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DIFFERENTIAL EQUATIONS AND
CONTROL PROCESSES N 2, 2004 Electronic Journal, reg. N P23275 at 07.03.97
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http://www.neva.ru/journal http://www.imop.csa.ru/ diff e-mail: diff@osipenko.stu.neva.ru
Computer modeling in dynamical and control systems
Numerical Explorations of the Ikeda mapping dynamics
George Osipenko Laboratory of Nonlinear Analysis St. Petersburg State Polytechnic University math@math.tu.neva.ru
1 Introduction
The Ikeda map I we study is given by
where C is the complex plane of the variable z — x+iy and R, C1 , C2, and C3 are real constants (mapping parameters). The Ikeda map occurs in the modeling of optical recording media (crystals) [1]. The numerical results obtained to date (see [2], [3], [4], [5], [6]) show that under certain parameter values the Ikeda map exhibits highly complicated dynamical behavior. In particular, the Ikeda map can have infinitely many hyperbolic periodic orbits, which are located in a bounded part of C, and a strange attractor (the Ikeda attractor). The aim of the paper is to give an analysis of the topological structure of orbits by symbolic dynamics methods (the package ASIDS) and by methods of curves iteration (the package Line). We also present an analysis of orbit behavior near fixed and periodic points and of bifurcations that lead to chaotic attractors as parameters vary.
I :z^R + C2z exp(i(Ci - C3/(l + \z\2)), zeC,
(1)
2 Analytical results
In this section we give some simple analytical results on the Ikeda map we need in the sequel. In the real notation the Ikeda map takes the form
I: (x,y) i—)► (R + C^x cost — y sinr), C^x sinr + y cost)), (2)
where r = C\ — Cs/(1 + x2 + y2). Some obvious properties of the Ikeda map are listed below.
1. The map I can be viewed as a composition of the three diffeomorphisms Ti, T2, and T3 of the plane onto itself:
/ = T3oT2oT1,
where T\(x,y) — {x cost—y sinr, x sinT-\-y cost) is a rotation through the angle r — r(r),r2 — x2 + y2, T^u, v) — (C2U, C2V) is a linear homothetic, and T3(s, t) = (R + s, t) is a translation along the real axis.
2. If C2 > 0 then I is an orientation preserving diffeomorphism of the plane onto itself.
3. If |C2| < 1 then the map I is dissipative, i.e. there exists an h > 0 such that
lim sup ||In(x,y)\\ < h
n—^ 00
for each point (x,y).
4. If IC2I < 1 then every disk Kr = {(x,y) : x2 + y2 < r2} with the radius r > |-R|/(1 — |C2|) is mapped into itself, i.e. I(Kr) C int Kr.
5. For every point (x, y) the Jacobian of I is of the form det DI(x, y) = C22. Thus, if IC2I < 1 then I contracts the area, i.e. for the Lebesque measure of every bounded measurable set U we have
mes I(U) < mes U. Let IC2I < 1. The properties listed above imply the following:
1. Every bounded invariant set U{I{U) — U) is contained in the disk K{r*) with the radius r* = |-R|/(1 — |C^D- Let Ag be the maximal bounded invariant set of I contained in K(r*):
00
= n in(K(r*)).
n=0
It is well known that the set Ag is closed connected and asymptotically stable in the large, i.e. Ag is a global attractor. By 5, Ag has measure zero: mes Ag — 0.
2. The behavior of orbits of I is entirely determined by the behavior of orbits from Ag. In particular, periodic, non-wandering, and chain- recurrent orbits of I are contained in Ag. Results of numerical explorations mentioned above indicate that under certain parameter values the diffeomorphism I can have infinitely many hyperbolic periodic orbits with periods tending to infinity. This leads to the existence of homoclinic orbits and indecomposable continua in Ag. The last means that Ag has a very intricate topological structure.
3 Numerical results
Numerical simulations of the dynamical behavior of the map I have been carried out with C\ = 0.4, C2 = 0.9, C3 = 6.0. The parameter R takes the values within the segment [0; 1.1] increasing by R — 0.01. For each value of R, phase portraits are indexed by small letters a), b), ? anew. Results of the numerical study are the following.
