A generalized scheme for designing multistable continuous dynamical systems

In this paper, a generalized scheme is proposed for designing multistable continuous dynamical systems. The scheme is based on the concept of partial synchronization of states and the concept of constants of motion. The most important observation is that by coupling two m-dimensional dynamical systems, multistable nature can be obtained if i number of variables of the two systems are completely synchronized and j number of variables keep a constant difference between them i.e., their differences are constants of motion, where i + j = m and 1 ≤ i, j ≤ m?1. The proposed scheme is illustrated by taking coupled Lorenz systems and coupled chaotic Lorenz-like systems. According to the scheme, two coupled systems reduce to single modified system with some initial condition-dependent parameters. Time evolution plots, phase diagrams, variation of maximum Lyapunov exponent and bifurcation diagrams of the systems are presented to show the multistable nature of the coupled systems.     

Projective synchronization of chaotic systems with bidirectional nonlinear coupling

This paper presents a new scheme for constructing bidirectional nonlinear coupled chaotic systems which synchronize projectively. Conditions necessary for projective synchronization (PS) of two bidirectionally coupled chaotic systems are derived using Lyapunov stability theory. The proposed PS scheme is discussed by taking as examples the so-called unified chaotic model, the Lorenz–Stenflo system and the nonautonomous chaotic Van der Pol oscillator. Numerical simulation results are presented to show the efficiency of the proposed synchronization scheme.     

Multiple dynamical time-scales in networks with hierarchically nested modular organization

Many natural and engineered complex networks have intricate mesoscopic organization, e.g., the clustering of the constituent nodes into several communities or modules. Often, such modularity is manifested at several different hierarchical levels, where the clusters defined at one level appear as elementary entities at the next higher level. Using a simple model of a hierarchical modular network, we show that such a topological structure gives rise to characteristic time-scale separation between dynamics occurring at different levels of the hierarchy. This generalizes our earlier result for simple modular networks, where fast intramodular and slow intermodular processes were clearly distinguished. Investigating the process of synchronization of oscillators in a hierarchical modular network, we show the existence of as many distinct time-scales as there are hierarchical levels in the system. This suggests a possible functional role of such mesoscopic organization principle in natural systems, viz., in the dynamical separation of events occurring at different spatial scales.