erio)q(d)c(2)p Fk(\031)h Fn(is)e(added)h(at)f Fk(t)i Fn(=)g(10)f(and)g(the)f(symmetry)d(of)j(the)g(solution)h(is)f (quic)o(kly)e(lost)j(as)f(exp)q(ected.)30 317 y(The)k(parameter)e(v)m (alues)h(are)h Fk(R)e Fn(=)g(4)p Fk(:)p Fn(2,)i Fk(\026)e Fn(=)g Fi(\000)p Fn(4)h(and)i Fk(\027)g Fn(=)c(2)p Fk(:)p Fn(8.)22 b(Note)15 b(that)h(only)f(the)h(real)f(part)h(of)30 390 y(the)g(solution)h(is)f(sho)o(wn.)103 462 y(Finally)l(,)d(w)o(e)g (consider)h(solutions)h(in)f(Fix)o(\(\006)916 469 y Fh(2)936 462 y Fn(\))g(whic)o(h)f(ha)o(v)o(e)g(spatial)i(p)q(erio)q(d)f Fk(\031)r Fn(,)g(are)g(ev)o(en)f(ab)q(out)30 534 y(the)j(origin)g(and)g (are)g(o)q(dd)h(ab)q(out)h Fk(\031)r(=)p Fn(4.)j(Recall)15 b(that)h(in)g(this)g(case)g(w)o(e)f(are)h(only)g(in)o(terested)f(in)g (the)30 606 y(Ly)o(apuno)o(v)g(exp)q(onen)o(ts)f(of)h(m)o(ultipli)o (cit)n(y)c(t)o(w)o(o)k(asso)q(ciated)g(with)f(the)g(t)o(w)o (o-dimensional)f(irreducible)30 679 y(represen)o(tation)j(of)g(\006)439 686 y Fh(2)459 679 y Fn(.)22 b(In)16 b(Fig.)g(7)g(these)g(dominan)o(t)g (Ly)o(apuno)o(v)g(exp)q(onen)o(ts)h(are)f(sho)o(wn.)22 b(W)l(e)16 b(see)30 751 y(that)h(o)o(v)o(er)e(the)h(parameter)g(range)h (sho)o(wn,)f(the)g(underlying)g(solution)h(c)o(hanges)f(from)g(p)q (erio)q(dic)g(or)30 823 y(quasip)q(erio)q(dic)i(to)g(c)o(haotic)f(and)h (bac)o(k)g(again)g(but)g(is)g(alw)o(a)o(ys)f(unstable)h(with)g(resp)q (ect)f(to)i(p)q(ertur-)30 895 y(bations)g(of)f(p)q(erio)q(d)g(2)p Fk(\031)r Fn(.)27 b(W)l(e)18 b(giv)o(e)e(an)j(example)d(of)i(suc)o(h)g (an)g(unstable)g(c)o(haotic)g(solution)g(in)g(Fig.)30 967 y(8.)k(A)15 b(small)g(p)q(erturbation)i(with)f(spatial)g(p)q(erio)q (d)h(2)p Fk(\031)h Fn(is)d(added)i(to)f(the)g(solution)h(at)f Fk(t)e Fn(=)f(0)p Fk(:)p Fn(3.)22 b(The)30 1040 y(parameter)15 b(v)m(alues)h(are)h Fk(R)d Fn(=)g(62,)i Fk(\026)e Fn(=)g Fi(\000)p Fn(4,)i Fk(\027)h Fn(=)d(2.)103 1112 y(F)l(or)23 b(all)e(con)o(tour)i(plots,)g(blac)o(k)f(con)o(tour)g(lines)g(corresp)q (ond)h(to)g(negativ)o(e)e(v)m(alues)h(and)h(grey)30 1184 y(con)o(tour)16 b(lines)g(to)g(p)q(ositiv)o(e)g(v)m(alues.)30 1344 y Fd(4.2)66 b(P)n(erio)r(d)23 b Fv(2)p Fc(\031)r(=)p Fv(3)h Fd(solutions)30 1448 y Fn(In)15 b(order)g(to)g(in)o(v)o (estigate)f(the)h(e\013ect)f(of)i(p)q(erturbations)g(three)e(times)g (the)g(p)q(erio)q(d)i(of)f(the)g(solution,)30 1520 y(w)o(e)k(computed)f (solutions)h(with)g(p)q(erio)q(d)h(2)p Fk(\031)r(=)p Fn(3,)g(initially)d(with)i(no)h(other)f(symmetri)o(es)d(imp)q(osed.)30 1592 y(The)d(dominan)o(t)g(Ly)o(apuno)o(v)g(exp)q(onen)o(ts)g(asso)q (ciated)i(with)e(the)g(t)o(w)o(o-dimensional)f(irreducible)f(rep-)30 1665 y(resen)o(tation)k(of)h Fe(Z)369 1672 y Fh(3)405 1665 y Fn(are)g(sho)o(wn)g(in)g(Fig.)f(9.)21 b(This)16 b(sho)o(ws)h(a)f(transition)g(to)g(c)o(haos)g(b)q(efore)g(and)g(after) 30 1737 y(whic)o(h)g(the)h(solution)h(is)f(unstable)g(with)g(resp)q (ect)g(to)g(p)q(erturbations)h(of)g(p)q(erio)q(d)g(2)p Fk(\031)r Fn(.)23 b(An)17 b(example)30 1809 y(of)h(an)g(unstable)g(c)o (haotic)f(solution)h(corresp)q(onding)h(to)f(this)g(parameter)e(range)i (is)g(sho)o(wn)g(in)g(Fig.)30 1881 y(10.)j(Note)13 b(that)h(at)g(appro) o(ximately)e Fk(t)h Fn(=)h(1)p Fk(:)p Fn(25,)g(the)g(solution)g(almost) e(has)j Fe(D)1450 1888 y Fh(3)1483 1881 y Fn(symmetry)10 b(but)k(then)30 1954 y(all)i(symmetry)d(is)j(so)q(on)h(quic)o(kly)d (lost)j(after)f(this)g(p)q(oin)o(t.)103 2026 y(W)l(e)i(next)g(consider) f(solutions)i(in)f(Fix)o(\(\006)882 2033 y Fh(3)902 2026 y Fn(\))g(whic)o(h)g(ha)o(v)o(e)f(spatial)h(p)q(erio)q(d)h(2)p Fk(\031)r(=)p Fn(3)g(and)f(are)g(also)30 2098 y(ev)o(en)f(ab)q(out)i (the)f(origin.)26 b(Again,)17 b(w)o(e)h(only)g(consider)f(the)h (dominan)o(t)f(\(m)o(ultiple)o(\))e(Ly)o(apuno)o(v)j(ex-)30 2170 y(p)q(onen)o(ts)c(asso)q(ciated)h(with)f(the)f(t)o(w)o (o-dimensional)f(irreducible)g(represen)o(tation)h(whic)o(h)g(are)h (sho)o(wn)30 2242 y(in)21 b(Fig.)f(11)i(as)g(a)g(function)e(of)i Fk(R)g Fn(for)f Fk(\026)h Fn(=)h Fi(\000)p Fn(4)e(and)h Fk(\027)j Fn(=)d(2.)36 b(W)l(e)21 b(see)g(that)h(for)f(this)g(range)h (of)30 2315 y(parameters)c(the)h(underlying)g(solution)h(is)f(either)f (p)q(erio)q(dic,)i(quasip)q(erio)q(dic,)f(or)h(c)o(haotic,)f(but)g(is) 30 2387 y(alw)o(a)o(ys)i(unstable)f(with)h(resp)q(ect)f(to)h(p)q (erturbations)h(of)f(p)q(erio)q(d)g(2)p Fk(\031)r Fn(.)34 b(W)l(e)21 b(sho)o(w)g(an)g(example)e(of)30 2459 y(suc)o(h)g(an)g (unstable)g(c)o(haotic)f(solution)h(in)g(Fig.)f(12)h(for)g(parameter)f (v)m(alues)h Fk(R)f 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Fh(1)1345 860 y Fn(=)1397 775 y Ff(")1447 826 y Fn(1)80 b(0)1447 898 y(0)42 b Fi(\000)p Fn(1)1597 775 y Ff(#)1634 860 y Fk(:)142 b Fn(\(3.3\))30 1011 y(F)l(or)16 b(all)g(of)h(the)f(one{dimensional)f(irreducible)f(represen)o(tations,) 829 1133 y Fk(r)851 1140 y Fj(!)890 1133 y Fn(=)g Fk(\032)967 1112 y Fh(2)1000 1133 y Fn(=)g Fk(I)t(;)30 1255 y Fn(so)22 b(these)f(isot)o(ypic)f(comp)q(onen)o(ts)g(are)i(not)f(of)h(use)f(for)g (studying)h(p)q(erio)q(d{tripling)f(instabilities.)30 1327 y(Ho)o(w)o(ev)o(er,)13 b Fk(r)261 1334 y Fj(!)301 1327 y Fn(do)q(es)j(not)g(act)f(trivially)e(for)j(either)e(of)h(the)g (t)o(w)o(o{dimensional)f(represen)o(tations)g(and)30 1399 y(so)h(these)g(are)g(relev)m(an)o(t.)k(W)l(e)c(note)g(that)g(for)g (the)f(underlying)g(solution)h(to)g(b)q(e)g(stable)g(with)f(resp)q(ect) 30 1471 y(to)19 b(p)q(erio)q(d-increasing)g(p)q(erturbations,)g(the)f (dominan)o(t)g(Ly)o(apuno)o(v)g(exp)q(onen)o(ts)h(asso)q(ciated)g(with) 30 1544 y Fl(b)n(oth)d 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Fe(D)668 2331 y Fj(n)692 2324 y Fn(.)49 2438 y(\(iii\))23 b(If)17 b(the)h(solution)h(is)e(ev)o(en)g(ab)q(out)j(the)d(origin)h (and)h(o)q(dd)g(ab)q(out)g Fk(\031)r(=)p Fn(\(2)p Fk(n)p Fn(\),)g(then)e(the)h(group)h(is)152 2510 y Fe(D)195 2517 y Fh(2)p Fj(n)236 2510 y Fn(.)30 2624 y(Since)e(all)f(the)i (dihedral)f(and)h(cyclic)d(groups)k(only)e(ha)o(v)o(e)f(one)i(and)g(t)o (w)o(o)f(dimensional)f(irreducible)30 2696 y(represen)o(tations,)f(the) h(metho)q(ds)g(used)h(here)e(are)i(easily)e(extended)g(to)i(higher)f(v) m(alues)g(of)h Fk(n)p Fn(.)936 2893 y(10)p eop %%Page: 9 11 9 10 bop 30 28 a Fn(where)17 b Fk(W)218 35 y Fh(1)252 28 y Fn(=)e(Fix\()p Fe(Z)430 35 y Fh(3)450 28 y Fn(\).)23 b(Since)16 b(the)h(t)o(w)o(o-dimensional)e(irreducible)g(represen)o (tation)h(is)h(not)h(abso-)30 101 y(lutely)h(irreducible,)f(there)h(is) h(no)g(further)g(decomp)q(osition)f(of)h(the)g(linear)f(op)q(erator)i Fk(g)1684 108 y Fj(A)1713 101 y Fn(\()p Fk(A)p Fn(\))e(in)o(to)30 173 y(t)o(w)o(o)f(diagonal)h(blo)q(c)o(ks,)f(as)h(o)q(ccurred)f(in)g (the)g(previous)g(section)f(with)h(the)g(group)i Fe(D)1636 180 y Fh(4)1655 173 y Fn(.)27 b(Ho)o(w)o(ev)o(er,)30 245 y(it)16 b(do)q(es)h(ha)o(v)o(e)e(a)i(complex)d(structure)i([15])g (in)g(that)688 396 y Fk(g)711 403 y Fj(A)740 396 y Fn(\()p Fk(A)p Fn(\))p Fi(j)829 403 y Fj(W)862 408 y Fa(2)895 396 y Fn(=)947 311 y Ff(")1000 362 y Fk(C)45 b Fi(\000)p Fk(D)996 434 y(D)86 b(C)1181 311 y Ff(#)1218 396 y Fk(;)30 550 y Fn(for)19 b(some)f(matrices)e Fk(C)22 b Fn(and)e Fk(D)q Fn(.)29 b(This)19 b(implies)d(that)j(if)f(there)g(is)g(a)h (solution)g Fk(\036)f Fn(=)g([)p Fk(u;)8 b(v)r Fn(])1720 532 y Fj(T)1764 550 y Fn(of)19 b(the)30 623 y(v)m(ariational)e (equation)803 729 y(_)791 742 y Fk(\036)d Fn(=)f Fk(g)908 749 y Fj(A)937 742 y Fn(\()p Fk(A)p Fn(\))p Fi(j)1026 749 y Fj(W)1059 754 y Fa(2)1078 742 y Fk(\036;)669 b Fn(\(3.1\))30 861 y(then)21 b(there)f(is)g(also)h(another)g(distinct)f (solution)h(of)g(\(3.1\))g(giv)o(en)f(b)o(y)g Fk(\036)h Fn(=)h([)p Fi(\000)p Fk(v)r(;)8 b(u)p Fn(])1621 843 y Fj(T)1646 861 y Fn(.)34 b(Th)o(us)21 b(the)30 934 y(Ly)o(apuno)o(v)c (exp)q(onen)o(ts)f(are)g(again)h(of)g(m)o(ultipli)o(cit)n(y)c(t)o(w)o (o)j(in)g(this)g(case.)103 1006 y(The)j(solution)h(with)f(spatial)g(p)q (erio)q(d)h(2)p Fk(\031)r(=)p Fn(3)f(will)f(b)q(e)i(stable)f(to)g(p)q (erturbations)h(of)g(p)q(erio)q(d)f(2)p Fk(\031)30 1078 y Fn(if)c(the)g(\(m)o(ultiple)o(\))e(dominan)o(t)h(Ly)o(apuno)o(v)h (exp)q(onen)o(ts)h(asso)q(ciated)g(with)f(the)g(isot)o(ypic)f(comp)q (onen)o(t)30 1150 y Fk(W)76 1157 y Fh(2)112 1150 y Fn(are)i(negativ)o (e.)103 1222 y(W)l(e)f(no)o(w)g(consider)g(solutions)g(whic)o(h)f(ha)o (v)o(e)g(some)g(re\015ectional)g(symmetri)o(es)e(and)j(ha)o(v)o(e)g(p)q (erio)q(d)30 1295 y(2)p Fk(\031)r(=)p Fn(3.)20 b(If)12 b(solutions)g(are)g(also)h(ev)o(en)d(ab)q(out)k(the)d(origin,)i(then)e (the)h(solutions)g(ha)o(v)o(e)g(symme)o(try)d(group)30 