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b(arian)m(t)30 b(co)s(ordinate)h(planes.)43 b(These)34 b(kinds)e(of)f(p)s(erturb)s(ed)i(systems)g(ha)m(v)m(e)g(b)s (een)g(studied)f(b)m(y)24 872 y(sev)m(eral)g(authors)f(within)f(the)i (con)m(text)g(of)f(symmetric)g(dynamical)e(systems.)44 b(Their)32 b(approac)m(h)24 992 y(relies)i(hea)m(vily)h(on)g(the)g (equiv)-5 b(arian)m(t)34 b(bifurcation)f(theory)j(and)f(symmetry)g (group)g(tec)m(hniques)24 1112 y(\(see)e(Sandstede)h(and)f(Sc)m(heel)g ([9],)g(Melb)s(ourne)g(and)f(Krupa)h([6]\).)170 1233 y(In)c(this)e(pap)s(er)i(w)m(e)g(will)d(consider)j(a)f(singular)e (transv)m(erse)31 b(p)s(erturbation)c(and,)j(for)d(our)h(pur-)24 1353 y(p)s(oses)i(the)h(symmetry)f(considerations)f(will)f(pla)m(y)h(a) h(minor)e(role.)42 b(A)m(t)30 b(\014rst)g(the)h(v)-5 b(ariable)28 b Fl(u)h Fp(w)m(as)24 1474 y(mean)m(t)37 b(to)g(mimic)e(the)j(dynamics)f(of)g(a)g(magnetic)g(instabilit)m(y)e (resulting)h(from)h(a)g(con)m(v)m(ectiv)m(e)24 1594 y(state)27 b(describ)s(ed)g(b)m(y)h(equations)f(\(1.2\).)41 b(The)27 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y(clinic)h(connections.)42 b(The)30 b(drop)f(p)s(oin)m(ts)f (where)i(the)g(orbits)e(jump)g(from)g(one)h(slo)m(w)f(manifold)e(to)24 992 y(another)34 b(happ)s(en)h(to)f(b)s(e)g(far)g(enough)g(from)f(the)i (Hopf)f(equilibria.)46 b(The)35 b(geometric)e(situation)24 1112 y(can)g(b)s(e)g(describ)s(ed)g(as)g(a)g(3-dimensional)d(\014gure)j Fj(S)40 b Fp(whose)34 b(geometry)f(is)g(the)g(result)g(of)f(adding)24 1233 y(the)h(new)g(transv)m(erse)i(direction)c(to)i(the)g(original)c (1-dimensional)g(hetero)s(clinic)i(connection.)170 1353 y(The)43 b(la)m(y)m(out)e(of)g(the)i(pap)s(er)e(is)h(as)g(follo)m(ws:) 60 b(in)41 b(Section)h(\(2\))f(w)m(e)i(presen)m(t)g(the)g(n)m(umerical) 24 1474 y(sim)m(ulations)37 b(that)i(induced)h(us)h(to)e(searc)m(h)i (for)e(c)m(haos.)65 b(Section)40 b(\(3\))f(is)g(dev)m(oted)i(to)e(a)g (bifur-)24 1594 y(cation)g(analysis)g(that)h(allo)m(ws)f(us)h(to)g (presen)m(t)h(a)f(geometric)f(mo)s(del)f(in)h(Section)h(\(4\).)65 b(In)41 b(the)24 1714 y(singular)28 b(limit)f(case,)k Fl(")d Fp(=)f(0,)k(w)m(e)g(can)f(solv)m(e)h Fl(u)e Fp(as)i(a)f(\(bi-v) -5 b(aluated\))28 b(function)h(of)h(the)h(slo)m(w)f(v)-5 b(ari-)24 1835 y(ables)36 b Fl(x)327 1850 y Fk(1)367 1835 y Fl(;)17 b(x)466 1850 y Fk(2)505 1835 y Fl(;)g(x)604 1850 y Fk(3)680 1835 y Fp(and)37 b(obtain)e(a)h(3-dimensional)e(system) j(of)f(equations.)55 b(It)37 b(will)d(b)s(e)i(assumed)24 1955 y(that)30 b(there)g(exists)h(a)f(v)m(ery)i(strong)e(con)m (tracting)g(direction)f(transv)m(erse)j(to)e(the)g(slo)m(w)g(manifolds) 24 2076 y(that)36 b(allo)m(ws)f(us)j(to)e(reduce)i(the)f(study)h(of)e (the)h(dynamics)f(to)g(a)h(t)m(w)m(o)g(dimensional)d(system)k(of)24 2196 y(equations.)59 b(The)39 b Fj(C)778 2160 y Fk(1)818 2196 y Fp(-appro)m(ximation)c(theory)j(dev)m(elop)s(ed)h(b)m(y)g(Misc)m (henk)m(o)h Fq(et)g(al.)59 b Fp([7)o(])38 b(will)e(b)s(e)24 2316 y(used)41 b(for)f(the)h(case)h Fl(")f Fj(6)p Fp(=)g(0.)67 b(The)42 b(pro)s(of)e(of)g(the)h(existence)h(of)e(c)m(haos)h(is)f(tac)m (kled)h(in)f(Section)24 2437 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2249 3256 a Fl(a)2300 3271 y Fh(i)2328 3256 y Fp(\(0\))170 3458 y(The)33 b(table)d(b)s(elo)m(w)i (presen)m(ts)i(some)d(parameter)g(v)-5 b(alues)32 b(where)h Fl(L)2610 3473 y Fk(1)2681 3458 y Fp(is)e(represen)m(tativ)m(e.)45 b(The)24 3578 y(\014xed)33 b(v)-5 b(alues)33 b(are)f Fl(\026)c Fp(=)f Fj(\000)p Fp(0)p Fl(:)p Fp(05)33 b(and)f Fl(")c Fp(=)f(0)p Fl(:)p Fp(01.)1793 5214 y(6)p eop %%Page: 7 7 7 6 bop 281 4563 a @beginspecial 0 @llx 0 @lly 612 @urx 792 @ury 3685 @rwi @setspecial %%BeginDocument: marania %!PS-Adobe-2.0 %%Title: %%Creator: dstool %%CreationDate: Sat Apr 26 17:52:01 1997 %%For: Ignacio Bosch %%Pages: 1 %%DocumentFonts: /Times-Roman %%BoundingBox: 0 0 612 792 %%EndComments % begin the prolog... /x0 -2.5 def /xf 2.5 def /y0 -2.5 def /yf 2.5 def /SetFonts { /LabelPtSize 10 def /TitlePtSize 14 def /LabelFont /Times-Roman findfont LabelPtSize scalefont def /TitleFont /Times-Roman findfont TitlePtSize scalefont def } def % declare global constants (edit with care). 72 points = 1 inch /BBoxSide 450 def % size (in pts) of bounding box /TickSize 5 def % size (in pts) of tick marks /xorigin 76 def % new origin (in pts) for x /yorigin 151 def % new origin (in pts) for y /linewidth 0.005 def % width of line for plotting (smallest resolution=0) /Small 1 def % typical sizes of symbols /Medium 2 def /Large 4 def /Huge 6 def /HeadPtSize 6 def % print the header in this font size /Yp 600 def % print the header beginning at this height % declare scalefactors /xscale BBoxSide xf x0 sub div def % horizontal scaling factor /yscale BBoxSide yf y0 sub div def % vertical scaling factor % define plotting symbols /xscalei 1 xscale div def % horizontal scaling factor inverse /yscalei 1 yscale div def % vertical scaling factor inverse % symbols are defined in two parts: a "glyph" which is an absolute symbol, and the command % for plotting the glyph, which scales the glyph into user coordinates. % To make symbols have different linewidths, uncomment the commands "linewidth setlinewidth" % and replace it with something like "0.1 setlinewidth". % Also, boxes and triangles can be filled in by adding the command % "fill" prior to the "stroke" command. % First, we define the glyphs... /Boxglyph {newpath dup scale %linewidth setlinewidth 0.5 0.5 moveto -0.5 0.5 lineto -0.5 -0.5 lineto 0.5 -0.5 lineto 0.5 0.5 lineto closepath stroke} def /Triangleglyph {newpath dup scale %linewidth setlinewidth 0 1 moveto 0.866 -0.5 lineto -0.866 -0.5 lineto closepath stroke} def /Circleglyph {newpath dup scale %linewidth setlinewidth 0 0 0.5 0 360 arc fill stroke} def /Crossglyph {newpath dup scale %linewidth setlinewidth 0.75 0 moveto -0.75 0 lineto 0 0.75 moveto 0 -0.75 lineto stroke} def % Now define the symbol commands... % procedure to draw a box. Called as: x y scalefactor Box -- /Box{gsave 3 1 roll translate xscalei yscalei scale Boxglyph grestore }def % procedure to draw a triangle. Called as: x y scalefactor Triangle -- /Triangle{gsave 3 1 roll translate xscalei yscalei scale Triangleglyph grestore }def % procedure to draw a cross. Called as: x y scalefactor Cross -- /Cross{gsave 3 1 roll translate xscalei yscalei scale Crossglyph grestore }def % procedure to draw a dot. Called as: x y scalefactor Dot -- /Dot{gsave 3 1 roll translate xscalei yscalei scale Circleglyph grestore }def % draws ticks on the bounding box. Called as: num_x_ticks num_y_ticks Ticks % For thicker tick marks, uncomment the end of the first line /Ticks{ gsave % 1 setlinewidth 1 add BBoxSide exch div dup 0 0 moveto dup BBoxSide{0 exch moveto TickSize 0 rlineto}for BBoxSide 0 moveto dup BBoxSide{BBoxSide exch moveto TickSize neg 0 rlineto}for 1 add BBoxSide exch div dup 0 0 moveto dup BBoxSide{0 moveto 0 TickSize rlineto}for 0 BBoxSide moveto dup BBoxSide{BBoxSide moveto 0 TickSize neg rlineto}for stroke grestore }def % draws box of side BBoxSide beginning at origin /BoundingBox{ gsave 1 setlinewidth 0 0 moveto BBoxSide 0 rlineto 0 BBoxSide rlineto BBoxSide neg 0 rlineto closepath stroke grestore }def % macros for string positioning and manipulation /Display{0 Yp moveto show /Yp Yp HeadPtSize sub def }def % display header info % centers arbitrary string at current point. Called as: (str) CenterString -- /CenterString { dup stringwidth pop 2 div currentpoint 3 -2 roll exch sub exch moveto }def % vertically centers arbitrary string at current point. Called as: fontheight (str) VCenterString -- /VCenterString { dup length .5 add 2 div 3 -1 roll mul currentpoint 3 -1 roll add moveto }def % prints figure titles centered at given pt % requires: CenterString; TitlePtSize; TitleFont % called as: x y (title) (caption) FigureTitle -- /FigureTitle { gsave TitleFont setfont 4 2 roll moveto gsave exch CenterString show grestore currentpoint TitlePtSize sub moveto CenterString show grestore }def % prints x-axis label % requires: CenterString; LabelPtSize; LabelFont % called as: (label) XLabel -- /XLabel { gsave LabelFont setfont BBoxSide 2 div LabelPtSize -1.5 mul moveto CenterString show grestore }def % prints y-axis label % requires: CenterString; LabelPtSize; LabelFont % called as: (label) YLabel -- /YLabel { gsave LabelFont setfont LabelPtSize -1.5 mul BBoxSide 2 div moveto LabelPtSize exch VCenterString LabelPtSize exch vshow grestore }def % The following procedure is from the PostScript Tutorial and Cookbook by Adobe Systems Inc., p. 