Time-delayed Nodes Research Articles

Analysis and Design of Adaptive Synchronization of a Complex Dynamical Network with Time-Delayed Nodes and Coupling Delays

This paper is devoted to the study of synchronization problems in uncertain dynamical networks with time-delayed nodes and coupling delays. First, a complex dynamical network model with time-delayed nodes and coupling delays is given. Second, for a complex dynamical network with known or unknown but bounded nonlinear couplings, an adaptive controller is designed, which can ensure that the state of a dynamical network asymptotically synchronizes at the individual node state locally or globally in an arbitrary specified network. Then, the Lyapunov-Krasovskii stability theory is employed to estimate the network coupling parameters. The main results provide sufficient conditions for synchronization under local or global circumstances, respectively. Finally, two typical examples are given, using the M-G system as the nodes of the ring dynamical network and second-order nodes in the dynamical network with time-varying communication delays and switching communication topologies, which illustrate the effectiveness of the proposed controller design methods.

Periodic Flows to Chaos Based on Discrete Implicit Mappings of Continuous Nonlinear Systems

This paper presents a semi-analytical method for periodic flows in continuous nonlinear dynamical systems. For the semi-analytical approach, differential equations of nonlinear dynamical systems are discretized to obtain implicit maps, and a mapping structure based on the implicit maps is employed for a periodic flow. From mapping structures, periodic flows in nonlinear dynamical systems are predicted analytically and the corresponding stability and bifurcations of the periodic flows are determined through the eigenvalue analysis. The periodic flows predicted by the single-step implicit maps are discussed first, and the periodic flows predicted by the multistep implicit maps are also presented. Periodic flows in time-delay nonlinear dynamical systems are discussed by the single-step and multistep implicit maps. The time-delay nodes in discretization of time-delay nonlinear systems were treated by both an interpolation and a direct integration. Based on the discrete nodes of periodic flows in nonlinear dynamical systems with/without time-delay, the discrete Fourier series responses of periodic flows are presented. To demonstrate the methodology, the bifurcation tree of period-1 motion to chaos in a Duffing oscillator is presented as a sampled problem. The method presented in this paper can be applied to nonlinear dynamical systems, which cannot be solved directly by analytical methods.

Chaotic synchronization in complex networks with delay nodes by non-delay and delay couplings

The synchronization in complex network model with time-delay nodes is investigated. The model is applicable for two sorts of complex networks. With linear feedback controller and adaptive feedback controller, positive definite functions are designed respectively. The sufficient conditions of synchronization are given by Lyapunov stability theory. Finally, numerical simulations show the effectiveness of the synchronization in complex networks.

Adaptive synchronization research on the uncertain complex networks with time-delay

The synchronization problem of uncertain dynamical network model with time-delay nodes has been researched, where the network structure is known or totally unknown. Based on Lyapunov stability theorem, the synchronization controller of complex networks and the adaptive controller of synchronization of complex networks are designed respectively under the circumstances of known parameters and unknown parameters. The sufficient condition of network synchronization is given to guarantee that the dynamical network asymptotically synchronizes at the individual node state in arbitrary specified network. Finally, numerical simulations are provided to verify the effectiveness of the developed methods.