1367 y(whic)o(h)15 b(w)o(e)f(call)h(\006)363 1374 y Fh(3)398 1367 y Fn(generated)g(b)o(y)g Fk(r)708 1374 y Fj(!)748 1367 y Fn(and)h Fk(s)865 1374 y Fh(1)900 1367 y Fn(and)g(so)g(is)f (isomorphic)e(to)j(the)f(dihedral)g(group)h Fe(D)1857 1374 y Fh(3)1876 1367 y Fn(.)30 1439 y(This)23 b(group)h(has)f(t)o(w)o (o)g(one{dimensional)f(irreducible)e(represen)o(tations)j Fk(r)1472 1446 y Fj(!)1522 1439 y Fn(=)i Fk(I)t(;)8 b(s)1656 1446 y Fh(1)1700 1439 y Fn(=)25 b Fk(I)h Fn(and)30 1511 y Fk(r)52 1518 y Fj(!)91 1511 y Fn(=)14 b Fk(I)t(;)8 b(s)214 1518 y Fh(1)247 1511 y Fn(=)13 b Fi(\000)p Fk(I)t Fn(,)i(and)i(one)g(t)o(w)o(o{dimensional)d(represen)o(tation)189 1660 y Fk(r)211 1667 y Fj(!)250 1660 y Fn(=)302 1574 y Ff( )362 1625 y Fn(cos)8 b Fk(!)44 b Fi(\000)8 b Fn(sin)g Fk(!)367 1698 y Fn(sin)g Fk(!)86 b Fn(cos)8 b Fk(!)677 1574 y Ff(!)731 1660 y Fn(=)787 1626 y(1)p 787 1648 25 2 v 787 1694 a(2)825 1574 y Ff( )888 1625 y Fi(\000)p Fn(1)42 b Fi(\000)1032 1584 y(p)p 1073 1584 V 1073 1625 a Fn(3)885 1656 y Fi(p)p 927 1656 V 42 x Fn(3)83 b Fi(\000)p Fn(1)1118 1574 y Ff(!)1215 1660 y Fn(and)17 b Fk(s)1333 1667 y Fh(1)1367 1660 y Fn(=)1418 1574 y Ff( )1479 1625 y Fn(1)80 b(0)1479 1698 y(0)41 b Fi(\000)p Fn(1)1628 1574 y Ff(!)1676 1660 y Fk(:)30 1808 y Fn(In)18 b(a)g(similar)d(w)o(a)o (y)j(to)g(the)g Fe(D)587 1815 y Fh(4)624 1808 y Fn(case)g(ab)q(o)o(v)o (e,)g Fk(r)905 1815 y Fj(!)948 1808 y Fn(acts)g(as)g(the)g(iden)o(tit)o (y)e(for)i(the)f(one{dimensional)30 1880 y(irreducible)h(represen)o (tations)h(and)i(so)f(p)q(erturbations)h(in)e(the)h(corresp)q(onding)h (isot)o(ypic)d(comp)q(o-)30 1952 y(nen)o(ts)e(ha)o(v)o(e)f(the)h(same)f (p)q(erio)q(d)i(as)g(the)f(solution.)21 b(Th)o(us,)c(only)f(the)f(t)o (w)o(o-dimensional)g(irreducible)30 2024 y(represen)o(tation)22 b(is)f(of)i(in)o(terest)e(and)h(since)g(it)f(is)h(also)h(absolutely)f (irreducible,)f(the)h(Ly)o(apuno)o(v)30 2097 y(exp)q(onen)o(ts)c(asso)q (ciated)g(with)g(the)f(corresp)q(onding)h(isot)o(ypic)f(comp)q(onen)o (t)g(will)f(ha)o(v)o(e)h(m)o(ultipli)o(cit)n(y)30 2169 y(t)o(w)o(o.)k(Again)16 b(n)o(umerically)d(it)j(is)g(su\016cien)o(t)f (to)h(consider)g(only)g(p)q(erturbations)i(for)e(whic)o(h)g Fk(s)1748 2176 y Fh(1)1781 2169 y Fn(=)e Fk(I)t Fn(.)103 2241 y(The)h(theory)g(is)f(again)i(similar)d(for)i(solutions)g(whic)o (h)f(ha)o(v)o(e)g(spatial)h(p)q(erio)q(d)g(2)p Fk(\031)r(=)p Fn(3)h(and)f(are)g(o)q(dd)30 2313 y(ab)q(out)j(the)e(origin.)103 2386 y(Finally)l(,)h(w)o(e)h(consider)h(solutions)g(whic)o(h)e(are)i (ev)o(en)e(ab)q(out)j(the)e(origin,)h(o)q(dd)g(ab)q(out)h Fk(\031)r(=)p Fn(6)e(and)30 2458 y(ha)o(v)o(e)d(p)q(erio)q(d)i(2)p Fk(\031)r(=)p Fn(3.)22 b(It)16 b(is)g(helpful)f(to)i(de\014ne)481 2577 y Fk(\014)s(A)p Fn(\()p Fk(x;)8 b(t)p Fn(\))k(:=)h Fk(s)755 2584 y Fh(2)775 2577 y Fk(s)798 2584 y Fh(1)818 2577 y Fk(r)840 2585 y Fj(\031)q(=)p Fh(3)898 2577 y Fk(A)p Fn(\()p Fk(x;)8 b(t)p Fn(\))13 b(=)h Fi(\000)p Fk(A)p Fn(\()p Fk(\031)r(=)p Fn(3)c Fi(\000)h Fk(x;)d(t)p Fn(\))p Fk(;)30 2696 y Fn(since)13 b(functions)g(\014xed)h(b)o(y)f Fk(\014)j Fn(are)d(o)q(dd)i(ab)q(out)g Fk(\031)r(=)p Fn(6.)20 b(The)14 b(symmetry)c(group)15 b(\006)1494 2703 y Fh(4)1527 2696 y Fn(of)f(these)f(solutions)30 2769 y(th)o(us)24 b(includes)e Fk(s)362 2776 y Fh(1)405 2769 y Fn(\(ev)o(en)h(ab)q(out)h(the)g(origin\),)g Fk(\014)i Fn(\(o)q(dd)f(ab)q(out)f Fk(\031)r(=)p Fn(6\))g(and)g Fk(r)1528 2776 y Fj(!)1577 2769 y Fn(\(p)q(erio)q(d)g(2)p Fk(\031)r(=)p Fn(3\).)948 2893 y(9)p eop %%Page: 8 12 8 11 bop 30 28 a Fn(represen)o(t)15 b(di\013eren)o(t)f(com)o(binations) h(of)h(re\015ectional)e(symmetrie)o(s)f(b)q(eing)j(brok)o(en)f(whic)o (h)g(preserv)o(e)30 101 y(the)k(p)q(erio)q(d,)i(whic)o(h)d(w)o(e)h (considered)g(in)h([5)o(].)31 b(There)19 b(is)g(also)h(one)g(t)o(w)o (o{dimensional)e(irreducible)30 173 y(represen)o(tation)e(of)g Fe(D)447 180 y Fh(4)483 173 y Fn(giv)o(en)f(b)o(y)253 314 y Fk(R)f Fn(=)356 229 y Ff(")405 280 y Fn(cos)9 b Fk(\031)r(=)p Fn(2)42 b Fi(\000)8 b Fn(sin)g Fk(\031)r(=)p Fn(2)411 352 y(sin)g Fk(\031)r(=)p Fn(2)83 b(cos)9 b Fk(\031)r(=)p Fn(2)813 229 y Ff(#)856 314 y Fn(=)908 229 y Ff(")958 280 y Fn(0)41 b Fi(\000)p Fn(1)958 352 y(1)80 b(0)1107 229 y Ff(#)1194 