As R increases from 0 to approximately 0.367, the global attractor Ag is a single asymptotically stable fixed point, i.e. I offers the convergence property.
3.1 # = 0.3
The Ikeda map has the fixed point ^4o(0.1766,0.2298). This fixed point attracts all other orbits.
3.2 R = 0A
The Ikeda map has three fixed points: the fixed point t4q(0.2280,0.2568), the hyperbolic saddle point i7o(3.0508, —1.6442), and the stable focus
5o(3.7763,0.8930) (see Fig. la where the global attractor of the map is shown). The unstable manifold Wu(Hq) of Hq consists of two séparatrices; the limit set of the left separatrix is the sink Aq* = Aq and the limit set of the right one is So. However, while So is a regular focus, the sink Aq* has a sufficiently complicated topological structure (see Fig. lb). The stable manifold Ws(Ho) of the saddle Hq separates the basins of attraction Ws{Aq*) and VFs(<So) of Aq* and Sq.
3.3 R = 0.5
While R increases from R = 0.4 to R = 0.5 the sink Aq bifurcates to the attractor A which when R = 0.5 contains the sink ^4o(0.2784,0.2734), the period 2 sink 5(0.0897, -0.7195), (0.6758,0.6141), and the period 2 hyperbolic saddle #(1.0017,0.0376), (-0.2517, -0.4987) (see Fig. 2a).
The unstable scparatriccs WU(H) of H ends at Aq and S. The closure of the unstable manifold WU(H) (colored dark) coincides with the attractor A = WU(H) + Aq + S. The stable manifold W8(H) (colored light) separates the basins of attraction of Aq and S. The basin boundary of A is formed by the stable manifold Ws(Hq) of the hyperbolic fixed point Hq at approximately (2.2330, -2.3346) (see Fig. 2b). The unstable manifold WU(H0) of H0 consists of two separatrices, the left one ends at A and the right one ends at the sink 50(3.5231, 2.1942). The closure of WU(H0) is the global attractor Ag — Wu(Hq)-\-A+So of the map. This form of the global attractor is preserved up to the parameter value R = 1, except that the structure of the attractor A varies over a wide range.
Figure 2: Ikeda map for R = 0.5.
Figure 3: Ikeda map for R = 0.6.
3.4 R = 0.6
The sink t40(0.3397, 0.2809), the period
2 hyperbolic orbit #(1.0094, -0.1100), (-0.2110, -0.4211), and the period 2 sink £(0.5997,0.6757), (0.2188, -0.7184) are contained in the attractor A. The unstable manifold WU(H) of each point of the orbit H is formed by two séparatrices, one of these séparatrices ends at the sink Aq, (see Fig. 3a), while the other one intersects the stable manifold WS(H), giving rise to a sequence of
homoclinic points. Some homoclinic points are listed in the following list:
x = 0.19290582 y--
x = 0.19345644 y =
x = -0.19643224 y =
x = -0.19769240 y =
x = —0.21104788 y~-
x = -0.20870386 y --
The Figure 3b, where depicted, indicates the near T.
-0.35802801 -0.35774567 -0.42426632 -0.42013205 -0.42113939 -0.42167093
x = -0.20891181 y = -0.42162378
x = —0.21068116 y = -0.42122255
x = -0.21070912 y = -0.42121621
x = -0.21099045 y = -0.42115241
x = -0.21099735 y = -0.42115085
x = -0.21103470 y = -0.42114238.
the stable WS(H) and unstable WU(H) manifolds are transverse character of intersections of these manifolds
Figure 4: Ikeda map for R = 0.6.
Since at R — 0.5 the manifolds WU(H) and WS(H) are disjoint then there exists a parameter value i?*, 0.5 < R* < 0.6, such that the manifold WU(H) is tangent to the manifold WS(H). The stable manifold WS(H) of the orbit H forms the boundary of the basins of attraction of the sink Aq and the period 2 attractor A2, which contains the period 2 sink S. In Fig. 4c is shown the basin of attraction of A2 (colored white grey). Its component containing the point (0.2188, -0.7184) of the sink S is shown in Fig. 4d.
