DIFFERENT TYPES OF CRITICAL BEHAVIOR IN CONSERVATIVELY COUPLED HÉNON MAPS

DIFFERENT TYPES OF CRITICAL BEHAVIOR IN CONSERVATIVELY COUPLED HÉNON MAPS D.V. Savin, *, A.P. Kuznetsov,, A.V. Savin and U. Feudel 3 Department of Nonlinear Processes, ernyshevsky Saratov State University, Saratov, Russia Kotel nikov Institute of Radioengineering and Electronics of RAS, Saratov branch, Saratov, Russia 3 Institute for emistry and Biology of the Marine Environment, Carl von Ossietzky University Oldenburg, Oldenburg, Germany * E-mail: savin.dmitry.v@gmail.com We study the dynamics of two conservatively coupled Hénon maps at different levels of dissipation. It is shown that the decrease of dissipation leads to changes in the parameter plane structure and scenarios of transition to chaos comparing with the case of infinitely strong dissipation. Particularly, the Feigenbaum line becomes divided into several fragments. Some of these fragments have critical points of different types, namely of C and H type, as their terminal points. Also the mechanisms of formation of these Feigenbaum line ruptures are described. Keywords: coupled maps, Hénon map, critical behaviour, Hamiltonian critical point, Feigenbaum line, periodic windows.. Introduction. Dynamics of systems with small dissipation level is a rather popular object for investigation, especially numerical, in nonlinear dynamics. There exist a number of well-known works concerning the critical behaviour in such systems and conservative-dissipative crossover (see e.g. [-3] and other works of these authors), multistable behaviour and mechanisms of attractor evolution with the change of dissipation both in model systems as standard map or Hénon map [4-7] and in more realistic ones [8] (see also the review in [9]), and other aspects of dynamics of such systems. Usually the investigation is carried out for a low-dimensional map or system of ordinary differential equations which is an autonomous system or a periodically driven system. At the same time the dynamics of coupled systems in the case of small dissipation level is not investigated in such detail. Meanwhile, since both coupled systems and weakly dissipative systems demonstrate a number of very interesting specific phenomena, combining these two classes of systems seems to be a promising way to obtain new interesting features of dynamics. Our idea is to take a simple example of coupled systems and to look out what is going on with the decrease of dissipation using numerical simulations. System of coupled logistic maps fits very well for this purpose, since the dynamics of such system is investigated very well. The logistic map xn+ = λ xn () is a universal model and can be considered as the limit case of the Hénon map x n+ = λ xn byn, yn+ = xn () at b=0, which physically corresponds to the infinite value of dissipation. Therefore, taking two coupled Hénon maps one can get a system which (i) has a very clear and well investigated extreme case of infinite dissipation, (ii) allows one to vary dissipation level up to the zero

and hence is a good object to explore changes which occur with structures typical for the coupled maps with the decrease of the dissipation. The paper is organized as follows: in Section we present the model system under investigation and an overview of the parameter plane evolution, Section 3 deals with the bifurcation structure of the parameter plane and the critical behaviour at the border of chaos, in Section 4 we try to investigate the parameter plane changes in more detail, and finally in Section 5 we give conclusions and discuss obtained results.. The model system and evolution of its parameter plane with decrease of dissipation. Let us now describe the system under investigation. According to the stated above we want to choose the system of two coupled Hénon maps. It is convenient to choose the coupling which will not introduce any extra dissipation in the system, because in this case it is quite simple to control the dissipation level. The simplest variant is the linear coupling by the first variable xn+ = λ xn byn + ε ( xn un ), yn+ = xn, (3) u = λ u bv + ε ( u x ), v = u. n+ The Jacobian of (3) is b, hence the system is conservative at b=. n n.5 In the case of infinite dissipation (3) turns infinity to the system of linearly coupled logistic maps. F The dynamics of the latter is investigated Q rather well [0-]. The structure of the (λ, ) parameter plane in this case is shown in 4 fig.. It is known that except of the transition F to chaos via the Feigenbaum period-doubling cascade the transition to chaos via the quasiperiodicity destruction also exists in the region close to the parameter plane diagonal. Thereby, there exist two regions in the parameter plane with different scenarios of transition to chaos (let us design them as F and Q -0.5-0.5 λ.5 respectively). On the border of these two regions the Feigenbaum line of transition to Fig.. Structure of the parameter plane of (3) at b=0, ε=0.4. F denote the line of transition to chaos terminates in the codimension critical chaos via Feigenbaum scenario, it is also point associated with the cycle of period in marked with the white line. Areas of different colour correspond to regions of the existence the renormgroup equation []. Such point is of the cycle of certain period (see numbers in usually referred to as the critical point of C- the figure), designates area with the positive value of the major Lyapunov exponent Let us now begin to decrease the dissipa- type [3]. (chaotic regime), Q area where Lyapunov tion level, i.e. to increase the parameter b towards. Structure of the parameter plane at exponent is close to 0 (quasiperiodical regime). White colour corresponds to the solution going to infinity. The dark rectangle is different b values is shown in fig.. One can enlarged in fig. 3. see that both F and Q regions change their structure with the decrease of dissipation. The Q area becomes much thicker but spreads into the region far from the parameter plane n n n+ n