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y Fk(2)1741 4767 y Fp(space)89 b(for)44 b Fl(\026)j Fp(=)h Fj(\000)p Fp(0)p Fl(:)p Fp(05,)91 b Fl(p)2866 4782 y Fk(1)2953 4767 y Fp(=)47 b Fj(\000)p Fp(0)p Fl(:)p Fp(07,)d Fl(p)3447 4782 y Fk(2)3534 4767 y Fp(=)24 4887 y(0)p Fl(:)p Fp(5,)32 b Fl(p)257 4902 y Fk(3)324 4887 y Fp(=)27 b(0)p Fl(:)p Fp(8)33 b(and)f Fl(")c Fp(=)f(0)p Fl(:)p Fp(01)p Fl(:)1793 5214 y Fp(7)p eop %%Page: 8 8 8 7 bop 281 4624 a @beginspecial 0 @llx 0 @lly 612 @urx 792 @ury 3685 @rwi @setspecial %%BeginDocument: lau %!PS-Adobe-2.0 %%Title: %%Creator: dstool %%CreationDate: Sun May 4 16:56:51 1997 %%For: Ignacio Bosch %%Pages: 1 %%DocumentFonts: /Times-Roman %%BoundingBox: 0 0 612 792 %%EndComments % begin the prolog... /x0 0 def /xf 200 def /y0 -4 def /yf 4 def /SetFonts { /LabelPtSize 10 def /TitlePtSize 14 def /LabelFont /Times-Roman findfont LabelPtSize scalefont def /TitleFont /Times-Roman findfont TitlePtSize scalefont def } def % declare global constants (edit with care). 72 points = 1 inch /BBoxSide 450 def % size (in pts) of bounding box /TickSize 5 def % size (in pts) of tick marks /xorigin 76 def % new origin (in pts) for x /yorigin 151 def % new origin (in pts) for y /linewidth 0.005 def % width of line for plotting (smallest resolution=0) /Small 1 def % typical sizes of symbols /Medium 2 def /Large 4 def /Huge 6 def /HeadPtSize 6 def % print the header in this font size /Yp 600 def % print the header beginning at this height % declare scalefactors /xscale BBoxSide xf x0 sub div def % horizontal scaling factor /yscale BBoxSide yf y0 sub div def % vertical scaling factor % define plotting symbols /xscalei 1 xscale div def % horizontal scaling factor inverse /yscalei 1 yscale div def % vertical scaling factor inverse % symbols are defined in two parts: a "glyph" which is an absolute symbol, and the command % for plotting the glyph, which scales the glyph into user coordinates. % To make symbols have different linewidths, uncomment the commands "linewidth setlinewidth" % and replace it with something like "0.1 setlinewidth". % Also, boxes and triangles can be filled in by adding the command % "fill" prior to the "stroke" command. % First, we define the glyphs... /Boxglyph {newpath dup scale %linewidth setlinewidth 0.5 0.5 moveto -0.5 0.5 lineto -0.5 -0.5 lineto 0.5 -0.5 lineto 0.5 0.5 lineto closepath stroke} def /Triangleglyph {newpath dup scale %linewidth setlinewidth 0 1 moveto 0.866 -0.5 lineto -0.866 -0.5 lineto closepath stroke} def /Circleglyph {newpath dup scale %linewidth setlinewidth 0 0 0.5 0 360 arc fill stroke} def /Crossglyph {newpath dup scale %linewidth setlinewidth 0.75 0 moveto -0.75 0 lineto 0 0.75 moveto 0 -0.75 lineto stroke} def % Now define the symbol commands... % procedure to draw a box. Called as: x y scalefactor Box -- /Box{gsave 3 1 roll translate xscalei yscalei scale Boxglyph grestore }def % procedure to draw a triangle. Called as: x y scalefactor Triangle -- /Triangle{gsave 3 1 roll translate xscalei yscalei scale Triangleglyph grestore }def % procedure to draw a cross. Called as: x y scalefactor Cross -- /Cross{gsave 3 1 roll translate xscalei yscalei scale Crossglyph grestore }def % procedure to draw a dot. Called as: x y scalefactor Dot -- /Dot{gsave 3 1 roll translate xscalei yscalei scale Circleglyph grestore }def % draws ticks on the bounding box. Called as: num_x_ticks num_y_ticks Ticks % For thicker tick marks, uncomment the end of the first line /Ticks{ gsave % 1 setlinewidth 1 add BBoxSide exch div dup 0 0 moveto dup BBoxSide{0 exch moveto TickSize 0 rlineto}for BBoxSide 0 moveto dup BBoxSide{BBoxSide exch moveto TickSize neg 0 rlineto}for 1 add BBoxSide exch div dup 0 0 moveto dup BBoxSide{0 moveto 0 TickSize rlineto}for 0 BBoxSide moveto dup BBoxSide{BBoxSide moveto 0 TickSize neg rlineto}for stroke grestore }def % draws box of side BBoxSide beginning at origin /BoundingBox{ gsave 1 setlinewidth 0 0 moveto BBoxSide 0 rlineto 0 BBoxSide rlineto BBoxSide neg 0 rlineto closepath stroke grestore }def % macros for string positioning and manipulation /Display{0 Yp moveto show /Yp Yp HeadPtSize sub def }def % display header info % centers arbitrary string at current point. Called as: (str) CenterString -- /CenterString { dup stringwidth pop 2 div currentpoint 3 -2 roll exch sub exch moveto }def % vertically centers arbitrary string at current point. Called as: fontheight (str) VCenterString -- /VCenterString { dup length .5 add 2 div 3 -1 roll mul currentpoint 3 -1 roll add moveto }def % prints figure titles centered at given pt % requires: CenterString; TitlePtSize; TitleFont % called as: x y (title) (caption) FigureTitle -- /FigureTitle { gsave TitleFont setfont 4 2 roll moveto gsave exch CenterString show grestore currentpoint TitlePtSize sub moveto CenterString show grestore }def % prints x-axis label % requires: CenterString; LabelPtSize; LabelFont % called as: (label) XLabel -- /XLabel { gsave LabelFont setfont BBoxSide 2 div LabelPtSize -1.5 mul moveto CenterString show grestore }def % prints y-axis label % requires: CenterString; LabelPtSize; LabelFont % called as: (label) YLabel -- /YLabel { gsave LabelFont setfont LabelPtSize -1.5 mul BBoxSide 2 div moveto LabelPtSize exch VCenterString LabelPtSize exch vshow grestore }def % The following procedure is from the PostScript Tutorial and Cookbook by Adobe Systems Inc., p. 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Dot 90.51 1.06247 Small Dot 90.52 1.07122 Small Dot 90.53 1.07974 Small Dot 90.54 1.08805 Small Dot 90.55 1.09614 Small Dot 90.56 1.10401 Small Dot 90.57 1.11167 Small Dot 90.58 1.11911 Small Dot 90.59 1.12636 Small Dot 90.6 1.13339 Small Dot 90.61 1.14023 Small Dot 90.62 1.14688 Small Dot 90.63 1.15333 Small Dot 90.64 1.15959 Small Dot 90.65 1.16566 Small Dot 90.66 1.17156 Small Dot 90.67 1.17728 Small Dot 90.68 1.18282 Small Dot 90.69 1.1882 Small Dot 90.7 1.19341 Small Dot 90.71 1.19846 Small Dot 90.72 1.20335 Small Dot 90.73 1.20809 Small Dot 90.74 1.21268 Small Dot 90.75 1.21713 Small Dot 90.76 1.22143 Small Dot 90.77 1.2256 Small Dot 90.78 1.22964 Small Dot 90.79 1.23355 Small Dot 90.8 1.23733 Small Dot 90.81 1.24099 Small Dot 90.82 1.24453 Small Dot 90.83 1.24796 Small Dot 90.84 1.25128 Small Dot 90.85 1.25449 Small Dot 90.86 1.2576 Small Dot 90.87 1.26061 Small Dot 90.88 1.26352 Small Dot 90.89 1.26633 Small Dot 90.9 1.26906 Small Dot 90.91 1.2717 Small Dot 90.92 1.27425 Small Dot 90.93 1.27672 Small Dot 90.94 1.27911 Small Dot 90.95 1.28143 Small Dot 90.96 1.28367 Small Dot 90.97 1.28584 Small Dot 90.98 1.28794 Small Dot 90.99 1.28997 Small Dot 91 1.29194 Small Dot 91.01 1.29385 Small Dot 91.02 1.29569 Small Dot 91.03 1.29748 Small Dot 91.04 1.29921 Small Dot 91.05 1.30089 Small Dot 91.06 1.30252 Small Dot 91.07 1.30409 Small Dot 91.08 1.30562 Small Dot 91.09 1.3071 Small Dot 91.1 1.30853 Small Dot 91.11 1.30992 Small Dot 91.12 1.31127 Small Dot 91.13 1.31257 Small Dot 91.14 1.31384 Small Dot 91.15 1.31507 Small Dot 91.16 1.31626 Small Dot 91.17 1.31741 Small Dot 91.18 1.31853 Small Dot 91.19 1.31961 Small Dot 91.2 1.32066 Small Dot 91.21 1.32168 Small Dot 91.22 1.32266 Small Dot 91.23 1.32362 Small Dot 91.24 1.32455 Small Dot 91.25 1.32545 Small Dot 91.26 1.32632 Small Dot 91.27 1.32716 Small Dot 91.28 1.32798 Small Dot 91.29 1.32878 Small Dot 91.3 1.32954 Small Dot 91.31 1.33029 Small Dot 91.32 1.33101 Small Dot 91.33 1.3317 