314 y Fn(and)59 b Fk(s)1354 321 y Fh(1)1387 314 y Fn(=)1439 229 y Ff(")1489 280 y Fn(1)80 b(0)1489 352 y(0)42 b Fi(\000)p Fn(1)1638 229 y Ff(#)30 455 y Fn(F)l(or)16 b(this)g(represen)o(tation)709 596 y Fk(r)731 603 y Fj(\031)769 596 y Fn(=)d Fk(R)857 575 y Fh(2)892 596 y Fn(=)943 510 y Ff(")993 561 y Fi(\000)p Fn(1)80 b(0)1032 634 y(0)42 b Fi(\000)p Fn(1)1182 510 y Ff(#)30 736 y Fn(and)18 b(so)g(all)f(p)q(erturbations)h(in)f(the)g (corresp)q(onding)i(isot)o(ypic)d(comp)q(onen)o(t)g(ha)o(v)o(e)h(minim) o(al)e(p)q(erio)q(d)30 809 y(2)p Fk(\031)j Fn(and)f(so)g(are)f (spatially)g(p)q(erio)q(d)h(doubling.)103 881 y(W)l(e)e(note)h(that)g (since)e(this)h(t)o(w)o(o-dimensional)f(irreducible)f(represen)o (tation)i(is)g(also)h(absolutely)30 953 y(irreducible,)23 b(the)g(linear)g(op)q(erator)i Fk(g)755 960 y Fj(A)784 953 y Fn(\()p Fk(A)p Fn(\))e(can)h(b)q(e)g(decomp)q(osed)f(further)g (in)o(to)g(t)o(w)o(o)h(iden)o(tical)30 1025 y(blo)q(c)o(ks)14 b(asso)q(ciated)i(with)e(the)h(spaces)g(on)g(whic)o(h)f Fk(s)974 1032 y Fh(1)1008 1025 y Fn(acts)h(as)g Fk(I)j Fn(or)d Fi(\000)p Fk(I)t Fn(.)20 b(This)15 b(results)f(in)h(Ly)o(apuno) o(v)30 1098 y(exp)q(onen)o(ts)j(of)h(m)o(ultipli)o(cit)n(y)c(t)o(w)o (o.)28 b(Moreo)o(v)o(er,)17 b(n)o(umericall)o(y)e(it)j(is)g(su\016cien) o(t)f(to)i(w)o(ork)f(with)g(only)30 1170 y(one)h(of)g(these)f(t)o(w)o (o)g(iden)o(tical)f(blo)q(c)o(ks)h(in)g(order)h(to)g(\014nd)g(just)g (one)f(of)h(the)g(Ly)o(apuno)o(v)f(exp)q(onen)o(ts.)30 1242 y(See)f(Aston)g(and)h(Dellnitz)e([4])g(for)i(more)e(details.)24 b(By)16 b(c)o(ho)q(osing)i(the)f(blo)q(c)o(k)g(asso)q(ciated)h(with)f (the)30 1314 y(space)e(on)g(whic)o(h)f Fk(s)386 1321 y Fh(1)420 1314 y Fn(=)f Fk(I)t Fn(,)i(w)o(e)f(ha)o(v)o(e)g(a)h(F)l (ourier)f(decomp)q(osition)g(of)h(the)g(p)q(erturbation)g(in)f(this)h (case)30 1386 y(giv)o(en)g(b)o(y)345 1482 y Fk(A)p Fn(\()p Fk(x;)8 b(t)p Fn(\))k(=)570 1420 y Fb(1)552 1434 y Ff(X)556 1541 y Fj(k)q Fh(=1)632 1482 y Fk(b)653 1489 y Fj(k)674 1482 y Fn(\()p Fk(t)p Fn(\))c(cos)h(\(2)p Fk(k)k Fi(\000)e Fn(1\))p Fk(x)g Fn(+)g Fk(i)1118 1420 y Fb(1)1099 1434 y Ff(X)1103 1541 y Fj(k)q Fh(=1)1180 1482 y Fk(c)1201 1489 y Fj(k)1222 1482 y Fn(\()p Fk(t)p Fn(\))d(cos)g(\(2)p Fk(k)14 b Fi(\000)d Fn(1\))p Fk(x:)30 1607 y Fn(The)k(alternativ)o(e)f (c)o(hoice)g(of)i(p)q(erturbation)g(whic)o(h)f(giv)o(es)f(the)i(second) f(iden)o(tical)f(Ly)o(apuno)o(v)h(exp)q(o-)30 1679 y(nen)o(t)h (consists)g(of)h(replacing)f(the)g(cosines)g(with)g(sines.)103 1751 y(Stabilit)o(y)22 b(of)i(the)f(c)o(haotic)f(solution)i(with)f (resp)q(ect)g(to)h(p)q(erio)q(d-doubling)g(p)q(erturbations)g(is)30 1824 y(determined)16 b(b)o(y)i(the)g(sign)h(of)g(the)f(dominan)o(t)g (Ly)o(apuno)o(v)h(exp)q(onen)o(t)f(asso)q(ciated)i(with)e(this)h(t)o (yp)q(e)30 1896 y(of)e(p)q(erturbation.)30 2054 y Fd(3.2)66 b(Spatial)24 b(p)r(erio)r(d)g(tripling)h(\()p Fc(n)17 b Fv(=)f(3)p Fd(\))30 2158 y Fn(When)h Fk(n)d Fn(=)g(3,)j(solutions)g (ha)o(v)o(e)e(spatial)i(p)q(erio)q(d)g(2)p Fk(\031)r(=)p Fn(3)g(and)g(so)h(are)e(\014xed)g(b)o(y)g(the)h(action)f(of)h Fk(r)1800 2166 y Fh(2)p Fj(\031)q(=)p Fh(3)1876 2158 y Fn(.)30 2230 y(F)l(or)j(ease)f(of)h(notation,)g(w)o(e)f(de\014ne)g Fk(!)j Fn(=)d(2)p Fk(\031)r(=)p Fn(3.)31 b(If)19 b(the)g(solutions)h (ha)o(v)o(e)f(no)h(other)f(symmetrie)o(s,)30 2303 y(then)g(they)g(are)g (con)o(tained)g(in)g(Fix)o(\()p Fe(Z)749 2310 y Fh(3)769 2303 y Fn(\).)30 b(There)19 b(are)g(only)g(t)o(w)o(o)h(irreducible)d (represen)o(tations)i(of)30 2375 y Fe(Z)64 2382 y Fh(3)100 2375 y Fn(whic)o(h)d(are)g(giv)o(en)f(b)o(y)583 2516 y Fk(r)605 2523 y Fj(!)644 2516 y Fn(=)f Fk(I)t(;)72 b(r)830 2523 y Fj(!)869 2516 y Fn(=)921 2431 y Ff(")971 2482 y Fn(cos)8 b Fk(!)44 b Fi(\000)8 b Fn(sin)g Fk(!)976 2554 y Fn(sin)g Fk(!)86 b Fn(cos)8 b Fk(!)1286 2431 y Ff(#)1323 2516 y Fk(:)30 2657 y Fn(The)16 b(corresp)q(onding)i(isot)o (ypic)d(decomp)q(osition)g(is)802 2769 y Fk(X)j Fn(=)c Fk(W)958 2776 y Fh(1)989 2769 y Fi(\010)d Fk(W)1085 2776 y Fh(2)1104 2769 y Fk(;)948 2893 y Fn(8)p eop %%Page: 7 13 7 12 bop 30 28 a Fn(isot)o(ypic)15 b(comp)q(onen)o(t.)20 b(Th)o(us,)60 173 y Fk(A)p Fn(\()p Fk(x;)8 b(t)p Fn(\))13 b Fi(2)h Fk(W)309 180 y Fh(1)370 173 y Fn(=)-8 b Fi(\))41 b Fk(A)p