The attractor A2 is a closure of the unstable manifold WU{P) of the period 6 hyperbolic orbit P(0.1869, -0.5785), (0.3556,0.7053), (0.2818, -0.7800), (0.6249,0.6969), (0.1343,-0.7635), (0.8751,0.4730). Each connected component of WU(P) consists of two séparatrices, the one ends at the sink 5, while the other one ends at the chaotic attractor A3 (see Fig. 4e). A3 contains the attractor A4, induced by the unstable manifold WU{Q) of the period 6 orbit Q. In Fig. 4f are shown the point (0.2056, —0.4874) of the orbit Q (depicted
Figure 5: Ikeda map for R = 0.7.
as a black dot) and its stable and unstable manifolds. The attractor A4 is a closure of the unstable manifold WU(Q), which ends at the period 12 sink G. Fig. 4f presents also two points (0.2022, -0.4816) and (0.2095,0.4953) of the orbit G. It is interesting to note that the stable and unstable manifolds are tangent at Q forming a sink. The global attractor A is a closure of the unstable manifold of the orbit H :A = WU{H) + A2 + A0. The stable manifold WS(H0) of the hyperbolic point #0(1.7660, —2.4891) is the common boundary of basins of attraction of A and the sink 5o(3.3064,2.8382). The displacement of A, Ho and Sq is similar to that in the cases R = 0.5 and R = 0.7.
3.5 R = 0.7
The Ikeda map with R = 0.7 has the inverse saddle fix point Ao(0.3804, 02817). The unstable manifold Wu(Aq) of Aq ends at the sink formed by a pair of the period 2 points 5(0.1548,0.2030), (0.6110,0.2118) which is a minimal attractor. The inverse saddle point Aq and the period 2 sink S arise from the sink Aq while R varies from R = 0.6 up to R = 0.7. A closure of the unstable manifold Wu(Aq) forms the attractor Ai = Wu(Aq) + S. The Ikeda map reverse the orientation of Wu(Aq) and hence the orientation of Ws(Aq) is also reversed since the Ikeda map is orientation preserving. There exists the period 2 hyperbolic orbit #i(0.5772,0.6788), (0.3102,-0.7009) with transverse intersection of the stable WU(H\) and unstable Wu(Hi) manifolds forming the chaotic attractor .A2 = WU{H\). The attractor A2 has two connected components derived from components of the unstable manifold WU{H\) for points of the orbit Hi. The
Figure 6: Ikeda map for R — 0.7.
attractor A2 can be viewed as a two-periodic attractor since the Ikeda map takes one connected component of A2 onto the other one. The unstable manifold WU{H) of the period 2 hyperbolic orbit #(-0.1364, -0.3495), (0.9931, -0.1676) is formed by two séparatrices WU(H)1 and WU(H)2, which ends at the attractors A\ and A2, respectively. Thus, the closure of WU(H) makes up the attractor A = Â! + WU(H) + A2 of the form A = S -h WU(A0) + WU{H) + Wu(Hl).
The stable manifold Ws(Hq) of the hyperbolic fixed point #o(l-5062, —2.5002) separates the basins of attraction of the attractor A and the sink 50(3.1580,3.2738). The unstable manifold WU{H0) of #0 is formed by two séparatrices, the left one ends at the attractor A while the right one ends at the sink So. The closure of Wu(Hq) generates the global attractor
Ag = A + Wu(H0) + S0.
3.6 R = 0.8
The Ikeda map has the inverse saddle Aq at approximately (0.4311,0.2761). Two unstable scparatriccs WU(H)S of
the period 2 orbit #(0.9429,-0.1339), (-0.0296,-0.2155) end at the period 2 sink 5(0.0387, —0.0345), (0.8467,-0.0013) while two other ones WU(H) intersect the stable manifolds Ws(Aq) and WS(H\) (colored light) of the saddle Aq and the period 2 hyperbolic orbit #1 (0.3844,-0.6761), (0.5798,0.6644). The unstable manifolds Wu(Aq) and WU{H\) (colored dark) intersect in turn the stable manifold WS(H), forming the heteroclinic cycle Aq —>- #1 —>- # —>• Aq (see Fig. 7a). The closure of unstable manifolds of the cycle generates the
Figure 7: Ikeda map for R = 0.8.
attractor A (see Fig. 7b).