Small Dot 91.34 1.33238 Small Dot 91.35 1.33303 Small Dot 91.36 1.33366 Small Dot 91.37 1.33427 Small Dot 91.38 1.33486 Small Dot 91.39 1.33543 Small Dot 91.4 1.33597 Small Dot 91.41 1.3365 Small Dot 91.42 1.33701 Small Dot 91.43 1.3375 Small Dot 91.44 1.33798 Small Dot 91.45 1.33843 Small Dot 91.46 1.33887 Small Dot 91.47 1.33929 Small Dot 91.48 1.33969 Small Dot 91.49 1.34007 Small Dot 91.5 1.34044 Small Dot 91.51 1.34079 Small Dot 91.52 1.34112 Small Dot 91.53 1.34144 Small Dot 91.54 1.34174 Small Dot 91.55 1.34202 Small Dot 91.56 1.34229 Small Dot 91.57 1.34254 Small Dot 91.58 1.34278 Small Dot 91.59 1.343 Small Dot 91.6 1.34321 Small Dot 91.61 1.3434 Small Dot 91.62 1.34358 Small Dot 91.63 1.34374 Small Dot 91.64 1.34388 Small Dot 91.65 1.34401 Small Dot 91.66 1.34413 Small Dot 91.67 1.34423 Small Dot 91.68 1.34431 Small Dot 91.69 1.34438 Small Dot 91.7 1.34444 Small Dot 91.71 1.34448 Small Dot 91.72 1.34451 Small Dot 91.73 1.34452 Small Dot 91.74 1.34452 Small Dot 91.75 1.3445 Small Dot 91.76 1.34447 Small Dot 91.77 1.34442 Small Dot 91.78 1.34436 Small Dot 91.79 1.34428 Small Dot 91.8 1.34419 Small Dot 91.81 1.34408 Small Dot 91.82 1.34396 Small Dot 91.83 1.34383 Small Dot 91.84 1.34368 Small Dot 91.85 1.34352 Small Dot 91.86 1.34334 Small Dot 91.87 1.34314 Small Dot 91.88 1.34293 Small Dot 91.89 1.34271 Small Dot 91.9 1.34247 Small Dot 91.91 1.34222 Small Dot 91.92 1.34195 Small Dot 91.93 1.34167 Small Dot 91.94 1.34137 Small Dot 91.95 1.34105 Small Dot 91.96 1.34073 Small Dot 91.97 1.34038 Small Dot 91.98 1.34002 Small Dot 91.99 1.33965 Small Dot 92 1.33926 Small Dot 92.01 1.33886 Small Dot 92.02 1.33844 Small Dot 92.03 1.338 Small Dot 92.04 1.33755 Small Dot 92.05 1.33708 Small Dot 92.06 1.3366 Small Dot 92.07 1.33611 Small Dot 92.08 1.33559 Small Dot 92.09 1.33506 Small Dot 92.1 1.33452 Small Dot 92.11 1.33396 Small Dot 92.12 1.33338 Small Dot 92.13 1.33279 Small Dot 92.14 1.33218 Small Dot 92.15 1.33156 Small Dot 92.16 1.33092 Small Dot 92.17 1.33026 Small Dot 92.18 1.32959 Small Dot 92.19 1.3289 Small Dot 92.2 1.32819 Small Dot 92.21 1.32747 Small Dot 92.22 1.32673 Small Dot 92.23 1.32598 Small Dot 92.24 1.32521 Small Dot 92.25 1.32442 Small Dot 92.26 1.32362 Small Dot 92.27 1.32279 Small Dot 92.28 1.32196 Small Dot 92.29 1.3211 Small Dot 92.3 1.32023 Small Dot 92.31 1.31934 Small Dot 92.32 1.31844 Small Dot 92.33 1.31751 Small Dot 92.34 1.31657 Small Dot 92.35 1.31562 Small Dot 92.36 1.31464 Small Dot 92.37 1.31365 Small Dot 92.38 1.31264 Small Dot 92.39 1.31162 Small Dot 92.4 1.31057 Small Dot 92.41 1.30951 Small Dot 92.42 1.30843 Small Dot 92.43 1.30734 Small Dot 92.44 1.30622 Small Dot 92.45 1.30509 Small Dot 92.46 1.30394 Small Dot 92.47 1.30277 Small Dot 92.48 1.30159 Small Dot 92.49 1.30038 Small Dot 92.5 1.29916 Small Dot 92.51 1.29792 Small Dot 92.52 1.29667 Small Dot 92.53 1.29539 Small Dot 92.54 1.2941 Small Dot 92.55 1.29279 Small Dot 92.56 1.29146 Small Dot 92.57 1.29011 Small Dot 92.58 1.28874 Small Dot 92.59 1.28736 Small Dot 92.6 1.28595 Small Dot 92.61 1.28453 Small Dot 92.62 1.28309 Small Dot 92.63 1.28163 Small Dot 92.64 1.28016 Small Dot 92.65 1.27866 Small Dot 92.66 1.27714 Small Dot 92.67 1.27561 Small Dot 92.68 1.27406 Small Dot 92.69 1.27249 Small Dot 92.7 1.2709 Small Dot 92.71 1.26929 Small Dot 92.72 1.26766 Small Dot 92.73 1.26601 Small Dot 92.74 1.26435 Small Dot 92.75 1.26266 Small Dot 92.76 1.26096 Small Dot 92.77 1.25923 Small Dot 92.78 1.25749 Small Dot 92.79 1.25573 Small Dot 92.8 1.25395 Small Dot 92.81 1.25215 Small Dot 92.82 1.25033 Small Dot 92.83 1.24849 Small Dot 92.84 1.24663 Small Dot 92.85 1.24475 Small Dot 92.86 1.24285 Small Dot 92.87 1.24094 Small Dot 92.88 1.239 Small Dot 92.89 1.23704 Small Dot 92.9 1.23507 Small Dot 92.91 1.23307 Small Dot 92.92 1.23106 Small Dot 92.93 1.22903 Small Dot 92.94 1.22697 Small Dot 92.95 1.2249 Small Dot 92.96 1.2228 Small Dot 92.97 1.22069 Small Dot 92.98 1.21856 Small Dot 92.99 1.21641 Small Dot 93 1.21423 Small Dot 93.01 1.21204 Small Dot 93.02 1.20983 Small Dot 93.03 1.2076 Small Dot 93.04 1.20535 Small Dot 93.05 1.20308 Small Dot 93.06 1.20079 Small Dot 93.07 1.19848 Small Dot 93.08 1.19614 Small Dot 93.09 1.19379 Small Dot 93.1 1.19142 Small Dot 93.11 1.18903 Small Dot 93.12 1.18662 Small Dot 93.13 1.18419 Small Dot 93.14 1.18174 Small Dot 93.15 1.17927 Small Dot 93.16 1.17678 Small Dot 93.17 1.17427 Small Dot 93.18 1.17174 Small Dot 93.19 1.16919 Small Dot 93.2 1.16662 Small Dot 93.21 1.16403 Small Dot 93.22 1.16142 Small Dot 93.23 1.15879 Small Dot 93.24 1.15615 Small Dot 93.25 1.15348 Small Dot 93.26 1.15079 Small Dot 93.27 1.14808 Small Dot 93.28 1.14535 Small Dot 93.29 1.1426 Small Dot 93.3 1.13983 Small Dot 93.31 1.13704 Small Dot 93.32 1.13424 Small Dot 93.33 1.13141 Small Dot 93.34 1.12856 Small Dot 93.35 1.1257 Small Dot 93.36 1.12281 Small Dot 93.37 1.1199 Small Dot 93.38 1.11698 Small Dot 93.39 1.11403 Small Dot 93.4 1.11107 Small Dot 93.41 1.10808 Small Dot 93.42 1.10508 Small Dot 93.43 1.10205 Small Dot 93.44 1.09901 Small Dot 93.45 1.09595 Small Dot 93.46 1.09287 Small Dot 93.47 1.08976 Small Dot 93.48 1.08664 Small Dot 93.49 1.08351 Small Dot 93.5 1.08035 Small Dot 93.51 1.07717 Small Dot 93.52 1.07397 Small Dot 93.53 1.07076 Small Dot 93.54 1.06752 Small Dot 93.55 1.06427 Small Dot 93.56 1.061 Small Dot 93.57 1.05771 Small Dot 93.58 1.0544 Small Dot 93.59 1.05107 Small Dot 93.6 1.04772 Small Dot 93.61 1.04436 Small Dot 93.62 1.04097 Small Dot 93.63 1.03757 Small Dot 93.64 1.03415 Small Dot 93.65 1.03072 Small Dot 93.66 1.02726 Small Dot 93.67 1.02379 Small Dot 93.68 1.02029 Small Dot 93.69 1.01678 Small Dot 93.7 1.01326 Small Dot 93.71 1.00971 Small Dot 93.72 1.00615 Small Dot 93.73 1.00257 Small Dot 93.74 0.998977 Small Dot 93.75 0.995363 Small Dot 93.76 0.991733 Small Dot 93.77 0.988086 Small Dot 93.78 0.984423 Small Dot 93.79 0.980742 Small Dot 93.8 0.977046 Small Dot 93.81 0.973333 Small Dot 93.82 0.969603 Small Dot 93.83 0.965858 Small Dot 93.84 0.962097 Small Dot 93.85 0.958321 Small Dot 93.86 0.954529 Small Dot 93.87 0.950721 Small Dot 93.88 0.946898 Small Dot 93.89 0.943061 Small Dot 93.9 0.939208 Small Dot 93.91 0.935341 Small Dot 93.92 0.931459 Small Dot 93.93 0.927563 Small Dot 93.94 0.923653 Small Dot 93.95 0.919729 Small Dot 93.96 0.915791 Small Dot 93.97 0.91184 Small Dot 93.98 0.907875 Small Dot 93.99 0.903897 Small Dot 94 0.899907 Small Dot 94.01 0.895904 Small Dot 94.02 0.891889 Small Dot 94.03 0.887861 Small Dot 94.04 0.883822 Small Dot 94.05 0.879771 Small Dot 94.06 0.875708 Small Dot 94.07 0.871635 Small Dot 94.08 0.867551 Small Dot 94.09 0.863456 Small Dot 94.1 0.859351 Small Dot 94.11 0.855236 Small Dot 94.12 0.851111 Small Dot 94.13 0.846977 Small Dot 94.14 0.842834 Small Dot 94.15 0.838682 Small Dot 94.16 0.834522 Small Dot 94.17 0.830353 Small Dot 94.18 0.826177 Small Dot 94.19 0.821994 Small Dot 94.2 0.817804 Small Dot 94.21 0.813607 Small Dot 94.22 0.809403 Small Dot 94.23 0.805194 Small Dot 94.24 0.800979 Small Dot 94.25 0.79676 Small Dot 94.26 0.792535 Small Dot 94.27 0.788306 Small Dot 94.28 0.784074 Small Dot 94.29 0.779838 Small Dot 94.3 0.775599 Small Dot 94.31 0.771357 Small Dot 94.32 0.767114 Small Dot 94.33 0.762869 Small Dot 94.34 0.758622 Small Dot 94.35 0.754375 Small Dot 94.36 0.750128 Small Dot 94.37 0.745882 Small Dot 94.38 0.741636 Small Dot 94.39 0.737392 Small Dot 94.4 0.73315 Small Dot 94.41 0.72891 Small Dot 94.42 0.724673 Small Dot 94.43 0.720441 Small Dot 94.44 0.716212 Small Dot 94.45 0.711988 Small Dot 94.46 0.70777 Small Dot 94.47 0.703558 Small Dot 94.48 0.699352 Small Dot 94.49 0.695154 Small Dot 94.5 0.690964 Small Dot 94.51 0.686783 Small Dot 94.52 0.682611 Small Dot 94.53 0.678449 Small Dot 94.54 0.674297 Small Dot 94.55 0.670158 Small Dot 94.56 0.66603 Small Dot 94.57 0.661915 Small Dot 94.58 0.657814 Small Dot 94.59 0.653728 Small Dot 94.6 0.649657 Small Dot 94.61 0.645601 Small Dot 94.62 0.641563 Small Dot 94.63 0.637542 Small Dot 94.64 0.633539 