3 4 infinity.7 4 infinity -0.5-0.5 λ 3 8 Q. 0. λ.5 3.5 infinity 3.3 infinity Q 4 4 c) d) -0.5.4-0.5 λ 3.5 0 λ.65 Fig.. Structure of the parameter plane (a, c) and its enlarged fragments (b, d) of (3) at ε=0.4; b=0.5 (a,, b=0.7 (c,d). Colour code and symbols as in fig., black rectangles in the left column are enlarged in the right one. diagonal. As for the Feigenbaum line, it undergoes a rupture, and the dynamics in corresponding region becomes sufficiently two-parameteric. At further increase of b the second Q area arises far from the parameter plane diagonal, which also causes new Feigenbaum line rupture. Finally, at b=0.7 the Feigenbaum line has 3 fragments in each half of the parameter plane versus in the case of small b (see sketch in fig. 3). Thereby two questions arise: (i) How does the bifurcation structure around the terminal points of the Feigenbaum line look like? (ii) What is the mechanism of the rupture formation? In the following sections we will try to answer these questions. 3. Bifurcation structure and critical behaviour. Let us examine bifurcation structure of the parameter plane in more detail. When b value is rather small, the Feigenbaum line is still continuous. In this case period-doubling

.5 3.3..4 0 λ.5 0 λ.65 Fig. 3. Sketch of the Feigenbaum critical line of (3) at b=0 ( and b=0.7 ( (magnification of the parameter plane parts marked with black rectangles in figs. and c). F, F, F3 designate fragments of the Feigenbaum line, C, C, C, C, C3, H, H critical points of C and H type correspondingly. lines of consequent periods from the period-doubling cascade terminate in the fold-flip points with pair of maximum in modulus multipliers (-, +) [4]. Coordinates of these points for the right end of the Feigenbaum line at b=0.3 are shown in Table. These sequences converge to certain limits which are the terminal points of the Feigenbaum line. Such sequences are known to have the critical points of the C-type as their limit [5]. We can conclude that in this case Feigenbaum line terminates by two C-type critical points, as in coupled logistic maps []. Table. Coordinates of the fold-flip points at the right end of the Feigenbaum line at b=0.3 Period λ 8.9367393.04376 6.35879.0783506 3.09967.0450455 64.555098.05983479 8.947400.0597370 56.37398.05774598 5.5007.05639539 04.335567.05734009 At increase of b the first rupture forms, and at b=0.5 (fig. a, the Feigenbaum line is already divided into two fragments. Terminal points of the PD lines are again fold-flip points, and these sequences also converge to the C-type critical points. Further increase of b leads to the formation of the new Q area, which can be regarded as the formation of the second rupture of the Feigenbaum line. It is formed due to the appearance of the Neimark-Sacker (NS) bifurcation lines for all periods from the perioddoubling cascade starting from the higher ones. Fig. 3 represents the structure of the Fei-