The attractor A contains the sink S and, hence, is not a minimal attractor. The basin of attraction WS(A) of A is bounded by the stable manifold Ws(Hq) of the saddle fixed point Hq(1.3219, —2.4527), the complement to the closure of WS{A) is the basin of attraction of the focus 5o(3.0614,1.6110). As above, the left unstable scparatrix Wu(Hq)1 of Ho ends at the attractor A while the right one Wu(Ho)r ends at the sink So. The global attractor Ag is the closure of the unstable manifold WU(H0) of the saddle H0 : Ag = WU{H0) + A + S0. We notice that at R = 0.7 the unstable manifold Wu(Aq) ends at the sink S, whereas at R — 0.8 the sink S is the limit of the unstable scparatrix WU(H)S, i.e. a bifurcation occurs.
3.7 R = 0.9
The Ikeda mapping with R = 0.9 has a chaotic minimal attractor named the Ikeda attractor. As R increases from R = 0.8 to R = 0.9, the following bifurcation occurs: the period 2 sink S and the period 2 hyperbolic orbit H disappear. The attractor A contains the inverse saddle Ao(0.4819,0.2645) and the period 2 hyperbolic orbit Hi(0.5964,0.6394), (0.4497, -0.6453). The stable Ws(Ao) and Ws(Hi) and unstable Wu(Aq) and Wu(Hi) manifolds (séparatrices) of these saddles intersect and form the heteroclinic cycle Aq —>• Hi —>• Aq (see. Fig. 8a), generating the chaotic attractor A which is the closure of the unstable manifolds Wu(Aq) or Wu(Hi). There exists a pair of the period 3 hyperbolic orbits P3(0.8091,0.7834), (0.9960,-1.0090), (-0.0280,-0.8758) and
Figure 10: Ikeda map for R = 1.0.
Q3(1.3512,-0.0707), (0.6568,-1.1932), (-0.2418,-0.4462), (see Fig. 8b). The stable and unstable manifolds of orbits P3 and Q3 intersect forming the hetero-clinic cycle, which also generates the attractor A. The closure of the unstable manifold of any one of the orbits Ao, Hi, P3 or is the attractor A (see Fig. 9c). Outside the attractor A there is the saddle Pq(1-1987, —2.3769) whose left sepa-ratrix Wu(Hq)1 ends at the attractor A. The right unstable separatrix Wu(Ho)r ends at the sink 50(3.0027,3.8945) (see Fig. 9d). The stable manifold WS(H0) of the saddle Hq separates the basin of attraction WS(A) of the attractor A and the basin of attraction VFs(5o) of the sink Sq. The closure of the unstable manifold WU(H0) generates the global attractor Ag = A + WU(H0) + So. The map has no other period 2 and period 3 orbits.
3.8 #=1.0
When R goes from 0.9 to R — 1.0 the period 1, period 2, and period 3 orbits survive, except that their coordinates vary: when R = 1.0 (see Fig. 10a) the inverse saddle Aq is approximately (0.5228,0.2469), the period 2 hyperbolic orbit Hx is approximately (0.6216,0.6059), (0.5098, -0.6084), and the period 3 hyperbolic orbits P3 and Q3 are approximately (0.7795,0.7672), (1.0140,-0.9832), (0.0858,-0.8832) and (0.6583,-1.1541), (1.3297,-0.1427), (—0.1353, —0.3756), respectively. The closure of unstable manifold of any orbit Ao, Hi, P3 or Q3 is an attractor A (see Fig. 10b). The basin of attraction of A is bounded by the stable manifold Ws(Hq) of the hyperbolic fixed point Hq(1 .1142, —2.2857) which is nearly tangent to A (see Fig. 10b). The enlarged
scale phase portraits (Figs. 11c and lid) show that the distance between A and Ws(Hq) near points B and C is yet positive.