Small Dot 94.65 0.629556 Small Dot 94.66 0.625593 Small Dot 94.67 0.621651 Small Dot 94.68 0.617731 Small Dot 94.69 0.613833 Small Dot 94.7 0.609959 Small Dot 94.71 0.60611 Small Dot 94.72 0.602286 Small Dot 94.73 0.598489 Small Dot 94.74 0.594719 Small Dot 94.75 0.590977 Small Dot 94.76 0.587264 Small Dot 94.77 0.583582 Small Dot 94.78 0.579931 Small Dot 94.79 0.576312 Small Dot 94.8 0.572727 Small Dot 94.81 0.569176 Small Dot 94.82 0.56566 Small Dot 94.83 0.56218 Small Dot 94.84 0.558738 Small Dot 94.85 0.555334 Small Dot 94.86 0.55197 Small Dot 94.87 0.548647 Small Dot 94.88 0.545365 Small Dot 94.89 0.542126 Small Dot 94.9 0.538932 Small Dot 94.91 0.535782 Small Dot 94.92 0.532678 Small Dot 94.93 0.529622 Small Dot 94.94 0.526614 Small Dot 94.95 0.523656 Small Dot 94.96 0.520748 Small Dot 94.97 0.517892 Small Dot 94.98 0.51509 Small Dot 94.99 0.512341 Small Dot 95 0.509648 Small Dot 95.01 0.507012 Small Dot 95.02 0.504433 Small Dot 95.03 0.501913 Small Dot 95.04 0.499453 Small Dot 95.05 0.497054 Small Dot 95.06 0.494718 Small Dot 95.07 0.492445 Small Dot 95.08 0.490237 Small Dot 95.09 0.488095 Small Dot 95.1 0.486019 Small Dot 95.11 0.484012 Small Dot 95.12 0.482074 Small Dot 95.13 0.480207 Small Dot 95.14 0.478411 Small Dot 95.15 0.476688 Small Dot 95.16 0.475039 Small Dot 95.17 0.473464 Small Dot 95.18 0.471966 Small Dot 95.19 0.470544 Small Dot 95.2 0.469201 Small Dot 95.21 0.467936 Small Dot 95.22 0.466752 Small Dot 95.23 0.465649 Small Dot 95.24 0.464628 Small Dot 95.25 0.463691 Small Dot 95.26 0.462837 Small Dot 95.27 0.462068 Small Dot 95.28 0.461386 Small Dot 95.29 0.46079 Small Dot 95.3 0.460282 Small Dot 95.31 0.459862 Small Dot 95.32 0.459531 Small Dot 95.33 0.459291 Small Dot 95.34 0.459141 Small Dot 95.35 0.459083 Small Dot 95.36 0.459117 Small Dot 95.37 0.459243 Small Dot 95.38 0.459463 Small Dot 95.39 0.459777 Small Dot 95.4 0.460185 Small Dot 95.41 0.460689 Small Dot 95.42 0.461287 Small Dot 95.43 0.461982 Small Dot 95.44 0.462772 Small Dot 95.45 0.463659 Small Dot 95.46 0.464642 Small Dot 95.47 0.465722 Small Dot 95.48 0.4669 Small Dot 95.49 0.468174 Small Dot 95.5 0.469545 Small Dot 95.51 0.471014 Small Dot 95.52 0.47258 Small Dot 95.53 0.474242 Small Dot 95.54 0.476001 Small Dot 95.55 0.477857 Small Dot 95.56 0.479809 Small Dot 95.57 0.481857 Small Dot 95.58 0.484 Small Dot 95.59 0.486238 Small Dot 95.6 0.488571 Small Dot 95.61 0.490997 Small Dot 95.62 0.493517 Small Dot 95.63 0.496128 Small Dot 95.64 0.498831 Small Dot 95.65 0.501625 Small Dot 95.66 0.504508 Small Dot 95.67 0.50748 Small Dot 95.68 0.51054 Small Dot 95.69 0.513685 Small Dot 95.7 0.516916 Small Dot 95.71 0.520231 Small Dot 95.72 0.523628 Small Dot 95.73 0.527105 Small Dot 95.74 0.530662 Small Dot 95.75 0.534297 Small Dot 95.76 0.538008 Small Dot 95.77 0.541793 Small Dot 95.78 0.54565 Small Dot 95.79 0.549579 Small Dot 95.8 0.553575 Small Dot 95.81 0.557639 Small Dot 95.82 0.561767 Small Dot 95.83 0.565957 Small Dot 95.84 0.570207 Small Dot 95.85 0.574516 Small Dot 95.86 0.578879 Small Dot 95.87 0.583296 Small Dot 95.88 0.587764 Small Dot 95.89 0.59228 Small Dot 95.9 0.596841 Small Dot 95.91 0.601445 Small Dot 95.92 0.60609 Small Dot 95.93 0.610772 Small Dot 95.94 0.615489 Small Dot 95.95 0.620238 Small Dot 95.96 0.625017 Small Dot 95.97 0.629822 Small Dot 95.98 0.63465 Small Dot 95.99 0.639499 Small Dot 96 0.644366 Small Dot 96.01 0.649247 Small Dot 96.02 0.65414 Small Dot 96.03 0.659043 Small Dot 96.04 0.663951 Small Dot 96.05 0.668862 Small Dot 96.06 0.673773 Small Dot 96.07 0.678681 Small Dot 96.08 0.683583 Small Dot 96.09 0.688476 Small Dot 96.1 0.693357 Small Dot 96.11 0.698224 Small Dot 96.12 0.703073 Small Dot 96.13 0.707902 Small Dot 96.14 0.712707 Small Dot 96.15 0.717487 Small Dot 96.16 0.722238 Small Dot 96.17 0.726958 Small Dot 96.18 0.731644 Small Dot 96.19 0.736294 Small Dot 96.2 0.740905 Small Dot 96.21 0.745474 Small Dot 96.22 0.75 Small Dot 96.23 0.75448 Small Dot 96.24 0.758912 Small Dot 96.25 0.763293 Small Dot 96.26 0.767622 Small Dot 96.27 0.771897 Small Dot 96.28 0.776116 Small Dot 96.29 0.780277 Small Dot 96.3 0.784379 Small Dot 96.31 0.788419 Small Dot 96.32 0.792397 Small Dot 96.33 0.79631 Small Dot 96.34 0.800158 Small Dot 96.35 0.80394 Small Dot 96.36 0.807654 Small Dot 96.37 0.811299 Small Dot 96.38 0.814875 Small Dot 96.39 0.81838 Small Dot 96.4 0.821814 Small Dot 96.41 0.825177 Small Dot 96.42 0.828468 Small Dot 96.43 0.831686 Small Dot 96.44 0.834831 Small Dot 96.45 0.837903 Small Dot 96.46 0.840902 Small Dot 96.47 0.843829 Small Dot 96.48 0.846682 Small Dot 96.49 0.849463 Small Dot 96.5 0.852172 Small Dot 96.51 0.854809 Small Dot 96.52 0.857375 Small Dot 96.53 0.85987 Small Dot 96.54 0.862296 Small Dot 96.55 0.864652 Small Dot 96.56 0.86694 Small Dot 96.57 0.869161 Small Dot 96.58 0.871316 Small Dot 96.59 0.873406 Small Dot 96.6 0.875432 Small Dot 96.61 0.877396 Small Dot 96.62 0.879299 Small Dot 96.63 0.881143 Small Dot 96.64 0.882928 Small Dot 96.65 0.884658 Small Dot 96.66 0.886333 Small Dot 96.67 0.887956 Small Dot 96.68 0.889527 Small Dot 96.69 0.891049 Small Dot 96.7 0.892525 Small Dot 96.71 0.893955 Small Dot 96.72 0.895343 Small Dot 96.73 0.89669 Small Dot 96.74 0.897998 Small Dot 96.75 0.89927 Small Dot 96.76 0.900507 Small Dot 96.77 0.901713 Small Dot 96.78 0.90289 Small Dot 96.79 0.90404 Small Dot 96.8 0.905165 Small Dot 96.81 0.906268 Small Dot 96.82 0.907352 Small Dot 96.83 0.90842 Small Dot 96.84 0.909473 Small Dot 96.85 0.910514 Small Dot 96.86 0.911546 Small Dot 96.87 0.912572 Small Dot 96.88 0.913595 Small Dot 96.89 0.914617 Small Dot 96.9 0.915641 Small Dot 96.91 0.91667 Small Dot 96.92 0.917706 Small Dot 96.93 0.918752 Small Dot 96.94 0.919812 Small Dot 96.95 0.920888 Small Dot 96.96 0.921983 Small Dot 96.97 0.923099 Small Dot 96.98 0.924241 Small Dot 96.99 0.925409 Small Dot 97 0.926608 Small Dot 97.01 0.927841 Small Dot 97.02 0.929109 Small Dot 97.03 0.930416 Small Dot 97.04 0.931766 Small Dot 97.05 0.93316 Small Dot 97.06 0.934601 Small Dot 97.07 0.936093 Small Dot 97.08 0.937638 Small Dot 97.09 0.939239 Small Dot 97.1 0.940899 Small Dot 97.11 0.942621 Small Dot 97.12 0.944406 Small Dot 97.13 0.946259 Small Dot 97.14 0.948182 Small Dot 97.15 0.950177 Small Dot 97.16 0.952248 Small Dot 97.17 0.954396 Small Dot 97.18 0.956624 Small Dot 97.19 0.958935 Small Dot 97.2 0.961331 Small Dot 97.21 0.963815 Small Dot 97.22 0.966389 Small Dot 97.23 0.969056 Small Dot 97.24 0.971818 Small Dot 97.25 0.974676 Small Dot 97.26 0.977634 Small Dot 97.27 0.980694 Small Dot 97.28 0.983856 Small Dot 97.29 0.987125 Small Dot 97.3 0.990501 Small Dot 97.31 0.993986 Small Dot 97.32 0.997582 Small Dot 97.33 1.00129 Small Dot 97.34 1.00511 Small Dot 97.35 1.00905 Small Dot 97.36 1.01311 Small Dot 97.37 1.01729 Small Dot 97.38 1.02158 Small Dot 97.39 1.026 Small Dot 97.4 1.03054 Small Dot 97.41 1.0352 Small Dot 97.42 1.03999 Small Dot 97.43 1.0449 Small Dot 97.44 1.04994 Small Dot 97.45 1.05511 Small Dot 97.46 1.0604 Small Dot 97.47 1.06582 Small Dot 97.48 1.07136 Small Dot 97.49 1.07704 Small Dot 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5536 4340 PD 5423 4284 PD 4569 3732 PD 5419 4265 PD 5326 4212 PD 2581 2493 PD 4842 3852 PD 4153 3361 PD 5248 4182 PD 4613 3703 PD 3941 3412 PD 5597 4341 PD 2549 2521 PD 4424 3685 PD 5417 4271 PD 3649 3018 PD 3566 2973 PD 3592 2998 PD 4141 3426 PD 4036 3324 PD 3618 3200 PD 4864 3983 PD 3179 2931 PD 5544 4290 PD 5877 4508 PD 5114 4031 PD 5755 4438 PD 2982 2741 PD 1801 1679 PD 4705 3866 PD 5272 4152 PD 5749 4435 PD 1365 1609 PD 5089 4051 PD 2595 2315 PD 2713 2407 PD 5051 4092 PD 1298 1548 PD 2413 2377 PD 5290 4204 PD 4646 3726 PD 2911 2712 PD 2433 2399 PD 2881 2549 PD 5216 4181 PD 5239 4167 PD 2782 2663 PD 2129 2168 PD 5387 4263 PD 5048 3999 PD 5577 4361 PD 4917 3902 PD 5823 4471 PD 5468 4305 PD 5382 4264 PD 4810 3954 PD 3921 3388 PD 2617 2529 PD 5460 4288 PD 5655 4356 PD 3135 2905 PD 5206 4107 PD 2631 2302 PD 5475 4257 PD 5466 4307 PD 2716 2413 PD 5490 4314 PD 4660 