3.74 3.69 3.643 3.688 0.73 λ 0.949 0.80 λ 0.86 Fig. 4. Structure of bifurcation lines of (3) at b=0.7. PD period-doubling line, NS line of Neimark- Sacker bifurcation, R Resonsnce : point, H location of the critical point of H type, numbers designate the period of the stable regime in the corresponding area. genbaum line in this case (fig. 3 comparing with the same for the logistic maps (fig. 3, and fig. 4 represents the picture of period-doubling lines and lines of NS bifurcation at the right border of Q area at b=0.7 (vicinity of H point from fig. 3. One could find that all period-doubling lines terminate at the Resonance : points, i.e. points with two maximum in modulus multipliers (-, -) [4]. Such sequence of points is known to be the route to the critical point of Hamiltonian type, or H-point [5, 6]. The coordinates of these terminal points are represented in Table. They accumulate to a certain limit, which we assume to be the critical point of the H type. Using corresponding scaling constant δ H =8.7. [7] we obtain the expected location of this point as λ =0.8083808, =3.6906587 and calculate the multipliers of cycles of periods 3 and 64 in it. We obtain μ =.06 and μ = 0.48 which is rather close to the universal values μ =.0574783 and μ = 0.486038 [5]. We conclude that the critical point of the H-type exists here as a terminal point of the Feigenbaum line. Dynamics on the other side of the Q area is the same, and there also exists the critical point of H type. It is worth mentioning here that the change of the type of NS-bifurcation (from subcritical to supercritical) between different periods from the period-doubling cascade, obtained in [6] for the system of another nature in similar situation, also takes place in our system, so it seems to be typical for the appearance of the H-type critical point in dissipative systems. At the borders of another Feigenbaum line rupture exist C-type critical points, similar to the picture at smaller values of b. Coordinates of the fold-flip terminal points from the converging sequence illustrating this fact are represented in Table 3. Finally, we have now three fragments of the Feigenbaum line (let us call it F, F and F3 respectively going from right to left, see fig. 3. F fragment has critical points of C- type at both sides as its terminal point while F and F3 have critical points of different types (C and H respectively) at their sides. This corresponds well with the suggestion made in [3] concerning the possible terminal points of the Feigenbaum line.

Table. Coordinates of the Resonance : points converging to the critical point of H type at b=0.7 (H point in designations of fig. 3 Period λ 6 0.8080895 3.68980 3 0.8038063 3.69094 64 0.8096 3.690445 8 0.808497 3.690634 56 0.808394 3.690656 Table 3. Coordinates of the fold-flip points at the left end of the Feigenbaum line rupture at b=0.7 (vicinity of C point in designations of fig. 3. Period λ 8 0.7090094.8854645 6 0.7044460.843864 3 0.7087390.8083558 64 0.70759608.8090086 8 0.7084760.808003 56 0.70873.80835790 5 0.7083656.80848 04 0.708859.8083500 048 0.7083307.808867 4. Formation of the rupture Let us now turn to the details of the formation of the first Feigenbaum line rupture. It manifests in the appearance of the area with a huge number of fragments of perioddoubling lines, Feigenbaum lines and multistability sheets instead of the usual structure with sequence of continuous smooth period-doubling lines without any multistability. Finally, the details of the transition to chaos turn out to be strongly dependent on the route in the parameter plane instead of sufficiently one-parametric dynamics at small values of b. In order to reveal the mechanism of this rupture formation we have computed a number of charts of dynamical regimes for different values of b. It is worth mentioning here that the chaotic region of the parameter space is filled by periodic windows of different periods, the most of which in two-dimensional parameter space look as based on the spring area and crossroad area structures [8, 9] (the latter sometimes are referred to as shrimp-shaped domains, or shrimps [0]). It turns out that with the increase of b merging of the main periodic area with one of these periodic windows occur. Parameter plane fragments at ε=0.4 and different b values illustrating different stages of this process are shown in fig. 5. In fig. 5a one can see the periodic window based on the spring area for 6-cycle together with the cascade of the periodic windows consisting of a number of spring areas for cycle of period 3, 64 etc. Figs. 5b and 5c show the evolution of these periodic windows with increase of b which results in merging of the whole structure with the main periodic area (fig. 5d). It is necessary to remind here briefly the structure of the typical periodic window. Crossroad area and spring area structures based on the cycle of period n are formed by