The stable manifold Ws(Hq) is a common boundary of the basins of attraction of the attractor A and the sink So(2.9721,4.1459). The stable and unstable manifolds of Hq are nearly tangent forming a sufficiently fine domain of attraction near points of "nearly tangency". The right scparatrix Wu(Ho)r ends at the sink Sq(2.9721,4.1459) and the left one Wu{Hq)1 approaches the chaotic attractor A.
b
Figure 13: Ikeda map for R — 1.1.
Figure 14: Ikeda map for R — 1.1.
3.9 R=1A
The mapping I has the following orbits with periods 1, 2, and 3: the inverse saddle A)(0.5837,0.2232), the period 2 orbit H2(0.6525,0.5641), (0.5670, -0.5643), and the period 3 orbits P3(0.1906, -0.8730), (1.0240, -0.9557), (0.7718,0.7342) and Q3(0.6660,-1.0738), (-0.0110,-0.2430), (1.2810,-0.1232). The relative positions of these orbits are similar to the case R = 1.0. The stable and unstable manifolds of the hyperbolic fixed point _£/o(l-05926, —2.1850) intersect transversally generating a homoclinic orbit (Fig. 13a).
Fig. 13b displays the manner in which the manifolds Ws(Hq) and Wu(Hq) intersect near Furthermore, Fig. 13a shows that the stable and unstable
manifolds of Aq and H$ intersect generating a heteroclinic cycle. Thus, the attractor A fails when R goes from 1.0 to 1.1. The global attractor Ag is the closure of the unstable manifolds of Hq or Aq. The right unstable scparatrix Wu(Ho)r ends at the focus 5o(2.9630,4.3773). Moreover, all other unstable manifolds stretching along Wu(Ho)r approach So as well. The set of chain recurrent points except for So is the closure of points of intersection of Ws(Hq) and Wu(Hq), Fig. 14d displays a neighborhood of the chain recurrent set.
4 Ikeda type mappings
In this section we consider some possible modifications of the Ikeda mapping. With this aim in view, let us rewrite the Ikeda mapping in the form
J : (x,y) (R + a(x cost — y sinr),b{x sinr + y cost)), (3)
where r = 0.4 — 6/(1 + x2 + y2). For the normal Ikeda mapping a = b = C2 and C2 € (0,1), i.e. the mapping is an orientation preserving contraction. Now we will not assume a = 6, in particular, a and b may be of opposite signs.
4.1 Ikeda type mappings preserving orientation 4.1.1 Inverse attraction: R = 3, a = b = —0.9
The mapping J has the hyperbolic fixed point (1.6030,0.8268) with nonempty intersection of stable and unstable manifolds: WS(H)C\WU(H). The stable and unstable manifolds are nearly tangent at a homoclinic point (Fig. 15a ). Since a = b < 0, J revises the orientation of WS(H) and WU(H) and H is an inverse saddle. There exists the period 2 sink 5(0.0320,0.3637), (3.3216,-0.0835), which is contained in the limit set of WU(H). The closure of WU(H) forms the global attractor Ag (Fig. 15b). The global attractor involves the chain recurrent set Q, which contains the orbits H and S and the points of intersection of WS(H) and WU(H) (homoclinic points). A neighborhood of Q obtained by the symbolic dynamics methods is shown in Fig. 15c. A neighborhood of S (colored dark) is a lower bound for a basin of attraction of S. The manifolds WS(H) and WU(H) and their intersection points are presented in Fig. 15d. The set of homoclinic points WS(H) fl WU(H) is a lower bound for the chain recurrent set
Q.
b
c d
Figure 15: Ikeda map for R = 3, a = b = —0.9.
4.1.2 Hyperbolic mapping: R = 1, a = 0.9, b = 1.2
There exists hyperbolic fixed point #(—0.1824, —2.3536) with nonempty intersection of the stable and unstable manifolds. The stable and unstable manifolds of H and the point F(0.0851,0.9643) homoclinic to H are shown in Fig. 16a. Table presents numerical results of successive computation of points H and F.