Fn(\()p Fk(x;)8 b(t)p Fn(\))k(=)703 139 y Fk(b)724 146 y Fh(0)744 139 y Fn(\()p Fk(t)p Fn(\))p 703 161 97 2 v 739 207 a(2)815 173 y(+)882 110 y Fb(1)864 125 y Ff(X)868 231 y Fj(k)q Fh(=1)944 173 y Fk(b)965 180 y Fj(k)986 173 y Fn(\()p Fk(t)p Fn(\))c(cos)h(2)p Fk(k)r(x)i Fn(+)g Fk(i)1288 87 y Ff(")1322 139 y Fk(c)1343 146 y Fh(0)1363 139 y Fn(\()p Fk(t)p Fn(\))p 1322 161 V 1358 207 a(2)1434 173 y(+)1502 110 y Fb(1)1483 125 y Ff(X)1487 231 y Fj(k)q Fh(=1)1564 173 y Fk(c)1585 180 y Fj(k)1606 173 y Fn(\()p Fk(t)p Fn(\))d(cos)g(2)p Fk(k)r(x)1823 87 y Ff(#)60 334 y Fk(A)p Fn(\()p Fk(x;)g(t)p Fn(\))13 b Fi(2)h Fk(W)309 341 y Fh(2)370 334 y Fn(=)-8 b Fi(\))41 b Fk(A)p Fn(\()p Fk(x;)8 b(t)p Fn(\))k(=)716 272 y Fb(1)698 287 y Ff(X)702 393 y Fj(k)q Fh(=1)778 334 y Fk(b)799 341 y Fj(k)820 334 y Fn(\()p Fk(t)p Fn(\))c(sin)g(2)p Fk(k)r(x)k Fn(+)f Fk(i)1135 272 y Fb(1)1117 287 y Ff(X)1120 393 y Fj(k)q Fh(=1)1197 334 y Fk(c)1218 341 y Fj(k)1239 334 y Fn(\()p Fk(t)p Fn(\))d(sin)g(2)p Fk(k)r(x)60 496 y(A)p Fn(\()p Fk(x;)g(t)p Fn(\))13 b Fi(2)h Fk(W)309 503 y Fh(3)370 496 y Fn(=)-8 b Fi(\))41 b Fk(A)p Fn(\()p Fk(x;)8 b(t)p Fn(\))k(=)716 434 y Fb(1)698 449 y Ff(X)702 555 y Fj(k)q Fh(=1)778 496 y Fk(b)799 503 y Fj(k)820 496 y Fn(\()p Fk(t)p Fn(\))c(cos)h(\(2)p Fk(k)k Fi(\000)e Fn(1\))p Fk(x)g Fn(+)g Fk(i)1264 434 y Fb(1)1245 449 y Ff(X)1249 555 y Fj(k)q Fh(=1)1326 496 y Fk(c)1347 503 y Fj(k)1368 496 y Fn(\()p Fk(t)p Fn(\))d(cos)g(\(2)p Fk(k)14 b Fi(\000)d Fn(1\))p Fk(x)60 658 y(A)p Fn(\()p Fk(x;)d(t)p Fn(\))13 b Fi(2)h Fk(W)309 665 y Fh(4)370 658 y Fn(=)-8 b Fi(\))41 b Fk(A)p Fn(\()p Fk(x;)8 b(t)p Fn(\))k(=)716 596 y Fb(1)698 611 y Ff(X)702 717 y Fj(k)q Fh(=1)778 658 y Fk(b)799 665 y Fj(k)820 658 y Fn(\()p Fk(t)p Fn(\))c(sin)g(\(2)p Fk(k)14 b Fi(\000)d Fn(1\))p Fk(x)g Fn(+)g Fk(i)1258 596 y Fb(1)1240 611 y Ff(X)1244 717 y Fj(k)q Fh(=1)1320 658 y Fk(c)1341 665 y Fj(k)1362 658 y Fn(\()p Fk(t)p Fn(\))d(sin)g(\(2)p Fk(k)14 b Fi(\000)d Fn(1\))p Fk(x)103 806 y Fn(W)l(e)k(in)o(v)o(estigate)e(solutions)j(in)e (Fix\(\006)811 813 y Fh(1)830 806 y Fn(\),)h(and)h(since)e(w)o(e)g(are) h(in)o(terested)f(in)g(p)q(erio)q(d{increasing)30 879 y(instabilities)20 b(w)o(e)h(only)g(calculate)f(the)h(dominan)o(t)g(Ly) o(apuno)o(v)g(exp)q(onen)o(ts)h(asso)q(ciated)g(with)f Fk(W)1870 886 y Fh(3)30 951 y Fn(and)16 b Fk(W)170 958 y Fh(4)206 951 y Fn(\(on)g(whic)o(h)f Fk(r)453 958 y Fj(\031)492 951 y Fn(acts)h(as)h Fi(\000)p Fk(I)t Fn(\))d(and)j(not)f (that)g(asso)q(ciated)h(with)e Fk(W)1424 958 y Fh(2)1460 951 y Fn(\(on)h(whic)o(h)f Fk(r)1707 958 y Fj(\031)1746 951 y Fn(acts)h(as)30 1023 y(the)g(iden)o(tit)o(y\).)103 1095 y(It)h(is)g(also)h(p)q(ossible)g(to)g(consider)f(solutions)h(with) f(p)q(erio)q(d)h Fk(\031)h Fn(whic)o(h)e(are)g(o)q(dd)i(functions)e(of) h Fk(x)p Fn(.)30 1168 y(Ho)o(w)o(ev)o(er,)c(this)i(is)f(v)o(ery)g (similar)f(to)i(the)g(previous)g(case)g(in)g(that)g(the)g(symmetry)c (of)17 b(the)e(solutions)30 1240 y(is)h(again)h(isomorphic)e(to)i Fe(Z)550 1247 y Fh(2)581 1240 y Fi(\002)10 b Fe(Z)664 1247 y Fh(2)701 1240 y Fn(and)16 b(so)h(w)o(e)f(do)h(not)g(consider)f (this)g(case)g(in)g(detail.)103 1312 y(The)d(other)g(com)o(bination)e (of)i(re\015ectional)e(symmetrie)o(s,)g(whic)o(h)g(w)o(e)i(considered)f (in)g([5],)g(is)h(when)30 1384 y(solutions)19 b(are)g(ev)o(en)e(ab)q (out)j(one)e(p)q(oin)o(t)h(and)g(o)q(dd)g(ab)q(out)h(another.)29 b(In)18 b(particular,)g(w)o(e)g(consider)30 1457 y(solutions)d(with)g (a)g(spatial)g(p)q(erio)q(d)h(of)f Fk(\031)h Fn(whic)o(h)f(are)f(ev)o (en)g(ab)q(out)i(the)f(origin)g(and)g(o)q(dd)h(ab)q(out)g Fk(\031)r(=)p Fn(4.)30 1529 y(T)l(o)h(help)e(in)h(the)g(follo)o(wing)g (discussion,)g(w)o(e)g(de\014ne)483 1644 y Fk(\021)r(A)p Fn(\()p Fk(x;)8 b(t)p Fn(\))13 b(:=)g Fk(s)753 1651 y Fh(2)773 1644 y Fk(s)796 1651 y Fh(1)815 1644 y Fk(r)837 1652 y Fj(\031)q(=)p Fh(2)896 1644 y Fk(A)p Fn(\()p Fk(x;)8 b(t)p Fn(\))13 b(=)h Fi(\000)p Fk(A)p Fn(\()p Fk(\031)r(=)p Fn(2)c Fi(\000)h Fk(x;)d(t)p Fn(\))p Fk(:)30 1759 y Fn(If)15 b Fk(A)p Fn(\()p Fk(x;)8 b(t)p Fn(\))14 b(is)h(\014xed)g(b)o(y)f Fk(\021)j Fn(then)f(it)e(is)h(o)q(dd)i(ab)q(out)f Fk(\031)r(=)p Fn(4.)21 b(Th)o(us,)15 b(the)g(group)i(of)e(symmetrie)o(s)e(of)i(these) 30 1832 