.06.03 6 3 6 3 6.96 0.8 λ 0.94.04 6.99 3 0.9 λ 0.94.5 3 8 6 c) 64 d) 3 3 8 0.89 λ 0.93 0.76 λ Fig. 5. Fragments of the parameter plane illustrating the process of the formation of the Feigenbaum line rupture. ε=0.4, b values: 0.3, 0.36, c)0.35, d)0.334. Colour code as in fig.. two fold lines for this cycle coming out from the cusp point [8, 9], hence the multistability exists and the parameter space becomes divided into two multistability sheets. On each of these multistability sheets there exists a period-doubling line. Since the periodic window usually consists of the cascade of such structures with periods n, n etc., at each level of this period-doubling cascade the new splitting of the parameter space in the multistability sheets occur, and finally at the border of chaos one have a very complicated fractal-like structure with the infinite number of Feigenbaum line fragments [5, ]. After the merging process has finished, one has all this hierarchy of bifurcation lines structures, which is normally located in small regions in the chaotic area and does not effect on the transition to chaos, connected to the main periodic area and hence determining the transition to chaos from the main periodic area in certain domain on the parameter plane. We found also that though the described mechanism of the rupture formation exists in certain interval of coupling parameter ε, the definite shape of the periodic window involved in the process could vary. E.g., for ε=0. the described process occurs not with the

.975 3.98 64 3 8 3 64.96.965 6.35 λ.45.35 λ.47 Fig. 6. Fragments of the parameter plane illustrating the process of the formation of the Feigenbaum line rupture at ε=0.. b values: 0.6558, 0.6575. Colour code as in fig.. spring area based structure but with a ring-shaped periodic window (see fig. 6). Moreover, varying ε and b one can find the transition from one type of the periodic window to another. E.g., fig. 7 represents fragments of the parameter plane showing the process of collision of this ring-shaped structure with another periodic window based on the crossroad area structure (fig. 7a,, which results in formation of two spring area based periodic windows (fig. 7c, d). It could be interpreted as existence of the connection between periodic windows in four-dimensional parameter space. 5. Conclusions and discussion In the present paper we have showed that the varying of dissipation in the system of conservatively coupled Hénon maps could lead to sufficient complication the transition to chaos. Instead of one-parametric dynamics in widespread domain in the parameter plane in the case of big dissipation one can fall into the region with sufficiently two-parametric dynamics: the transitions which the system undergoes while moving towards chaos and the location of the chaos border depend strongly on the route chosen in the parameter plane. The Feigenbaum line of transition to chaos becomes divided into three fragments. We also reported the existence of the Feigenbaum line fragment with different scaling properties on different sides, namely of H and C type, which seems to be the typical situation for maps with more than one dimensions of the parameter space according to [3] but to the best of our knowledge it is the first case of obtaining such phenomenon in numerical simulations. We have to mention here that the described scenario of merging of the periodic window with the main periodic area is not a trivial one for two-parameter presentation, but it seems that in multidimensional parameter space it could be rather general. Indeed, the transition from coupled logistic maps to the coupled Hénon maps could be regarded as an adding of a new parameter to the system, so full parameter space becomes fourdimensional. In this 4D parameter space the main periodic area looks as some volume with tails bristling from it. Of course, in a section by two-dimensional plane these tails could be separated from the main periodic area in certain interval of the parameters. Moving the secant plane, one can reach the situation when the separation vanishes. The 8 64 3

.5.05 0.87 λ.3.5 c) d).05 0.87 λ.3 Fig. 7. Fragments of the parameter plane illustrating the process of the collision of two periodic windows. Pictures in the right column are the enlargements of the black parallelograms from the left one. ε=0.37, b values:, 0.98, c), d)0.304. Colour code reversed comparing to fig. (white chaotic region, grey etc. periodic windows), the main period of the concerning windows is 6. computation of the unstable cycles from the period-doubling cascades with the help of continuation software confirms that the periodic window involved into the merging process is connected with the main periodic area via unstable cycles. From this point of view it is not very surprising fact that the particular structure of the merging periodic window differs for different coupling values, because it can be regarded as moving in the another region of the 4D parameter space. Moreover, it looks like if we will add some new parameters to the system, in this multidimensional parameter space we could some other periodic windows organized in a kind of a net connected with the main periodic area. In conclusion we have to note that all bifurcation lines presented in the paper were obtained by the means of the software CONTENT [].