Step
Fixed point
Homoclinic point
30 x=-0.18235986. -2.35361944 x=-0.08509742. ,y= =0.96427872
31 x=-0.18235987, >y= -2.35361803 x=-0.08144479. ,y= =0.96428413
32 x=-0.18235936. -2.35361106 x=-0.08519972. ,y= =0.96428226
The mapping has the hyperbolic fixed point H\(0.5153,0.2835) and the period 2 hyperbolic orbit P(0.3708,0.6824), (0.5505,-0.7136). The stable and
Figure 16: Ikeda map for R = 1, a = 0.9, b = 1.2.
unstable manifolds of H} Hi and P intersect generating heteroclinic cycles (see Fig. 16b). Fig. 17c shows how the stable and unstable manifolds of P are situated. The set of points homoclinic to H (constructed as an intersection of WS(H) and WU(H)) is a lower bound of the chain-recurrent set Q and is depicted in Fig. 17d. A neighborhood of Q (an upper bound) obtained by localization using symbolic dynamics methods is displayed in Fig. 17e. The stable manifold WS(H) of H and stable manifolds of all other orbits from Q start from the source £(-2.9622,5.8918), see Fig. 17f.
4.1,3 Expansion: R = 1, a = b = 1.2
The mapping J increases an area by a2 = 1.44 and has a global repeller Rg. This repeller contains the hyperbolic fixed point #(0.4368,0.3100) which stable and unstable manifolds intersect generating a homoclinic contour. The fixed point H is an inverse saddle, i.e. the map J reverses orientation on WS(H) and WU(H). In addition, there exists the 2-periodic orbit H\(0.5132, —0.7463), (0.1850,0.7191) whose stable Ws(Hi) and unstable WU(H{) manifolds intersect each other and stable WS(H) and unstable WU(H) manifolds of H generating a heteroclinic contour (see Fig. 18a). The closure of WS(H) (or WS(H\)) forms the repeller R. Fig. 18b presents the repeller R and the manifolds WS(H) and WU{H).
The set of points (colored dark) of intersection of stable and unstable manifolds of H and H\ (a lower bound for Q) is depicted in Fig. 18b. Obtained
by symbolic dynamics methods, a neighborhood of the chain-recurrent set Q (an upper bound) containing R is shown in Fig. 19c. It seems likely that R — Q. Outside R there exists a hyperbolic fixed point H$(—1.2588, —2.5318) (see Fig. 19d), the left separatrix of which starts from R and the right one starts from the source ¿'(-3.7022,2.3228).
4.2 Ikeda type mappings reversing orientation
4.2.1 Contraction: R = l, a = 0.9, b = -0.9
The map J decreases an area and has a global attractor Ag. There exist two hyperbolic fixed points #o(0.5726,0.6602) and Hx(0.5606, -0.5692) whose stable and unstable manifolds intersect forming a heteroclinic cycle. In addition, there is a unique 2-periodic hyperbolic orbit P(0.9391,-0.2036), (0.1539,0.1791) whose stable (unstable) manifold intersects Wu(Hq) and Wu(Hi) (Ws(Hq) and Ws(Hq)) forming a heteroclinic cycle (Fig. 20a). Points of intersection of stable and unstable manifolds of these orbits (colored dark in Fig. 20b) yield a lower bound for the chain-recurrent set Q.
An upper bound for Q obtained by symbolic dynamics methods is depicted in Fig. 20c. Near Ho the manifold Ws(Hq) bounds Q, with the left separatrix Wu(Ho)l involved in Q and the right one Wu{Ho)r going to the right (Figs. 20a,b and d). Near Hi the manifold Wu{Hi) bounds Q, with the right separatrix Wu(Hi)r involved in Q and the left one Wu{H\)l going to infinity (Figs. 20a,b and d). Stretching along the right separatrix Wu(Ho)r, unstable manifolds start from Q and end at the sink S(9.7301, —1.5751). Stable manifolds start from Q and along the left separatrix Ws{Hq)1 reach infinity in the form of "rabbit ears" (Fig. 20d and Fig. 21). The global attractor Ag is the closure of Wu(Hq) (Fig. 21).
