Coupled Networks, Patterns and Complexity - Abstract

Kinzel, Wolfgang

Chaos in nonlinear networks with time-delayed couplings

Nonlinear units, interacting by time-delayed couplings, can develop chaos. In the limit of large delay times we distinguish two types of chaos: Strong chaos, where the maximal Lyapunov exponent is identical to the largest sub-exponent of the units, and weak chaos where it decreases with the inverse delay time. Strong chaos has a positive, and weak chaos a negative sub-exponent. For lasers and for some electronic circuits we find transitions from weak to strong to weak chaos by increasing the coupling strength. Networks of such units can synchronize to a common chaotic trajectory. The eigenvalues of the coupling matrix and the properties of a single unit with self-feedback determine the stability of complete synchronization. We find that only networks with weak chaos (i. e. with negative sub-exponents ) can synchronize in the limit of large delay times. If an eigenvalue gap exists, the network can synchronize completely. Otherwise, cluster synchronization is possible. The global loop structure of the graph of the network determines the patterns of synchronized clusters. If the greatest common divisor of all loop lengths is one, complete synchronization occurs for sufficiently weak chaos. Otherwise, the GCD gives the number of chaotic clusters. Physically, the GCD is related to mixing information about the trajectories of the individual units. Finally, we address the question of how many chaotic attractors are possible for a network of N units. Combining statistical mechanics of neural networks with nonlinear dynamics, we construct a model where the number of chaotic attractors increases exponentially with the size of the network. newline References: newline [1] S. Heiligenthal, T. Dahms, S. Yanchuk, T. Jüngling, V. Flunkert, I. Kanter, E.Schöll and W. Kinzel, “Strong and Weak Chaos in Nonlinear Networks with Time-Delayed Couplings", Phys. Rev. Lett. 107, 234102 (2011). newline [2] I. Kanter, M. Zigzag, A. Englert, F. Geissler and W. Kinzel, “Synchronization of unidirectional time delay chaotic networks and the greatest common divisor", Europhys. Lett. 93, 60003 (2011). newline [3] Y. Peleg, M. Zigzag, W. Kinzel and I. Kanter, “Coexistence of exponentially many chaotic spin-glass attractors", Phys. Rev. E 84, 066204 (2011).