y(solutions)i(includes)e Fk(s)444 1839 y Fh(1)480 1832 y Fn(\(ev)o(en)g(ab)q(out)j(the)e(origin\),)f Fk(\021)j Fn(\(o)q(dd)g(ab)q(out)f Fk(\031)r(=)p Fn(4\))g(and)f Fk(r)1531 1839 y Fj(\031)1571 1832 y Fn(\(p)q(erio)q(d)h Fk(\031)r Fn(\).)103 1904 y(De\014ning)f Fk(R)e Fi(\021)g Fk(\021)r(s)450 1911 y Fh(1)484 1904 y Fn(w)o(e)h(see)g(that)h(this)f (group,)h(whic)o(h)e(w)o(e)h(call)g(\006)1316 1911 y Fh(2)1336 1904 y Fn(,)g(is)g(generated)g(b)o(y)g Fk(R)g Fn(and)h Fk(s)1870 1911 y Fh(1)30 1976 y Fn(whic)o(h)g(satisfy)621 2091 y Fk(R)658 2071 y Fh(4)692 2091 y Fn(=)d Fk(I)t(;)41 b(s)847 2073 y Fh(2)847 2104 y(1)880 2091 y Fn(=)14 b Fk(I)t(;)41 b(s)1036 2098 y Fh(1)1055 2091 y Fk(R)15 b Fn(=)f Fk(R)1196 2071 y Fb(\000)p Fh(1)1243 2091 y Fk(s)1266 2098 y Fh(1)1286 2091 y Fk(;)30 2206 y Fn(and)23 b(so)h(is)e(isomorphic)f(to)i Fe(D)615 2213 y Fh(4)635 2206 y Fn(.)41 b(W)l(e)22 b(note)h(that)g Fk(\021)k Fn(=)d Fk(Rs)1180 2213 y Fh(1)1223 2206 y Fn(and)g Fk(r)1347 2213 y Fj(\031)1395 2206 y Fn(=)h Fk(R)1495 2188 y Fh(2)1515 2206 y Fn(.)41 b(There)22 b(are)h(four)30 2279 y(one{dimensional)15 b(irreducible)f(represen)o(tations)i(of)h Fe(D)1074 2286 y Fh(4)1110 2279 y Fn(giv)o(en)e(b)o(y)758 2379 y Fk(R)g Fn(=)f Fk(I)92 b(s)999 2386 y Fh(1)1032 2379 y Fn(=)14 b Fk(I)758 2451 y(R)h Fn(=)f Fi(\000)p Fk(I)53 b(s)999 2458 y Fh(1)1032 2451 y Fn(=)14 b Fk(I)758 2523 y(R)h Fn(=)f Fk(I)92 b(s)999 2530 y Fh(1)1032 2523 y Fn(=)14 b Fi(\000)p Fk(I)758 2595 y(R)h Fn(=)f Fi(\000)p Fk(I)53 b(s)999 2602 y Fh(1)1032 2595 y Fn(=)14 b Fi(\000)p Fk(I)t(;)30 2696 y Fn(but)21 b(since)f Fk(r)270 2703 y Fj(\031)314 2696 y Fn(==)i Fk(R)449 2678 y Fh(2)490 2696 y Fn(=)f Fk(I)j Fn(for)d(all)f(of)h(these,)g(p)q(erturbations)h(in)e(the)h (corresp)q(onding)g(isot)o(ypic)30 2769 y(comp)q(onen)o(ts)i(are)h(not) g(useful)f(when)h(considering)g(spatial)g(p)q(erio)q(d)g(doubling.)44 b(Indeed,)24 b(these)948 2893 y(7)p eop %%Page: 6 14 6 13 bop 30 28 a Fn(The)16 b(F)l(ourier)g(decomp)q(ositions)g(of)g (solutions)h(in)f Fk(W)1002 35 y Fh(1)1038 28 y Fn(and)h Fk(W)1179 35 y Fh(2)1215 28 y Fn(are)136 174 y Fk(A)p Fn(\()p Fk(x;)8 b(t)p Fn(\))13 b Fi(2)h Fk(W)385 181 y Fh(1)418 174 y Fn(=)-8 b Fi(\))14 b Fk(A)p Fn(\()p Fk(x;)8 b(t)p Fn(\))40 b(=)779 140 y Fk(b)800 147 y Fh(0)820 140 y Fn(\()p Fk(t)p Fn(\))p 779 162 97 2 v 815 208 a(2)891 174 y(+)959 111 y Fb(1)940 126 y Ff(X)944 232 y Fj(k)q Fh(=1)1012 174 y Fi(f)p Fk(b)1058 181 y Fj(k)1079 174 y Fn(\()p Fk(t)p Fn(\))8 b(cos)h(2)p Fk(k)r(x)i Fn(+)g Fk(c)1377 181 y Fj(k)1398 174 y Fn(\()p Fk(t)p Fn(\))d(sin)g(2)p Fk(k)r(x)p Fi(g)774 341 y Fn(+)p Fk(i)837 256 y Ff(")871 308 y Fk(d)896 315 y Fh(0)916 308 y Fn(\()p Fk(t)p Fn(\))p 871 330 101 2 v 909 375 a(2)988 341 y(+)1055 279 y Fb(1)1037 294 y Ff(X)1041 400 y Fj(k)q Fh(=1)1109 341 y Fi(f)p Fk(d)1159 348 y Fj(k)1180 341 y Fn(\()p Fk(t)p Fn(\))g(cos)h(2)p Fk(k)r(x)i Fn(+)g Fk(e)1480 348 y Fj(k)1501 341 y Fn(\()p Fk(t)p Fn(\))d(sin)g(2)p Fk(k)r(x)p Fi(g)1737 256 y Ff(#)136 503 y Fk(A)p Fn(\()p Fk(x;)g(t)p Fn(\))13 b Fi(2)h Fk(W)385 510 y Fh(2)418 503 y Fn(=)-8 b Fi(\))14 b Fk(A)p Fn(\()p Fk(x;)8 b(t)p Fn(\))40 b(=)793 441 y Fb(1)774 456 y Ff(X)778 562 y Fj(k)q Fh(=1)846 503 y Fi(f)p Fk(b)892 510 y Fj(k)913 503 y Fn(\()p Fk(t)p Fn(\))8 b(cos)h(\(2)p Fk(k)k Fi(\000)e Fn(1\))p Fk(x)g Fn(+)g Fk(c)1334 510 y Fj(k)1355 503 y Fn(\()p Fk(t)p Fn(\))d(sin)g(\(2)p Fk(k)14 b Fi(\000)d Fn(1\))p Fk(x)p Fi(g)774 665 y Fn(+)p Fk(i)855 603 y Fb(1)837 618 y Ff(X)841 724 y Fj(k)q Fh(=1)909 665 y Fi(f)p Fk(d)959 672 y Fj(k)981 665 y Fn(\()p Fk(t)p Fn(\))d(cos)g(\(2)p Fk(k)14 b Fi(\000)c Fn(1\))p Fk(x)i Fn(+)f Fk(e)1404 672 y Fj(k)1425 665 y Fn(\()p Fk(t)p Fn(\))d(sin)g(\(2)p Fk(k)13 b Fi(\000)e Fn(1\))p Fk(x)p Fi(g)30 817 y Fn(In)18 b(this)g(case,)g(the)g(solution)g(with)g(p)q(erio)q(d)h Fk(\031)g Fn(will)e(b)q(e)i(stable)f(with)f(resp)q(ect)h(to)h(p)q (erturbations)g(of)30 890 y(p)q(erio)q(d)h(2)p Fk(\031)h Fn(if)d(the)h(dominan)o(t)g(Ly)o(apuno)o(v)g(exp)q(onen)o(t)g(asso)q (ciated)h(with)f(p)q(erturbations)h(in)f Fk(W)1818 897 y Fh(2)1857 890 y Fn(is)30 962 y(negativ)o(e)c(and)i(unstable)g (otherwise.)103 1034 y(If)12 b(w)o(e)h(ha)o(v)o(e)f(solutions)h(whic)o (h)f(ha)o(v)o(e)g(p)q(erio)q(d)i