3794 PD 1779 1914 PD 2309 2340 PD 5675 4405 PD 4696 3787 PD 5774 4445 PD 5262 4203 PD 4034 3428 PD 5007 4035 PD 3505 3096 PD 4645 3726 PD 4223 3567 PD 5035 4075 PD 5347 4181 PD 1581 1547 PD 5559 4346 PD 4269 3535 PD 2206 2224 PD 5540 4342 PD 5029 3978 PD 2936 2780 PD 3908 3262 PD 3276 2951 PD 2244 1971 PD 5507 4286 PD 2056 2101 PD 5323 4201 PD 2633 2574 PD 3296 2861 PD 5387 4225 PD 1547 1814 PD 1420 1668 PD 5850 4500 PD 2438 2164 PD 4256 3493 PD 3008 2546 PD 3850 3192 PD 5194 4166 PD 5347 4246 PD 4560 3706 PD 3889 3207 PD 5550 4349 PD 5579 4340 PD 1313 1564 PD 5552 4350 PD 4434 3559 PD 5585 4360 PD 4738 3812 PD 5888 4515 PD 4709 3825 PD 4097 3367 PD 2358 2389 PD 4571 3644 PD 3563 3178 PD 4376 3697 PD 5598 4351 PD 4967 4028 PD 3299 2981 PD 4569 3643 PD 1120 1497 PD 4044 3328 PD 3701 3243 PD 5223 4171 PD 5067 4035 PD 5208 4078 PD 1090 1456 PD 5885 4511 PD 4566 3726 PD 5341 4242 PD 2884 2672 PD 4668 3826 PD 1869 1977 PD 4733 3868 PD 5106 4122 PD 4847 3870 PD 5681 4415 PD 3866 3194 PD 5666 4408 PD 5399 4271 PD 4894 3910 PD 3132 2871 PD 1952 1751 PD 5402 4229 PD 3171 2919 PD 4944 3927 PD 4728 3897 PD 3997 3259 PD 5566 4356 PD 4455 3605 PD 4565 3811 PD 1479 1734 PD 5578 4360 PD 5670 4397 PD 2365 2120 PD 3170 2696 PD 2504 2461 PD 5189 4141 PD 5584 4362 PD 5392 4269 PD 5495 4298 PD 5097 4057 PD 3684 3123 PD 4839 3929 PD 2563 2232 PD 5432 4255 PD 5056 4037 PD 4434 3603 PD 4493 3764 PD 5053 4083 PD 4431 3592 PD 4331 3651 PD 1079 1452 PD 4760 3912 PD 1228 1522 PD 3740 3172 PD 2965 2793 PD 3503 2988 PD 5625 4380 PD 5107 4003 PD 5468 4307 PD 3929 3285 PD 1664 1882 PD 1192 1502 PD 2617 2339 PD 4861 3922 PD 5183 4130 PD 4848 3845 PD 1157 1489 PD 2091 1863 PD 2364 2071 PD 5066 4045 PD 5318 4173 PD 5127 4081 PD 5223 4185 PD 1128 1482 PD 4630 3817 PD 1715 1923 PD 5278 4213 PD 5007 3939 PD 4307 3527 PD 1960 2087 PD 4660 3789 PD 3252 2952 PD 3979 3276 PD 5092 4078 PD 3842 3340 PD 3127 2907 PD 5543 4310 PD 5874 4504 PD 4879 3857 PD 4613 3792 PD 3452 3070 PD 4980 3928 PD 5250 4100 PD 1770 1949 PD 2401 2365 PD 4423 3595 PD 4564 3811 PD 1411 1690 PD 5360 4244 PD 5262 4184 PD 2383 2350 PD 1732 1918 PD 4001 3419 PD 3905 3391 PD 5833 4489 PD 5055 3987 PD 4148 3557 PD 5551 4350 PD 3852 3314 PD 5262 4205 PD 5430 4284 PD 5075 4003 PD 2809 2673 PD 3364 2812 PD 5309 4208 PD 5488 4315 PD 5524 4331 PD 4668 3796 PD 2343 2329 PD 4083 3517 PD 3403 2832 PD 3990 3406 PD 4709 3870 PD 5607 4362 PD 2421 2422 PD 5170 4080 PD 3166 2698 PD 1844 1943 PD 5506 4319 PD 4374 3666 PD 5549 4335 PD 4826 3921 PD 1522 1488 PD 3140 2904 PD 5313 4133 PD 4271 3632 PD 3081 2669 PD 3507 3150 PD 4191 3584 PD 4764 3899 PD 5418 4271 PD 2870 2530 PD 3073 2828 PD 3234 2967 PD 5389 4242 PD 1929 1795 PD 3308 2985 PD 4465 3576 PD 5740 4417 PD 4152 3513 PD 2285 2287 PD 5674 4395 PD 5319 4225 PD 4071 3471 PD 1933 2052 PD 2527 2472 PD 4260 3617 PD 1521 1723 PD 3793 3303 PD 4432 3713 PD 5645 4396 PD 4184 3468 PD 4278 3623 PD 5740 4444 PD 3980 3261 PD 5262 4201 PD 3714 3277 PD 3010 2762 PD 5663 4397 PD 3934 3295 PD 2114 1936 PD 1503 1730 PD 2162 2199 PD 5541 4288 PD 5346 4230 PD 4335 3637 PD 4030 3287 PD 5154 4147 PD 1705 1663 PD 2773 2605 PD 1372 1606 PD 3062 2677 PD 3829 3323 PD 4595 3791 PD 5575 4309 PD 5373 4257 PD 5432 4246 PD 2878 2676 PD 5624 4345 PD 1322 1574 PD 2069 2183 PD 4289 3631 PD 5392 4224 PD 3134 2865 PD 4671 3835 PD 5131 4074 PD 3473 2953 PD 4983 4003 PD 5257 4151 PD 4614 3829 PD 4595 3823 PD 3400 3062 PD 5720 4424 PD 3495 2930 PD 3057 2631 PD 5665 4405 PD 4009 3301 PD 4038 3486 PD 2438 2234 PD 2732 2438 PD 5339 4242 PD 4418 3623 PD 1838 2020 PD 3876 3332 PD 5696 4420 PD 4522 3784 PD 4217 3556 PD 2764 2647 PD 3989 3411 PD 4036 3481 PD 2031 2106 PD 5175 4159 PD 5260 4203 PD 2256 2306 PD 3556 3137 PD 5486 4269 PD 4380 3664 PD 5224 4125 PD 5141 4080 PD 4637 3699 PD 1375 1666 PD 4939 3951 PD 4563 3786 PD 2416 2170 PD 3465 3114 PD 5666 4402 PD 2338 2310 PD 5267 4133 PD 2298 2086 PD 4521 3766 PD 3779 3136 PD 5801 4468 PD 4789 3824 PD 3522 2944 PD 3086 2646 PD 3554 3124 PD 4877 3861 PD 5080 4071 PD 3993 3439 PD 5707 4427 PD 4749 3795 PD 4392 3653 PD 4843 3894 PD 5765 4448 PD 3975 3337 PD 2196 2262 PD 5471 4309 PD 4304 3493 PD 5307 4203 PD 5382 4264 PD 4848 3971 PD 2993 2799 PD 2381 2155 PD 3727 3265 PD 4398 3712 PD 5256 4201 PD 3938 3284 PD 2371 2384 PD 4269 3585 PD 4411 3695 PD 5298 4144 PD 5004 3949 PD 5269 4205 PD 4687 3808 PD 1584 1789 PD 5181 4097 PD 1501 1455 PD 4971 4016 PD 3923 3364 PD 5240 4173 PD 5016 4055 PD 3858 3353 PD 2914 2714 PD 1594 1806 PD 4126 3538 PD 5822 4464 PD 5857 4501 PD 5110 4018 PD 3549 3158 PD 5569 4354 PD 5848 4498 PD 3570 2994 PD 5108 4001 PD 5635 4384 PD 4259 3523 PD 5230 4156 PD 4334 3573 PD 3444 3058 PD 3825 3225 PD 2498 2439 PD 5130 4115 PD 4464 3703 PD 4624 3771 PD 2639 2577 PD 4600 3789 PD 1924 1808 PD 4587 3706 PD 5088 4011 PD 2646 2541 PD 5223 4167 PD 5434 4291 PD 5167 4062 PD 4327 3619 PD 3771 3188 PD 3056 2843 PD 3697 3105 PD 5614 4381 PD 5533 4331 PD 3807 3169 PD 5146 4142 PD 2401 2190 PD 5768 4458 PD 5578 4333 PD 2326 2331 PD 1121 1466 PD 3612 3224 PD 2107 2216 PD 3389 2828 PD 4895 3941 PD 2910 2710 PD 5379 4262 PD 4955 3909 PD 3158 2730 PD 3310 3029 PD 5517 4326 PD 1611 1845 PD 5420 4284 PD 4691 3725 PD 4812 3925 PD 5537 4340 PD 5011 3967 PD 3900 3363 PD 4434 3692 PD 1573 1503 PD 2475 2158 PD 2512 2228 PD 3697 3047 PD 3373 2916 PD 5266 4198 PD 4175 3432 PD 5538 4336 PD 2930 2754 PD 4084 3390 PD 5803 4476 PD 3959 3283 PD 3593 3189 PD 5134 4115 PD 5649 4379 PD 3995 3285 PD 4523 3784 PD 4445 3692 PD 4595 3705 PD 4462 3572 PD 1124 1478 PD 4869 3988 PD 2713 2426 PD 4485 3738 PD 5387 4224 PD 1976 2072 PD 4929 3994 PD 3670 3239 PD 4099 3525 PD 4533 3650 PD 3184 2944 PD 5255 4186 PD 5218 4160 PD 4269 3631 PD 5594 4370 PD 4827 3858 PD 5478 4312 PD 4267 3534 PD 3596 2985 PD 3133 2666 PD 5879 4497 PD 2025 2139 PD 5430 4266 PD 3678 3087 PD 5712 4429 PD 4910 3902 PD 5402 4275 PD 2602 2303 PD 4584 3800 PD 4070 3398 PD 5433 4276 PD 4331 3481 PD 3925 3364 PD 4955 4032 PD 2603 2349 PD 5425 4285 PD 4995 3946 PD 5674 4411 PD 5275 4139 PD 2340 2355 PD 5120 4089 PD 2158 2205 PD 5658 4403 PD 4776 3835 PD 5194 4168 PD 5573 4361 PD 3773 3102 PD 4988 3922 PD 2341 2077 PD 2210 2255 PD 5752 4437 PD 2035 1809 PD 2852 2526 PD 5450 4293 PD 4560 3670 PD 3232 2794 PD 3573 3032 PD 5333 4240 PD 3899 3256 PD 3014 2822 PD 3870 3227 PD 4553 3756 PD 5260 4193 PD 3444 2937 PD 5002 4014 PD 5250 4180 PD 5902 4517 PD 4240 3492 PD 4963 3959 PD 5494 4278 PD 2438 2388 PD 4331 3525 PD 5054 4079 PD 1582 1543 PD 2383 2121 PD 5087 4105 PD 5238 4191 PD 5420 4273 PD 3297 2828 PD 3511 2986 PD 5497 4307 PD 5579 4360 PD 4980 4036 PD 5318 4224 PD 3937 3400 PD 1983 2081 PD 3428 3049 PD 3790 3195 PD 3753 3252 PD 5340 4231 PD 2409 2120 PD 5151 4059 PD 5372 4256 PD 5047 3998 PD 3953 3433 PD 4341 3551 PD 5425 4215 PD 5523 4281 PD 5275 4211 PD 5576 4362 PD 1751 1935 PD 3621 2998 PD 3190 2698 PD 4019 3446 PD 1362 1663 PD 3330 2977 PD 4944 4023 PD 4540 3791 PD 4477 3745 PD 4750 3774 PD 4961 4037 PD 5092 4056 PD 4668 3777 PD 5649 4399 PD 3466 2913 PD 2938 2546 PD 3288 2803 PD 1636 1506 PD 3548 3131 PD 5194 4109 PD 5370 4180 PD 3148 2895 PD 3875 3218 PD 4664 3866 PD 4141 3542 PD 2094 2166 PD 4771 3929 PD 3569 3159 PD 5685 4416 PD 4875 3861 PD 3981 3420 PD 2384 2376 PD 5656 4398 PD 5060 4004 PD 4745 3910 PD 1470 1691 PD 4193 3563 PD 4145 3532 PD 5088 3987 PD 5182 4114 PD 4318 3651 PD 4094 3490 PD 4978 4039 PD 2707 2411 PD 5399 4216 PD 5893 4506 PD 4764 3847 PD 2263 2244 PD 5729 4436 PD 4776 3808 PD 2794 2617 PD 1377 1654 PD 5130 4102 PD 1909 2004 PD 5778 4437 PD 4941 3979 PD 4448 3615 PD 5498 4323 PD 2107 1966 PD 3692 3274 PD 5491 4310 PD 3944 3249 PD 5079 4095 PD 2847 2488 PD 5452 4273 PD 5557 4340 PD 1967 1823 PD 5290 4166 PD 4973 4044 PD 5421 4255 PD 4697 3858 PD 3578 3058 PD 4866 3973 PD 4468 3753 PD 4954 4014 PD 5650 4394 PD 4962 3947 PD 2541 2456 PD 5369 4255 PD 3939 3388 PD 3547 3181 PD 3302 3007 PD 5362 4241 PD 5279 4208 PD 4385 3690 PD 5202 4138 PD 5439 4293 PD 5419 4278 PD 4698 3810 PD 