Variety of regimes of starlike networks of Hénon maps

Abstract

In this paper we categorize dynamical regimes demonstrated by starlike networks with chaotic nodes. This analysis is done in view of further studying of chaotic scale-free networks, since a starlike structure is the main motif of them. We analyze starlike networks of Hénon maps. They are found to demonstrate a huge diversity of regimes. Varying the coupling strength we reveal chaos, quasiperiodicity, and periodicity. The nodes can be both fully and phase synchronized. The hub node can be either synchronized with the subordinate nodes or oscillate separately from fully synchronized subordinates. There is a range of wild multistability where the zoo of regimes is the most various. One can hardly predict here even a qualitative nature of the expected solution, since each perturbation of the coupling strength or initial conditions results in a new character of dynamics.

Cluster synchronization of starlike networks with normalized Laplacian coupling: master stability function approach

“…We already considered this type of coupling matrices for scale-free and starlike networks of Hénon maps. 6,8 Substituting this L to equation for a cell B mi of the Jacobian matrix (6) one gets:…”

“…This approach is often used for analysis of multistability. 5,6 Figure 4(a) shows the first Lyapunov exponent of the Ikeda network with N = 5 and unit weights of links computed for various initial conditions vs. . One can observe that chaotic regimes dominate, but also there are areas where dynamics is periodic.…”

“…The emergence of the rich multistability area below the full synchronization regime was already reported by Kuptsov and Kuptsova for a starlike network of Hénon maps. 6 An example of dynamics in this area is shown in Fig. 5(a).…”

“…Moreover, the ranges of existence of particular attractors can be narrow so that the qualitative behavior of the system can change dramatically when its parameters are slightly varied. [4][5][6] In the present paper we suggest a generalized model of starlike network taking into account non-additive coupling and nonlinear transformation of coupling variables. For this model we develop an analysis of stability of synchronized clusters generalizing the idea of master stability function.…”

Radial and circular synchronization clusters in extended starlike network of van der Pol oscillators

“…Ma et al [27] study formation of synchronized clusters in such networks and derive a sufficient condition of existence and asymptotic stability of a cluster synchronization invariant manifold. Kuptsov and Kuptsova [28] show that starlike networks of chaotic maps can demonstrate wild multistability, i.e., multistability including hardly predictable number of states with narrow basins of attraction. For such networks the generalization of master stability function approach is developed and synchronized clusters of chaotic nodes are studied [29].…”