Acknowledgments The work was in part supported by the RFBR, project -0-3089. D.S. also would like to thank German Academic Exchange Service and the Direction of development of the National Research University ernyshevsky Saratov State University for financial support of his visits in Oldenburg and Prof. Feudel and her group of Complex Systems for their hospitality. References. Zisook A.B. Universal effects of dissipation in two-dimensional mappings. //Physical Review A, 98, 4, 3, pp. 640 64.. Reinout G., Quispel W. Analytical crossover results for the Feigenbaum constants: Crossover from conservative to dissipative systems. //Physical Review A, 985, 3, 6, pp. 394 398. 3. en C., Gyorgyi G., Schmidt G. Universal transition between Hamiltonian and dissipative chaos. //Physical Review A, 986, 34, 3, pp. 568 570. 4. Feudel U., Grebogi C., Hunt B.R., Yorke J.A. Map with more than 00 coexisting low-period periodic attractors. //Physical Review E, 996, 54,, pp. 7 8. 5. Feudel U., Grebogi C. Why are chaotic attractors rare in multistable systems? //Physical Review Letters, 003, 9, 3, 340. 6. Rech P., Beims M., Gallas J. Basin size evolution between dissipative and conservative limits. //Physical Review E, 005, 7,, 070. 7. Martins L.С., Gallas J.A.C. Multistability, phase diagrams and statistical properties of the kicked rotor: a map with many coexisting attractors. //International Journal of Bifurcation and aos, 008, 8, 6, pp. 705 77. 8. de Freitas M.S.T., Viana R.L. and Grebogi C. Basins of Attraction of Periodic Oscillations in Suspension Bridges. //Nonlinear Dynamics, 004, 37, pp. 07 6. 9. Feudel U. Complex dynamics in multistable systems. //International Journal of Bifurcation and aos, 008, 8, 6, pp. 607 66. 0. Juan J.-M., Tung M., Feng D.H., Narducci L.M. Instability and irregular behavior of coupled logistic equations. //Physical Review A, 983, 8, 3, pp. 66 666.. Satoii K., Aihara T. Numerical study on a coupled-logistic map as a simple model for a predator-prey system. //Journal of the Physical Society of Japan, 990, 59, 4, pp. 84 98.. Kuznetsov A.P., Sataev I.R., Sedova J.V. Dynamics of coupled non-identical systems with period-doubling cascade. //Regular and chaotic dynamics, 3,, 008, pp. 9 8. 3. Kuznetsov S.P., Sataev I.R. New types of critical dynamics for two-dimensional maps. //Physics Letters A, 99, 6, 3, pp. 36-4. 4. Kuznetsov Yu. A. Elements of Applied Bifurcation Theory. New York: Springer-Verlag, 995, 495 p.

5. Kuznetsov S.P., Kuznetsov A.P., Sataev I.R.. Multiparameter Critical Situations, Universality and Scaling in Two-Dimensional Period-Doubling Maps. //Journal of Statistical Physics, 005,, 5-6, pp. 697 748. 6. Savin D.V., Savin A.V., Kuznetsov A.P., Kuznetsov S.P., U. Feudel. The selfoscillating system with compensated dissipation the dynamics of the approximate discrete map. //Dynamical Systems: An International Journal, 7, 0, 7-9. 7. Reichl L.E.. The Transition to aos in Conservative Classical Systems: Quantum Manifestations, Springer-Verlag, Berlin, 99. 8. Mira C., Carcasses J.P., Bosch M., Simo C., Tatjer J. C. Crossroad area spring area transition. (II) Foliated parametric representation. //International Journal of Bifurcation and aos, 99,,, pp. 339 348. 9. Mira C., Carcasses J.P. On the crossroad area saddle area and crossroad area spring area transitions. //International Journal of Bifurcations and aos, 99,, 3, pp. 64 655. 0. Gallas J.A.C. Structure of the Parameter Space of the Hénon Map. Physicasl Review Letters, 993, 70, 8, pp. 74-77.. Kuznetsov A.P., Kuznetsov S.P., Sataev I.R. and ua L.O. Two-parameter study of transition to chaos in ua s circuit: renormalization group, universality and scaling. //International Journal of Bifurcation and aos, 993, 3, pp. 943 96.. Yu.A. Kuznetsov, V.V. Levitin, CONTENT: A multiplatform environment for analyzing dynamical systems, Dynamical Systems Laboratory, Centrum voor Wiskunde en Informatica, Amsterdam, 997. Available at http://www.math.uu.nl/people/kuznet/content/