2854 2684 PD 1497 1722 PD 5844 4492 PD 4708 3819 PD 2041 2086 PD 4086 3505 PD 5192 4107 PD 4584 3738 PD 5372 4250 PD 5115 4093 PD 1855 1985 PD 5511 4308 PD 4616 3685 PD 1178 1536 PD 4378 3699 PD 5597 4350 PD 5250 4162 PD 5223 4185 PD 4351 3574 PD 5234 4172 PD 4582 3657 PD 5355 4250 PD 4442 3643 PD 3398 2886 PD 3977 3274 PD 4988 4027 PD 3174 2871 PD 3387 3014 PD 2609 2255 PD 2851 2708 PD 4956 4034 PD 4496 3622 PD 5463 4289 PD 4799 3942 PD 3666 3220 PD 5563 4352 PD 4947 4011 PD 4891 4000 PD 3076 2695 PD 2716 2334 PD 5750 4448 PD 4369 3588 PD 5339 4238 PD 4993 3956 PD 4289 3602 PD 2131 2147 PD 4790 3898 PD 5444 4294 PD 5088 3996 PD 5151 4146 PD 5108 4088 PD 2778 2631 PD 1350 1596 PD 5386 4267 PD 3099 2881 PD 5189 4071 PD 1601 1548 PD 3864 3347 PD 3894 3379 PD 1672 1865 PD 3989 3264 PD 5692 4419 PD 5167 4134 PD 2421 2130 PD 2163 2195 PD 3429 2857 PD 4995 4052 PD 3204 2701 PD 2430 2432 PD 2393 2140 PD 3498 3118 PD 5705 4422 PD 5034 3981 PD 3295 2991 PD 5236 4116 PD 5866 4488 PD 2982 2785 PD 3601 3164 PD 5429 4254 PD 2367 2391 PD 5174 4103 PD 5548 4345 PD 4981 3965 PD 4358 3509 PD 2065 2172 PD 5287 4207 PD 4865 3980 PD 3406 3039 PD 3941 3306 PD 5590 4360 PD 4319 3499 PD 5294 4221 PD 3389 3039 PD 4776 3854 PD 1665 1850 PD 5359 4226 PD 3361 2899 PD 5243 4164 PD 4616 3799 PD 1303 1584 PD 5545 4335 PD 5386 4265 PD 4459 3658 PD 4558 3721 PD 5832 4491 PD 5328 4238 PD 3418 2870 PD 2587 2469 PD 5559 4328 PD 2555 2484 PD 2045 2151 PD 4376 3555 PD 3962 3441 PD 4278 3606 PD 4104 3338 PD 5138 4116 PD 5012 4022 PD 5255 4200 PD 4514 3609 PD 4250 3458 PD 1174 1540 PD 4025 3472 PD 1302 1559 PD 1416 1673 PD 5464 4253 PD 5080 4105 PD 3095 2835 PD 3274 2937 PD 3867 3385 PD 4939 3895 PD 1237 1551 PD 1383 1633 PD 4614 3823 PD 5276 4209 PD 5116 4021 PD 3449 3114 PD 5028 4046 PD 3248 2979 PD 5367 4243 PD 5631 4387 PD 4748 3845 PD 5871 4499 PD 3784 3111 PD 4893 3903 PD 4386 3691 PD 2841 2507 PD 5554 4319 PD 4453 3737 PD 5569 4326 PD 2509 2458 PD 1928 2076 PD 2925 2706 PD 4491 3603 PD 1757 1892 PD 5466 4307 PD 2128 1925 PD 3015 2792 PD 5201 4139 PD 3000 2798 PD 1803 1655 PD 2248 2236 PD 5315 4231 PD 1924 2043 PD 4433 3683 PD 2417 2429 PD 5309 4210 PD 4634 3690 PD 4010 3274 PD 5470 4308 PD 5171 4139 PD 3896 3383 PD 1504 1711 PD 3983 3337 PD 5153 4070 PD 5887 4512 PD gs 899 388 5359 4226 MR c np gr 6 w end eplot %%EndObject graph 1 epage end showpage %%Trailer %%EOF %%EndDocument @endspecial 24 2581 a Fp(Figure)25 b(4:)40 b(2D)25 b(pro)5 b(jection)26 b(of)g(a)g(3D)f(P)m(oincare)h(section)h Fl(u)g Fp(=)g(0)f(for)g(the)h(same)f(parameter)f(v)-5 b(alues)24 2701 y(as)32 b(in)g(Figure)g(2)p 170 2857 2688 4 v 168 2977 4 121 v 1569 2977 V 2856 2977 V 168 3097 V 504 3061 a(P)m(arameter)g(v)-5 b(alues)p 1569 3097 V 547 w(Ly)m(apuno)m(v)34 b(Exp)s(onen)m(t)p 2856 3097 V 168 3218 V 1569 3218 V 2856 3218 V 170 3221 2688 4 v 170 3238 V 168 3358 4 121 v 1569 3358 V 2856 3358 V 168 3478 V 259 3442 a Fl(p)308 3457 y Fk(1)375 3442 y Fp(=)27 b(0)p Fl(:)p Fp(53)32 b Fl(p)733 3457 y Fk(2)800 3442 y Fp(=)c(0)p Fl(:)p Fp(1)k Fl(p)1110 3457 y Fk(3)1177 3442 y Fp(=)27 b Fj(\000)p Fp(1)p Fl(:)p Fp(6)p 1569 3478 V 530 w Fl(L)2078 3457 y Fk(1)2118 3442 y Fp(=0.017)p 2856 3478 V 168 3599 V 1569 3599 V 2856 3599 V 168 3719 V 1569 3719 V 2856 3719 V 170 3723 2688 4 v 168 3843 4 121 v 1569 3843 V 2856 3843 V 168 3963 V 220 3927 a Fl(p)269 3942 y Fk(1)336 3927 y Fp(=)g(0)p Fl(:)p Fp(23)32 b Fl(p)694 3942 y Fk(2)761 3927 y Fp(=)c Fj(\000)p Fp(0)p Fl(:)p Fp(2)33 b Fl(p)1149 3942 y Fk(3)1216 3927 y Fp(=)27 b Fj(\000)p Fp(1)p Fl(:)p Fp(6)p 1569 3963 V 516 w Fl(L)2103 3942 y Fk(1)2143 3927 y Fp(=0.34)p 2856 3963 V 168 4084 V 1569 4084 V 2856 4084 V 168 4204 V 1569 4204 V 2856 4204 V 170 4207 2688 4 v 168 4328 4 121 v 1569 4328 V 2856 4328 V 168 4448 V 220 4412 a Fl(p)269 4427 y Fk(1)336 4412 y Fp(=)g(0)p Fl(:)p Fp(33)32 b Fl(p)694 4427 y Fk(2)761 4412 y Fp(=)c Fj(\000)p Fp(0)p Fl(:)p Fp(1)33 b Fl(p)1149 4427 y Fk(3)1216 4412 y Fp(=)27 b Fj(\000)p Fp(1)p Fl(:)p Fp(5)p 1569 4448 V 516 w Fl(L)2103 4427 y Fk(1)2143 4412 y Fp(=0.72)p 2856 4448 V 168 4568 V 1569 4568 V 2856 4568 V 168 4689 V 1569 4689 V 2856 4689 V 170 4692 2688 4 v 168 4813 4 121 v 1569 4813 V 2856 4813 V 168 4933 V 259 4897 a Fl(p)308 4912 y Fk(1)375 4897 y Fp(=)g(0)p Fl(:)p Fp(73)32 b Fl(p)733 4912 y Fk(2)800 4897 y Fp(=)c(0)p Fl(:)p Fp(3)k Fl(p)1110 4912 y Fk(3)1177 4897 y Fp(=)27 b Fj(\000)p Fp(1)p Fl(:)p Fp(1)p 1569 4933 V 555 w Fl(L)2103 4912 y Fk(1)2143 4897 y Fp(=0.69)p 2856 4933 V 168 5053 V 1569 5053 V 2856 5053 V 168 5174 V 1569 5174 V 2856 5174 V 170 5177 2688 4 v 168 5297 4 121 v 1569 5297 V 2856 5297 V 168 5418 V 259 5382 a Fl(p)308 5397 y Fk(1)375 5382 y Fp(=)g(1)p Fl(:)p Fp(03)32 b Fl(p)733 5397 y Fk(2)800 5382 y Fp(=)c(0)p Fl(:)p Fp(6)k Fl(p)1110 5397 y Fk(3)1177 5382 y Fp(=)27 b Fj(\000)p Fp(0)p Fl(:)p Fp(8)p 1569 5418 V 555 w Fl(L)2103 5397 y Fk(1)2143 5382 y Fp(=4.45)p 2856 5418 V 168 5538 V 1569 5538 V 2856 5538 V 168 5659 V 1569 5659 V 2856 5659 V 170 5662 2688 4 v 1793 5214 a(9)p eop %%Page: 10 10 10 9 bop 24 631 a Fm(3)215 b(Bifurcation)54 b(analysis)24 850 y Fp(In)40 b(order)g(to)g(understand)h(the)g(c)m(hange)g(in)e(the)h (orbit)f(structure)i(of)f(the)g(\015o)m(w)h(\(1.1\))f(one)g(can)24 970 y(restrict)27 b(atten)m(tion)g(to)g(a)h(small)d(neigh)m(b)s(ourho)s (o)s(d)h(of)i(those)g(parameter)f(v)-5 b(alues)27 b(for)g(whic)m(h)h (there)24 1091 y(is)i(a)g(non)h(h)m(yp)s(erb)s(olic)g(equilibrium)c(p)s (oin)m(t.)42 b(The)32 b(n)m(umerical)d(sim)m(ulations)g(induce)i(us)g (to)g(study)24 1211 y(the)36 b(dynamics)g(with)g(the)g(use)h(of)f(fast) g(slo)m(w)g(tec)m(hniques.)56 b(In)36 b(fact,)h(w)m(e)h(will)33 b(sho)m(w)38 b(that)d(there)24 1332 y(are)40 b(slo)m(w)g(manifolds)e (that)i(lo)s(ose)f(normal)f(h)m(yp)s(erb)s(olicit)m(y)-8 b(.)66 b(As)41 b(the)f(parameters)g(are)h(v)-5 b(aried,)24 1452 y(bifurcations)41 b(tak)m(e)i(place)f(in)f(whic)m(h)i(the)g(real)e (part)i(of)e(the)i(eigen)m(v)-5 b(alues)43 b(asso)s(ciated)f(to)g(the) 24 1572 y(b)s(eha)m(viour)g(normal)f(to)h(the)h(in)m(v)-5 b(arian)m(t)42 b(planes)g(cross)i(zero.)74 b(The)44 b(corresp)s(onding) f(in)m(v)-5 b(arian)m(t)24 1693 y(manifolds)30 b(are)i(the)h(core)g(of) f(a)h(c)m(haotic)f(in)m(v)-5 b(arian)m(t)31 b(set.)24 1980 y Fe(3.1)180 b(The)44 b(bifurcated)h(equilibrium)h(p)t(oin)l(ts)24 2164 y Fi(Prop)s(osition)35 b(3.1)49 b Fq(The)34 b(non)h(trivial)f(e)-5 b(quilibria)35 b(for)f(the)h(system)g(\(1.1\))f(ar)-5 b(e)35 b(as)f(fol)5 b(lows)34 b(:)139 2376 y(1.)48 b(If)35 b Fl(u)27 b Fp(=)h(0)p Fq(,)34 b(then)h Fl(X)976 2340 y Ff(\006)1062 2376 y Fp(=)63 b(\()p Fj(\006)1316 2277 y Fd(q)p 1399 2277 135 4 v 99 x Fj(\000)1487 2337 y Fk(1)p 1486 2353 38 4 v 1486 2410 a Fh(a)1534 2376 y Fl(;)17 b Fp(0)p Fl(;)g Fp(0)p Fl(;)g Fp(0\))p Fq(,)33 b Fl(Y)1992 2340 y Ff(\006)2079 2376 y Fp(=)63 b(\(0)p Fl(;)17 b Fj(\006)2426 2277 y Fd(q)p 2508 2277 135 4 v 2508 2376 a Fj(\000)2596 2337 y Fk(1)p 2595 2353 38 4 v 2595 2410 a Fh(a)2643 2376 y Fl(;)g Fp(0)p Fl(;)g Fp(0\))267 2537 y Fl(Z)7 b Fj(\006)29 b Fp(=)62 b(\(0)p Fl(;)17 b Fp(0)p Fl(;)g Fj(\006)886 2438 y Fd(q)p 968 2438 135 4 v 968 2537 a Fj(\000)1056 2498 y Fk(1)p 1055 2514 38 4 v 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2111 3393 214 4 v 2111 3485 a(y)e Fp(+)d Fl(c)2334 3416 y Fp(\))p Fl(:)170 3681 y Fp(After)33 b(simpli\014cations)c(the)k(appro)m(ximated)f(P)m (oincar)m(\023)-46 b(e)33 b(map)e(reads:)993 3935 y Fl(F)1056 3950 y Fk(0)1096 3935 y Fp(\()p Fl(y)t Fp(\))26 b(=)i Fl(A)1427 3893 y Fk(\(1)p Ff(\000)p Fh(k)r Fk(\))1615 3935 y Fp(\()p Fl(y)d Fp(+)d Fl(c)p Fp(\))1904 3893 y Fh(k)1963 3935 y Fp(cos)q(\()p Fl(!)e Fp(ln)o(\()2342 3867 y Fl(y)25 b Fp(+)d Fl(c)p 2342 3911 V 2412 4003 a(A)2565 3935 y Fp(\)\))24 4134 y(where)1143 4255 y Fl(k)31 b Fp(=)1339 4187 y Fl(\026)1398 4202 y Fk(1)p 1339 4231 99 4 v 1339 4323 a Fl(\026)1398 4338 y Fk(2)1457 4187 y Fl(!)1518 4202 y Fk(2)p 1457 4231 101 4 v 1457 4323 a Fl(!)1518 4338 y Fk(1)1594 4255 y Fl(>)d Fp(0)65 b(and)97 b Fl(!)31 b Fp(=)d(2)2321 4187 