Fk(\031)g Fn(and)f(in)g(addition)g (are)g(ev)o(en)e(functions)i(of)g Fk(x)p Fn(,)g(then)30 1106 y(the)18 b(solutions)h(can)g(b)q(e)g(found)g(b)o(y)f(solving)g (for)h Fk(A)f Fn(on)h(the)f(in)o(terv)m(al)g([0)p Fk(;)8 b(\031)r(=)p Fn(2])18 b(with)g(homogeneous)30 1178 y(Neumann)d(b)q (oundary)i(conditions)g(and)f(are)h(con)o(tained)f(in)f(Fix\(\006)1283 1185 y Fh(1)1303 1178 y Fn(\))h(where)734 1300 y(\006)769 1307 y Fh(1)803 1300 y Fn(=)e Fi(f)p Fk(I)t(;)8 b(r)950 1307 y Fj(\031)973 1300 y Fk(;)g(s)1018 1307 y Fh(1)1037 1300 y Fk(;)g(r)1081 1307 y Fj(\031)1105 1300 y Fk(s)1128 1307 y Fh(1)1147 1300 y Fi(g)p Fk(;)30 1423 y Fn(whic)o(h)20 b(is)g(isomorphic)f(to)h Fe(Z)575 1430 y Fh(2)609 1423 y Fi(\002)14 b Fe(Z)696 1430 y Fh(2)736 1423 y Fn(as)21 b Fk(r)822 1430 y Fj(\031)866 1423 y Fn(and)g Fk(s)988 1430 y Fh(1)1028 1423 y Fn(are)f(b)q(oth)h(generators)h(for)e Fe(Z)1585 1430 y Fh(2)1605 1423 y Fn(.)34 b(This)20 b(group)30 1495 y(has)c(four)f(one{dimensional)e(irreducible)g(represen)o (tations,)h(corresp)q(onding)i(to)f(the)g(four)g(p)q(ossible)30 1567 y(com)o(binations)h(of)i Fk(r)407 1574 y Fj(\031)448 1567 y Fn(and)g Fk(s)567 1574 y Fh(1)604 1567 y Fn(b)q(eing)g Fi(\006)p Fk(I)t Fn(,)e(and)i(there)f(are)h(four)f(corresp)q(onding)i (isot)o(ypic)d(comp)q(o-)30 1639 y(nen)o(ts)g(giv)o(en)f(b)o(y)286 1761 y Fk(W)332 1768 y Fh(1)394 1761 y Fn(=)41 b Fi(f)p Fk(A)13 b Fi(2)h Fk(X)19 b Fn(:)13 b Fk(A)718 1768 y Fj(x)740 1761 y Fn(\(0)p Fk(;)8 b(t)p Fn(\))13 b(=)h(0)j(and)f Fk(A)1079 1768 y Fj(x)1101 1761 y Fn(\()p Fk(\031)r(=)p Fn(2)p Fk(;)8 b(t)p Fn(\))13 b(=)h(0)p Fi(g)31 b Fn(=)f(Fix)o(\(\006) 1595 1768 y Fh(1)1615 1761 y Fn(\))286 1846 y Fk(W)332 1853 y Fh(2)394 1846 y Fn(=)41 b Fi(f)p Fk(A)13 b Fi(2)h Fk(X)19 b Fn(:)13 b Fk(A)p Fn(\(0)p Fk(;)8 b(t)p Fn(\))13 b(=)h(0)j(and)g Fk(A)p Fn(\()p Fk(\031)r(=)p Fn(2)p Fk(;)8 b(t)p Fn(\))13 b(=)g(0)p Fi(g)286 1931 y Fk(W)332 1938 y Fh(3)394 1931 y Fn(=)41 b Fi(f)p Fk(A)13 b Fi(2)h Fk(X)19 b Fn(:)13 b Fk(A)718 1938 y Fj(x)740 1931 y Fn(\(0)p Fk(;)8 b(t)p Fn(\))13 b(=)h(0)j(and)f Fk(A)p Fn(\()p Fk(\031)r(=)p Fn(2)p Fk(;)8 b(t)p Fn(\))13 b(=)h(0)p Fi(g)286 2015 y Fk(W)332 2022 y Fh(4)394 2015 y Fn(=)41 b Fi(f)p Fk(A)13 b Fi(2)h Fk(X)19 b Fn(:)13 b Fk(A)p Fn(\(0)p Fk(;)8 b(t)p Fn(\))13 b(=)h(0)j(and)g Fk(A)1058 2022 y Fj(x)1079 2015 y Fn(\()p Fk(\031)r(=)p Fn(2)p Fk(;)8 b(t)p Fn(\))13 b(=)h(0)p Fi(g)p Fk(;)30 2137 y Fn(W)l(e)h(note)h(that)f(functions)h(in)f(eac)o(h)f(isot)o(ypic)h (decomp)q(osition)f(can)i(b)q(e)f(distinguished)g(b)o(y)g(di\013eren)o (t)30 2210 y(b)q(oundary)22 b(conditions.)36 b(This)22 b(observ)m(ation)g(has)g(b)q(een)f(exploited)f(n)o(umerically)d(in)k (bifurcation)30 2282 y(problems)15 b(in)h([3].)103 2354 y(By)g(expanding)g Fk(A)p Fn(\()p Fk(x;)8 b(t)p Fn(\))16 b(as)h(a)f(F)l(ourier)g(series)g(it)g(is)g(easy)h(to)f(see)g(whic)o(h)g (mo)q(des)g(o)q(ccur)g(in)h(eac)o(h)948 2893 y(6)p eop %%Page: 5 15 5 14 bop 30 28 a Fn(corresp)q(onding)19 b(to,)g(resp)q(ectiv)o(ely)l(,) d(a)i(rotation)h(of)g(the)f(complex)e(amplitude,)g(space)j (translation,)30 101 y(time)f(translation)j(and)h(a)e(spatial)h (re\015ection.)33 b(W)l(e)20 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Fn(\).)21 b(The)16 b(corresp)q(onding)i(isot)o(ypic)d(decomp)q (osition)g(is)h(simply)809 2048 y Fk(X)i Fn(=)c Fk(W)965 2055 y Fh(1)996 2048 y Fi(\010)c Fk(W)1091 2055 y Fh(2)30 2170 y Fn(where)528 2292 y Fk(W)574 2299 y Fh(1)635 2292 y Fn(=)41 b Fi(f)p Fk(A)14 b Fi(2)g Fk(X)k Fn(:)13 b Fk(r)944 2299 y Fj(\031)968 2292 y Fk(A)g Fn(=)h Fk(A)p Fi(g)29 b Fn(=)h(Fix\()p Fe(Z)1354 2299 y Fh(2)1374 2292 y Fn(\))528 2377 y Fk(W)574 2384 y Fh(2)635 2377 y Fn(=)41 b Fi(f)p Fk(A)14 b Fi(2)g Fk(X)k Fn(:)13 b Fk(r)944 2384 y Fj(\031)968 2377 y Fk(A)g Fn(=)h Fi(\000)p Fk(A)p Fi(g)p Fk(:)948 2893 y Fn(5)p eop %%Page: 4 16 4 15 bop 30 28 a Fn(that)17 b Fk(g)h Fn(satis\014es)f(the)f(equiv)m (ariance)f(condition)625 151 y Fk(\015)s(g)r Fn(\()p Fk(A)p Fn(\))f(=)g Fk(g)r Fn(\()p Fk(\015)s(A)p Fn(\))64 b(for)17 b(all)e Fk(\015)i Fi(2)d Fn(\000)p Fk(;)504 b Fn(\(2.2\))30 273 y(where)19 b(\000)h(is)g(a)g(compact)e(Lie)h (group.)33 b(F)l(or)19 b(an)o(y)h(subgroup)h(\006)f(of)f(\000,)i(w)o(e) 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b(w)o(e)g(can)h(asso)q(ciate)h(Ly)o(apuno)o(v)f(exp)q(onen)o(ts)f(with) h(a)g(particular)f(isot)o(ypic)f(comp)q(onen)o(t.)30 1711 y(There)i(are)g(t)o(w)o(o)g(imp)q(ortan)o(t)g(consequences)f(of)i (this)f(decomp)q(osition)f(whic)o(h)h(are)g(as)h(follo)o(ws:)90 1825 y(1.)24 b(the)13 b(Ly)o(apuno)o(v)h(exp)q(onen)o(ts)f(can)g(b)q(e) h(calculated)e(for)i(p)q(erturbations)g(in)f(eac)o(h)g(of)g(the)g(isot) o(ypic)152 1897 y(comp)q(onen)o(ts)i(indep)q(enden)o(tly;)90 2011 y(2.)24 b(the)14 b(motion)f(in)g(Fix\(\006\))h(will)f(b)q(e)h (stable)g(if)f(the)h(dominan)o(t)f(Ly)o(apuno)o(v)h(exp)q(onen)o(t)g (asso)q(ciated)152 2083 y(with)i(eac)o(h)g(of)g(the)g(isot)o(ypic)f (comp)q(onen)o(ts)h(other)g(than)h(the)f(trivial)f(one)h(\(whic)o(h)g (is)g(Fix)o(\(\006\)\))152 2155 y(are)g(negativ)o(e.)103 2269 y(W)l(e)g(apply)g(these)h(ideas)f(to)h(the)f(CGL)h(equation)f (\(1.1\))h(whic)o(h)f(has)h(a)f(n)o(um)o(b)q(er)f(of)i(symmetri)o(es)30 2341 y(giv)o(en)e(b)o(y)571 2405 y Fk(\022)q(A)p Fn(\()p Fk(x;)8 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y(systems)k(with)h(symmetry)d(for)k(the)f(sak)o(e)f(of)i (completeness.)i(F)l(or)e(more)d(details,)i(see)g([4,)f(5)q(].)103 2498 y(W)l(e)h(consider)g(a)h(general)f(ev)o(olution)f(equation)h(of)h (the)f(form)682 2597 y Fk(A)719 2604 y Fj(t)748 2597 y Fn(=)d Fk(g)r Fn(\()p Fk(A)p Fn(\))p Fk(;)73 b(g)16 b Fn(:)d Fk(X)18 b Fi(!)c Fk(X)q(;)561 b Fn(\(2.1\))30 2696 y(where)15 b Fk(g)j Fn(is)e(assumed)f(to)h(b)q(e)f(a)h(nonlinear)g (op)q(erator)h(in)o(v)o(olving)d(spatial)i(deriv)m(ativ)o(es)e(and)i Fk(X)k Fn(is)c(an)30 2769 y(appropriate)g(Hilb)q(ert)e(space)h(whic)o (h)g(incorp)q(orates)h(the)f(b)q(oundary)i(conditions.)k(W)l(e)15 b(also)h(assume)948 2893 y(3)p eop %%Page: 2 18 2 17 bop 30 28 a Fm(1)81 b(In)n(tro)r(duction)30 150 y Fn(The)16 b(formation)g(of)g(patterns)h(in)f(the)g(solutions)h(of)f (partial)g(di\013eren)o(tial)f(equations)i(whic)o(h)e(mo)q(del)30 222 y(man)o(y)i(ph)o(ysical)h(systems)f(has)i(b)q(een)g(the)f(sub)s 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b(There)21 b(are)h(solutions)g(on)g(this)f(branc)o(h)h(whic)o(h)e(are)i (stable)f(with)h(resp)q(ect)f(to)30 1884 y(p)q(erio)q(dic)14 b(p)q(erturbations)i(whose)f(p)q(erio)q(d)g(is)g Fl(any)g Fn(in)o(teger)e(m)o(ultiple)f(of)j(the)f(p)q(erio)q(d)h(of)g(the)g (solution)30 1956 y(\(see)20 b(the)g(n)o(umerical)e(results)i(in)g ([12]\))g(and)h(so)g(w)o(e)f(w)o(ould)g(exp)q(ect)g(to)g(see)g(this)h (solution)f(in)g(the)30 2028 y(ph)o(ysical)15 b(systems)g(mo)q(delled)g (b)o(y)g(the)h(Kuramoto-Siv)m(ashinsky)g(equation.)103 2100 y(In)i(a)g(previous)g(pap)q(er)h([5])e(w)o(e)h(in)o(v)o(estigated) f(c)o(haotic)g(solutions)i(of)f(the)g(complex)d(Ginzburg{)30 2172 y(Landau)j(\(CGL\))f(equation)369 2290 y Fk(A)406 2297 y Fj(t)434 2290 y Fn(=)d Fk(RA)d Fn(+)g(\(1)g(+)g Fk(i\027)s Fn(\))p Fi(r)828 2269 y Fh(2)847 2290 y Fk(A)g Fi(\000)g Fn(\(1)g(+)g Fk(i\026)p Fn(\))p Fk(A)p Fi(j)p Fk(A)p Fi(j)1215 2269 y Fh(2)1234 2290 y Fk(;)82 b(x)14 b Fi(2)g Fn([0)p Fk(;)8 b Fn(2)p Fk(\031)r Fn(\))238 b(\(1.1\))30 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b(and)i(Chaos)g(in)g(the)303 375 y(Complex)f(Ginzburg{Landau)i(Equation.)473 500 y(I)r(I:)f(T)-7 b(ranslati)q(ona)q(l)29 b(symmetries)497 662 y Fv(Philip)16 b(J.)k(Aston)888 641 y Fu(\003)927 662 y Fv(and)f(Carlo)g(R.)h(Laing) 1424 641 y Fu(y)406 752 y Fv(Departmen)n(t)e(of)i(Mathematics)e(and)h (Statistics,)696 842 y(Univ)n(ersit)n(y)f(of)i(Surrey)-5 b(,)691 931 y(Guildford)17 b(GU2)i(5XH,)746 1021 y(United)f(Kingdom)763 1142 y(Marc)n(h)h(11,)i(1999)860 1391 y Ft(Abstract)220 1487 y Fs(The)e(complex)h(Ginzburg{Landau)f(\(CGL\))f(equation)h(on)g (a)f(1{dimensional)j(domain)152 1555 y(with)i(p)q(erio)q(dic)i(b)q (oundary)e(conditions)g(has)g(a)f(n)o(um)o(b)q(er)h(of)f(di\013eren)o (t)h(symmetries,)h(and)152 1622 y(solutions)16 b(of)e(the)h(CGL)g(ma)o (y)f(or)h(ma)o(y)f(not)g(b)q(e)i(\014xed)g(b)o(y)e(the)i(action)f(of)f (these)h(symmetries.)152 1690 y(In)20 b(this)h(pap)q(er)f(w)o(e)f(in)o (v)o(estigate)h(the)g(stabilit)o(y)h(of)e(c)o(haotic)h(solutions)g (that)f(are)h(spatially)152 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