4.2.2 Contraction: R = 2, a = -0.9, b = 0.9.
The map J decreases an area and the global attractor Ag. There exists the unique hyperbolic fixed point H( 1.3815, —2.4746) (Fig. 22a) whose stable and unstable manifolds WS(H) and WU(H) intersect (Fig. 22a and b). In addition, there is the unique periodic orbit P2(0.2378, -0.7031), (1.9995,0.6681) stable and unstable manifolds of which intersect WS(H) and WU(H) forming a heteroclinic cycle(Fig. 22a). The global attractor Ag is a closure of WU(H) or WU(P) (Fig. 22b). The set WS(H) f] WU(H) is a lower bound for the chain-recurrent
b
Figure 22: Ikeda map for R = 2, a = -0.9, b = 0.9.
set Q. Fig. 22c presents a neighborhood of Q constructed by symbolic dynamics methods. Since Ag contains all -limits points, stable manifolds of orbits from Ag cover the plane R2.
Using symbolic dynamics methods we obtain the 6-period hyperbolic orbit P6 (1.0847,-1.0732), (2.7889,-1.1242), (-0.2626,-1.4846), (3.3560,0.0508), (—1.0124,-0.2235), (1.3964,0.7116). Its Lyapunov exponents are calculated by A = | • ln|7|, where by are the eigenvalues of the differential of Ikeda mapping along the orbit Pq. We obtain: 71 = —23.098, 72 = —0.012 and Ai = 0.523 and A2 = —0.734. The attractor has the 2-periodic orbit P2 (0.2385,-0.7024), (1.9989,0.6691). The eigenvalues of the differential along P2 are Ai = —0.134, 72 = —4.888, and the Lyapunov exponents A = \ • ln|7| are Ai = —1.004, A2 = 0.793. There exists the 4-periodic orbit P4(-0.6836,-0.6319), (0.7312,-0.9389), (1.6003,0.72792), (3.0613,-0.1713) with the Lyapunov exponents Ai = —0.843, A2 = 0.633.
4.2.3 Hyperbolic mapping: R = 1, a = —0.9, b = 1.2
The map J has the hyperbolic fixed point H(—0.0950,2.1937) stable manifold WS(H) of which can be bijectively projected on the The map J re-
verses orientation on WS(H). The unstable manifold WU(H) can be bijectively projected on the y-axis near H, however, the lower part of WU(H) offers a complicated structure (Fig. 23a). Such a behavior of WU(H) results from the fact that WU(H) intersects the stable manifold WS(Q2) of the 2-periodic hyperbolic orbit Q2(—1-5584, -1.9046), (3.0088, -1.2438), which in turn has a homoclinic point of transverse intersection of stable and unstable manifolds WU(Q2) and Ws(Q2) (Fig. 23b). Fig. 24c shows the manner in which WU{Q2) and WS{Q2) intersect near Q2(—1.5584, —1.9046). Besides Q2 there is another 2-periodic hyperbolic orbit P2(-0.2554,-0.9207), (1.1152,1.1362) with homoclinic intersection of its stable and unstable manifolds WU{P2) and WS(P2). Fig. 24d shows the manner in which WU(P2) and WS(P2) intersect near P2(-0.2554, -0.9207). Stable and unstable manifolds of orbits Q2 and P2 intersect forming a hetero-clinic cycle. This leads to the chaotic chain-recurrent set Q. Fig. 24e depicts a neighborhood (an upper bound) of Q. The set WU(Q2) fl WS(Q2) gives a lower bound for Q. Fig. 24f shows the displacement of WU(Q2), WS(Q2), and their points of intersection. The stable manifold WS(H) is in the closure of WS(Q2). The closure of WS(Q2) forms the set looking like a "Napoleon" hat (Fig. 24f).
V'
e f
Figure 24: Ikeda map for R = 1, a = —0.9, b = 1.2.
References
[1] K. Ikeda, Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system, Opt. Commun30, 257-261 (1979).
[2] H.E.Nusse, J.A.Yorke, Dynamics: Numerical Explorations, 1997.
[3] C.Robinson, Dynamical Systems: Stability, Symbolic Dynamics and Chaos, 1995.
[4] J.Guckenheimer, P.Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, 1983.
[5] R.L.Devaney, An Introduction to Chaotic Dynamical Systems, 1990.
[6] S.Wiggins, Introduction to Applied non linear Dynamical Systems and Chaos, 1990.