y Fl(!)2382 4202 y Fk(2)p 2321 4231 V 2322 4323 a Fl(\026)2381 4338 y Fk(2)2463 4255 y Fl(:)24 4488 y Fp(T)-8 b(aking)38 b(^)-55 b Fl(y)25 b Fp(+)d Fl(c)28 b Fp(=)703 4445 y Fh(y)r Fk(+)p Fh(c)p 703 4464 123 4 v 738 4522 a(A)868 4488 y Fp(and)33 b(dropping)f(the)h (hat)f(the)h(\014nal)f(expression)i(yields:)1083 4728 y Fl(F)1146 4743 y Fk(0)1186 4728 y Fp(\()p Fl(y)t Fp(\))27 b(=)g Fl(A)p Fp(\()p Fl(y)e Fp(+)d Fl(c)p Fp(\))1806 4687 y Fh(k)1866 4728 y Fp(cos\()p Fl(!)e Fp(ln)o(\()p Fl(y)26 b Fp(+)c Fl(c)p Fp(\)\))p Fl(:)1768 5214 y Fp(18)p eop %%Page: 19 19 19 18 bop 24 631 a Fi(Prop)s(osition)35 b(5.1)49 b Fq(:)170 799 y(L)-5 b(et)44 b Fp(\003)h(=)g Fj(f)p Fl(p)g Fj(2)g Fp(\006)906 814 y Fk(1)975 799 y Fj([)29 b Fp(\006)1140 814 y Fk(2)1225 799 y Fp(:)90 b Fl(F)1419 762 y Fh(n)1405 823 y Fk(0)1465 799 y Fp(\()p Fl(p)p Fp(\))45 b Fj(2)g Fp(\006)1816 814 y Fk(1)1885 799 y Fj([)30 b Fp(\006)2051 814 y Fk(2)2135 799 y Fq(for)35 b(e)-5 b(ach)34 b(n)45 b Fj(2)g Fi(N)p Fj(g)f Fq(that)g(is,)i Fp(\003)e Fq(is)g(the)24 919 y(maximal)33 b(invariant)h(set)h(of)g(the)g(map)f Fl(F)1545 934 y Fk(0)1619 919 y Fq(r)-5 b(estricte)g(d)35 b(to)g Fp(\006)2232 934 y Fk(1)2294 919 y Fj([)23 b Fp(\006)2453 934 y Fk(2)2493 919 y Fq(.)45 b(Then)139 1100 y(1.)j Fp(\003)35 b Fq(is)g(a)f(Cantor)h(set.)139 1296 y(2.)48 b Fl(P)14 b(er)s Fp(\()p Fl(F)537 1311 y Fk(0)604 1296 y Fj(j)632 1311 y Fk(\003)685 1296 y Fp(\))35 b Fq(ar)-5 b(e)34 b(dense)g(in)h Fp(\003)p Fq(.)139 1492 y(3.)48 b Fl(F)330 1507 y Fk(0)405 1492 y Fq(is)35 b(tr)-5 b(ansitive)34 b(on)h Fp(\003)p Fq(.)24 1733 y Fi(Pro)s(of)48 b Fp(:)170 1854 y(The)37 b(zero)s(es)h(of)e Fl(F)847 1869 y Fk(0)886 1854 y Fp(\()p Fl(y)t Fp(\))g(are)g Fl(y)1268 1817 y Fh(m)1264 1878 y Fk(0)1369 1854 y Fp(=)e Fl(e)1535 1790 y Fg(\031)p 1534 1802 41 4 v 1534 1843 a(!)1585 1817 y Fk(\(1)p Fh(=)p Fk(2+)p Fh(m)p Fk(\))1892 1854 y Fj(\000)25 b Fl(c)p Fp(,)37 b Fl(m)e Fj(2)g Fl(N)10 b Fp(.)56 b(Without)36 b(loss)g(of)g(generalit)m(y)24 1983 y(w)m(e)d(can)g(restrict)g(our)f (analysis)g(to)g(the)h(in)m(terv)-5 b(al)31 b Fj(I)1916 1998 y Fk(0)1984 1983 y Fp(=)c([)33 b Fl(y)2199 1947 y Fk(0)2195 2007 y(0)2270 1983 y Fl(;)49 b(y)2398 1947 y Fk(1)2394 2007 y(0)2469 1983 y Fp(])28 b(=)g([)p Fl(e)2726 1919 y Fg(\031)p 2710 1931 72 4 v 2710 1972 a Fc(2)p Fg(!)2817 1983 y Fj(\000)23 b Fl(c;)17 b(e)3059 1919 y Fc(3)p Fg(\031)p 3058 1931 V 3058 1972 a Fc(2)p Fg(!)3165 1983 y Fj(\000)23 b Fl(c)p Fp(].)170 2103 y(No)m(w)33 b(compute)495 2297 y Fl(F)558 2312 y Fk(0)597 2297 y Fp(\()p Fl(y)t Fp(\))725 2256 y Ff(0)831 2297 y Fp(=)83 b Fl(A)p Fp(\()p Fl(y)25 b Fp(+)d Fl(c)p Fp(\))1352 2256 y Fh(k)r Ff(\000)p Fk(1)1485 2297 y Fp(\()p Fl(k)d Fp(cos)q(\()p Fl(!)h Fp(ln)o(\()p Fl(y)25 b Fp(+)d Fl(c)p Fp(\)\))g Fj(\000)h Fl(!)d Fp(sin\()p Fl(!)g Fp(ln)o(\()p Fl(y)25 b Fp(+)d Fl(c)p Fp(\)\)\))831 2442 y(=)83 b Fl(A)p Fp(\()p Fl(y)25 b Fp(+)d Fl(c)p Fp(\))1352 2401 y Fh(k)r Ff(\000)p Fk(1)1485 2356 y Fj(p)p 1568 2356 318 4 v 86 x Fl(k)1622 2414 y Fk(2)1683 2442 y Fp(+)h Fl(!)1847 2414 y Fk(2)1902 2442 y Fp(cos\()p Fl(!)d Fp(ln\()p Fl(y)25 b Fp(+)d Fl(c)p Fp(\))g(+)g Fl(\022)s Fp(\))170 2648 y(where)35 b Fl(\022)d Fp(=)d(arctan)933 2608 y Fh(!)p 933 2624 47 4 v 937 2682 a(k)1018 2648 y Fp(=)g(arctan\(2)1491 2608 y Fh(!)1535 2617 y Fc(1)p 1491 2624 79 4 v 1492 2682 a Fh(\026)1534 2691 y Fc(1)1579 2648 y Fp(\),)34 b(no)m(w)g Fl(F)1945 2663 y Fk(0)1985 2648 y Fp(\()p Fl(y)2075 2611 y Ff(\003)2113 2648 y Fp(\))2151 2611 y Ff(0)2204 2648 y Fp(=)29 b(0)g Fj(\))f Fl(y)2567 2611 y Ff(\003)2635 2648 y Fp(=)h Fl(e)2785 2611 y Fk(3)2847 2584 y Fg(\031)p 2831 2596 72 4 v 2831 2637 a Fc(2)p Fg(!)2912 2611 y Ff(\000)2982 2584 y Fg(\022)p 2977 2596 41 4 v 2977 2637 a(!)3054 2648 y Fj(\000)23 b Fl(c)p Fp(.)47 b(Because)34 2729 y Fh(\031)p 34 2745 43 4 v 38 2802 a Fk(2)124 2768 y Fl(>)38 b(\022)j(>)c Fp(0,)j Fl(y)605 2732 y Ff(\003)682 2768 y Fj(2)e(I)46 b Fp(is)38 b(the)h(unique)g(p)s(oin)m(t)e(in)h Fj(I)1921 2783 y Fk(0)1999 2768 y Fp(where)i(the)f(deriv)-5 b(ativ)m(e)38 b(v)-5 b(anishes.)62 b(Hence)24 2888 y Fl(F)87 2903 y Fk(0)126 2888 y Fp(\()p Fl(y)t Fp(\))23 b(is)g(strictly)g(monotone)g (increasing)g(on)g(the)h(in)m(terv)-5 b(al)23 b([)p Fl(e)2306 2824 y Fg(\031)p 2290 2836 72 4 v 2290 2878 a Fc(2)p Fg(!)2379 2888 y Fj(\000)t Fl(c;)17 b(y)2598 2852 y Ff(\003)2637 2888 y Fp([)23 b(and)h(strictly)f(monotone)24 3018 y(decreasing)33 b(on)f(the)h(in)m(terv)-5 b(al)31 b(])p Fl(y)1230 2981 y Ff(\003)1269 3018 y Fl(;)17 b(e)1369 2954 y Fc(3)p Fg(\031)p 1368 2966 V 1368 3007 a Fc(2)p Fg(!)1475 3018 y Fj(\000)23 b Fl(c)p Fp(])p Fl(:)170 3138 y Fp(Next)33 b(compute)g(that)1237 3269 y Fj(j)p Fl(F)1328 3284 y Fk(0)1367 3269 y Fp(\()p Fl(y)1457 3227 y Ff(\003)1495 3269 y Fp(\))p Fj(j)28 b Fp(=)f Fl(Ae)1823 3200 y Fg(k)p 1820 3212 41 4 v 1820 3253 a(!)1871 3227 y Fk(\()1908 3200 y Fc(3)p Fg(\031)p 1908 3212 69 4 v 1927 3253 a Fc(2)1987 3227 y Ff(\000)p Fh(\022)r Fk(\))2141 3269 y Fj(j)17 b Fp(sin)e Fl(\022)s Fj(j)24 3432 y Fp(w)m(e)34 b(claim)c(that)j Fj(j)p Fl(F)731 3447 y Fk(0)770 3432 y Fp(\()p Fl(y)860 3396 y Ff(\003)899 3432 y Fp(\))p Fj(j)28 b Fl(>)g(L)33 b Fp(where)i Fl(L)28 b Fp(=)h Fl(e)1749 3368 y Fg(\031)p 1733 3380 72 4 v 1733 3421 a Fc(2)p Fg(!)1818 3432 y Fp(\()p Fl(e)1901 3396 y Fh(\031)1971 3432 y Fj(\000)23 b Fp(1\))32 b(denotes)j(the)e(length)f(of)h(the)g(in) m(terv)-5 b(al)24 3563 y Fj(I)78 3578 y Fk(0)117 3563 y Fp(.)56 b(First)36 b(notice)g(that)h Fl(Ae)1080 3499 y Fg(k)p 1077 3511 41 4 v 1077 3553 a(!)1128 3527 y Fk(\(3)1200 3499 y Fg(\031)p 1201 3511 38 4 v 1205 3553 a Fc(2)1248 3527 y Ff(\000)p Fh(\022)r Fk(\))1407 3563 y Fj(j)17 b Fp(sin)e Fl(\022)s Fj(j)35 b Fl(>)f(Ae)1939 3499 y Fg(k)p 1937 3511 41 4 v 1937 3553 a(!)1997 3499 y(\031)p 1997 3511 38 4 v 2001 3553 a Fc(2)2086 3563 y Fj(j)17 b Fp(sin)f Fl(\022)s Fj(j)p Fp(,)37 b(hence)i(w)m(e)e(need)h(to)e(sho)m (w)i(that)24 3695 y Fl(Ae)155 3631 y Fg(k)p 152 3643 41 4 v 152 3684 a(!)212 3631 y(\031)p 212 3643 38 4 v 216 3684 a Fc(2)299 3695 y Fj(j)17 b Fp(sin)f Fl(\022)s Fj(j)64 b Fl(>)h(e)833 3631 y Fg(\031)p 816 3643 72 4 v 816 3684 a Fc(2)p Fg(!)902 3695 y Fp(\()p Fl(e)985 3659 y Fh(\031)1055 3695 y Fj(\000)24 b Fp(1\),)34 b(that)h(is)e Fl(Ae)1735 3659 y Fh(k)1813 3695 y Fj(j)17 b Fp(sin)e Fl(\022)s Fj(j)65 b Fl(>)g Fp(\()p Fl(e)2358 3659 y Fh(\031)2429 3695 y Fj(\000)23 b Fp(1\))34 b(.)49 b(Replacing)33 b(terms)h(this)24 3816 y(means)e(that)1460 3892 y Fl(A)p 1243 3936 508 4 v 1243 3953 a Fd(q)p 1326 3953 425 4 v 90 x Fp(1)22 b(+)g(\()1565 4001 y Fh(\026)1607 4010 y Fc(1)p 1543 4020 121 4 v 1543 4078 a Fk(2)p Fh(w)1629 4087 y Fc(1)1673 4043 y Fp(\))1711 4015 y Fk(2)1789 3959 y Fl(>)27 b(e)1937 3918 y Ff(\000)p Fh(k)2035 3959 y Fp(\()p Fl(e)2118 3918 y Fh(\031)2187 3959 y Fj(\000)c Fp(1\))p Fl(:)1009 b Fp(\(5.4\))24 4247 y(Thanks)28 b(to)f Fq(Hyp)-5 b(othesis)29 b(2)e Fp(this)f(inequalit)m(y)g(will)f(b)s(e)i(satis\014ed)g(as)g(long) f(as)h Fl(A)h(>)f(e)3028 4211 y Fh(\031)3075 4156 y Fd(q)p 3158 4156 425 4 v 91 x Fp(1)22 b(+)g(\()3397 4204 y Fh(\026)3439 4213 y Fc(1)p 3375 4224 121 4 v 3375 4282 a Fk(2)p Fh(w)3461 4291 y Fc(1)3506 4247 y Fp(\))3544 4218 y Fk(2)3583 4247 y Fp(.)170 4396 y(No)m(w,)48 b(there)e(exists)f(t)m(w)m(o)g(p)s(oin)m (ts)53 b(\026)-57 b Fl(p)1528 4411 y Fh(i)1556 4396 y Fl(;)17 b(i)48 b Fp(=)g(1)p Fl(;)17 b Fp(2)44 b(satisfying)g Fl(F)2501 4411 y Fk(0)2540 4396 y Fp(\()8 b(\026)-57 b Fl(p)2627 4411 y Fh(i)2655 4396 y Fp(\))49 b(=)f Fl(e)2937 4332 y Fg(\031)p 2921 4344 72 4 v 2921 4386 a Fc(2)p Fg(!)3006 4396 y Fp(\()p Fl(e)3089 4360 y Fh(\031)3167 4396 y Fj(\000)31 b Fp(1\).)79 b(Let)24 4526 y(us)43 b(denote)g Fj(I)537 4541 y Fk(1)621 4526 y Fp(=)h([)p Fl(e)840 4462 y Fg(\031)p 823 4474 V 823 4515 a Fc(2)p Fg(!)937 4526 y Fj(\000)30 b Fl(c;)36 b Fp(\026)-68 b Fl(p)1179 4541 y Fk(1)1218 4526 y Fp(])42 b(and)h Fj(I)1541 4541 y Fk(2)1625 4526 y Fp(=)h([)20 b(\026)-69 b Fl(p)1821 4541 y Fk(2)1861 4526 y Fl(;)17 b(e)1961 4462 y Fc(3)p Fg(\031)p 1959 4474 V 1959 4515 a Fc(2)p Fg(!)2074 4526 y Fj(\000)29 b Fl(c)p Fp(].)73 b(Finally)40 b(let)h Fj(I)52 b Fp(=)45 b Fj([)3134 4541 y Fh(i)p Fk(=1)p Fh(;)p Fk(2)3307 4526 y Fj(I)3361 4541 y Fh(i)3390 4526 y Fp(.)73 b(By)24 4646 y(construction)32 b(w)m(e)i(ha)m(v)m(e)g Fj(I)h(\032)28 b Fl(F)1208 4661 y Fk(0)1248 4646 y Fp(\()p Fj(I)1340 4661 y Fh(i)1368 4646 y Fp(\))33 b Fl(i)28 b Fp(=)f(1)p Fl(;)17 b Fp(2)32 b(for)g(all)f Fl(y)f Fj(2)e(I)i(\\)23 b Fl(F)2484 4605 y Ff(\000)p Fk(1)2470 4668 y(0)2578 4646 y Fp(\()p Fj(I)7 b Fp(\).)24 4844 y Fi(Lemma)37 b(5.2)49 b Fq(Under)41 b(hyp)-5 b(othesis)41 b(H\(1\)-H\(2\),)i(ther)-5 b(e)41 b(exists)g(a)h(r)-5 b(e)g(al)41 b(numb)-5 b(er)41 b Fl(m)f Fj(\025)h Fp(1)p Fq(,)h(such)24 4965 y(that)35 b Fj(j)p Fl(F)314 4980 y Fk(0)353 4965 y Fp(\()p Fl(y)t Fp(\))481 4929 y Ff(0)503 4965 y Fj(j)28 b Fl(>)f(m)36 b Fq(for)e(al)5 b(l)35 b Fl(y)30 b Fj(2)f(I)1329 4898 y Fd(T)1415 4965 y Fl(F)1492 4923 y Ff(\000)p Fk(1)1478 4986 y(0)1586 4965 y Fp(\()p Fj(I)7 b Fp(\))p Fl(:)1768 5214 y Fp(19)p eop %%Page: 20 20 20 19 bop 24 631 a Fi(Pro)s(of)48 b Fp(:)316 954 y Fl(F)379 969 y Fk(0)419 954 y Fp(\()p Fl(y)t Fp(\))547 912 y Ff(00)616 954 y Fp(=)27 b Fl(A)792 867 y Fj(p)p 875 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Fp(\(6.5\))1768 5214 y(22)p eop %%Page: 23 23 23 22 bop 24 631 a Fi(Lemma)37 b(6.3)49 b Fq(:)170 799 y(Assume)38 b Fl(\015)588 814 y Fk(0)627 799 y Fp(\()p Fl(s)p Fp(\))g Fq(is)g(a)g Fj(C)1041 762 y Fh(r)1117 799 y Fq(\()p Fl(r)e Fj(\025)f Fp(3)p Fq(\))i(curve)i(with)e Fj(j)p Fl(\015)2040 739 y Fb(0)2035 823 y Fk(0)2074 799 y Fp(\()p Fl(s)p Fp(\))p Fj(j)c(\024)i Fl(\024)j Fq(wher)-5 b(e)37 b Fl(\024)i Fq(is)f(a)g(smal)5 b(l)37 b(p)-5 b(ositive)24 919 y(c)g(onstant,)34 b(then)h Fl(\015)715 859 y Fb(0)710 944 y Fk(1)749 919 y Fp(\()p Fl(s)p Fp(\))f Fq(is)h(also)f Fj(C)1267 883 y Fh(r)1340 919 y Fq(and)h(ob)-5 b(eys)34 b Fj(j)p Fl(\015)1870 859 y Fb(0)1865 944 y Fk(1)1904 919 y Fp(\()p Fl(s)p Fp(\))p Fj(j)27 b(\024)h Fl(\024)35 b Fq(for)g Fl(s)28 b Fj(2)g(I)7 b Fq(.)24 1188 y Fi(Pro)s(of)71 b Fp(W)-8 b(e)23 b(will)e(denote)i Fl(G)1081 1203 y Fh(s;t)1186 1188 y Fp(and)g Fl(F)1429 1203 y Fh(s;t)1534 1188 y Fp(the)g(partial)e (deriv)-5 b(ativ)m(e)22 b(with)g(resp)s(ect)j(to)d(the)h(v)-5 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b(some)g(constan)m(ts)i Fl(c)891 2287 y Fk(1)930 2272 y Fl(;)17 b(c)1016 2287 y Fk(2)1056 2272 y Fl(;)g(c)1142 2287 y Fk(3)1181 2272 y Fp(.)170 2393 y(Using)35 b(the)i(estimates)e (\(6.7\))g(and)h(\(6.9\))f(w)m(e)i(ha)m(v)m(e)g(that)f(the)g(map)f Fl(\015)2753 2357 y Fh(\033)2793 2365 y Fg(n)2748 2417 y Fh(n)2840 2393 y Fp(\()p Fl(s)p Fp(\))e Fj(!)g Fl(\015)3184 2342 y Fh(\033)3224 2351 y Fg(n)p Fc(+1)3179 2415 y Fh(n)p Fk(+1)3348 2393 y Fp(\()p Fl(s)3432 2408 y Fk(1)3471 2393 y Fp(\))j(is)24 2513 y(a)31 b(con)m(traction)g(in)g Fj(C)786 2477 y Fk(1)826 2513 y Fp(.)43 b(It)32 b(then)g(follo)m(ws)e (in)h(a)g(standard)h(w)m(a)m(y)h(that)f(the)g(iterates)f(will)e(con)m (v)m(erge)24 2634 y(to)k(a)h(unique)g(solution)e(of)i(the)g(functional) e(equation)i(\(6.5\).)47 b(It)34 b(can)g(b)s(e)g(seen)h(from)e(\(6.8\)) g(that)24 2754 y(higher)e(deriv)-5 b(ativ)m(es)31 b(are)h(estimated)f (in)g(the)h(same)f(w)m(a)m(y)i(and)f(w)m(e)g(obtain)f(a)g Fj(C)2903 2718 y Fk(3)2974 2754 y Fp(in)m(v)-5 b(arian)m(t)31 b(curv)m(e)24 2874 y(for)h(the)h(2-D)e(mapping.)42 b(By)33 b(construction)g(this)f(curv)m(e)i(satis\014es:)952 3094 y Fj(H)q Fp(\()p Fl(s;)17 b(\015)5 b Fp(\()p Fl(s)p Fp(\)\))60 b(=)28 b(\()p Fl(F)14 b Fp(\()p Fl(s;)j(\015)5 b Fp(\()p Fl(s)p Fp(\)\))p Fl(;)17 b(\015)25 b Fj(\016)d Fl(F)14 b Fp(\()p Fl(s;)j(\015)5 b Fp(\()p Fl(s)p Fp(\)\)\))p Fl(:)24 3314 y Fp(This)32 b(ends)i(the)f(pro)s(of)f(of)g(the)h(Theorem) g(\(6.2\).)p 3568 3314 43 61 v 170 3506 a(Using)41 b(this)f(in)m(v)-5 b(arian)m(t)40 b(curv)m(e)i Fl(\015)5 b Fp(\()p Fl(s)p Fp(\),)43 b(w)m(e)f(ha)m(v)m(e)g(that)f(the)g(restricted)h(dynamics)f (is)f(of)h(the)24 3626 y(form)1256 3746 y Fl(s)27 b Fj(7!)h Fl(F)14 b Fp(\()p Fl(s)p Fp(\))21 b(+)h Fl(")1821 3705 y Fk(2)p Fh(=)p Fk(3)1953 3721 y Fp(~)1931 3746 y Fl(F)14 b Fp(\()p Fl(s;)j(\015)5 b Fp(\()p Fl(s)p Fp(\)\))p Fl(:)24 3921 y Fp(Since)41 b(the)h(p)s(erturbation)f(is)g Fj(C)1210 3885 y Fk(3)1291 3921 y Fp(one)g(can)h(compute)f(the)h(Sc)m(h)m(w)m (artzian)h(deriv)-5 b(ativ)m(e)41 b(whic)m(h)h(is)24 4041 y(clearly)36 b(negativ)m(e)i(when)h Fl(")f Fp(is)f(su\016cien)m (tly)h(small.)57 b(This)38 b(pro)m(v)m(es)h(the)f(existence)i(of)d(a)g (c)m(haotic)24 4161 y(motion)30 b(in)i(this)g(class)h(of)f(mo)s(del.) 170 4282 y(Moreo)m(v)m(er)25 b(these)f(orbits)f(are)g(con\014ned)i(in)d (a)h(exp)s(onen)m(tially)g(small)e(region)h(around)h(the)h(orbit)24 4402 y(of)35 b(the)g(singular)f(system.)53 b(This)35 b(do)s(es)h(not)f(preclude)h(the)g(existence)h(of)e(a)g(more)g (complicated)24 4523 y(in)m(v)-5 b(arian)m(t)25 b(c)m(haotic)h(set.)42 b(In)27 b(fact)f(the)h(enco)s(ding)f(of)f(the)i(in)m(tersection)g(of)e (these)j(in)m(v)-5 b(arian)m(t)25 b(curv)m(es)24 4643 y(with)36 b(the)h Fl(t)p Fp(-axis,)h Fl(t)d Fp(=)g Fl(\015)958 4607 y Fh(\033)998 4615 y Fg(n)953 4668 y Fh(n)1045 4643 y Fp(\()p Fl(s)p Fp(\))h(pro)m(vides)i(a)e(w)m(a)m(y)i(to)f(describ)s (e)g(in)f(detail)g(the)h(foliation)c(of)j(this)24 4763 y(unstable)28 b(attractor.)42 b(A)28 b(complete)g(study)i(of)e(this)g (set)h(is)f(b)s(ey)m(ond)h(the)g(scop)s(e)h(of)e(this)g(w)m(ork)h(and) 24 4884 y(is)j(the)h(sub)5 b(ject)34 b(of)e(a)g(forthcoming)f(pap)s (er.)1768 5214 y(24)p eop %%Page: 25 25 25 24 bop 24 631 a Fm(References)72 850 y Fp([1])49 b(F.H)35 b(Busse.)h(K.E)f(Heik)m(es)h Fq(Conve)-5 b(ction)35 b(in)i(a)f(r)-5 b(otating)37 b(layer:)49 b(A)37 b(simple)f(c)-5 b(ase)36 b(of)g(turbu-)224 970 y(lenc)-5 b(e.)32 b Fp(Science,)h(v)m(ol)f(208.)g (\(1980\))72 1174 y([2])49 b(P)-8 b(.)33 b(Chossat,)h(M.)f(Krupa,)g(I.) g(Melb)s(ourne,)g(A.)g(Sc)m(heel,)g(P)-8 b(.)33 b(Chossat.)h Fq(T)-7 b(r)i(ansverse)34 b(bifur)-5 b(c)g(a-)224 1294 y(tions)35 b(of)f(homo)-5 b(clinic)33 b(cycles)p Fp(.)g(Ph)m(ysica)g(D) f Fi(100)p Fp(,)h(85-100.)e(\(1997\))72 1498 y([3])49 b(B.)d(Deng.)g Fq(Spir)-5 b(al-Plus-Sadd)5 b(le)45 b(attr)-5 b(actors)48 b(and)e(elementary)h(me)-5 b(chanisms)45 b(for)i(chaos)224 1618 y(gener)-5 b(ation.)32 b Fp(Journal)f(of)h (Bifurcation)f(and)i(Chaos.)g(V)-8 b(ol.6,)31 b(No.)p Fi(3)p Fp(.)i(pp)g(513-527.)e(\(1996\))72 1821 y([4])49 b(J.)30 b(Guc)m(k)m(enheimer.)g Fq(T)-7 b(owar)i(ds)30 b(a)i(glob)-5 b(al)31 b(mo)-5 b(del)31 b(the)-5 b(ory)33 b(of)e(singularly)h(p)-5 b(erturb)g(e)g(d)32 b(dynami-)224 1942 y(c)-5 b(al)31 b(systems.)e Fp(Progress)h(in)f(Nonlinear)e (Di\013eren)m(tial)g(Equations)i(and)h(their)e(applications,)224 2062 y(v)m(ol.)k Fi(19)p Fp(,)h(213-225.)e(\(1996\))72 2266 y([5])49 b(J.)30 b(Guc)m(k)m(enheimer,)h(P)-8 b(.)30 b(Holmes.)f Fq(Structur)-5 b(al)5 b(ly)34 b(stable)e(heter)-5 b(o)g(clinic)31 b(cycles.)e Fp(Math.)h(Pro)s(c.)224 2386 y(Cam)m(b.)j(Phil.)e(So)s(c.)i Fi(103)p Fp(,)g(189-192.)d(\(1988\))72 2589 y([6])49 b(M.)33 b(Krupa,)f(I.)h(Melb)s(ourne.)g Fq(Asymptotic)i(Stability)g(of)f(Heter)-5 b(o)g(clinic)35 b(Cycles)f(in)g(systems)224 2710 y(with)h(Symmmetry.)d Fp(Erg.Th.Dyn.Sys.)j Fi(15)e Fp(121-147.)d(\(1995\))72 2913 y([7])49 b(E.F)22 b(Misc)m(henk)m(o,)27 b(Y)-8 b(u.)22 b(S)g(Koleso)m(v,)j(A.)d(Y)-8 b(u)22 b(Koleso)m(v,)j(N.)d(Kh)g(Rozo)m (v.)h Fq(Asymptotic)i(Metho)-5 b(ds)224 3033 y(in)44 b(Singularly)g(Perturb)-5 b(e)g(d)44 b(Systems.)f Fp(Monographs)g(in)f (Con)m(temp)s(orary)g(Mathematics.)224 3154 y(\(1993\))72 3357 y([8])49 b(C.)35 b(Robinson.)e Fq(Dynamic)-5 b(al)35 b(systems;)h(stability,)h(symb)-5 b(olic)35 b(dynamics)g(and)g(chaos.)e Fp(CR)m(C)224 3478 y(Press,)h(Clev)m(eland,)f(OH.)g(\(1994\))72 3681 y([9])49 b(B.)31 b(Sandstede,)h(A.)f(Sc)m(heel)g Fq(F)-7 b(or)i(c)g(e)g(d)31 b(symmetry)i(br)-5 b(e)g(aking)32 b(of)g(homo)-5 b(clinic)31 b(cycles.)f Fp(Nonlin-)224 3801 y(earit)m(y)j Fi(8)p Fp(,)f(333-365.)f(\(1995\))1768 5214 y(25)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF