Grazing Bifurcation in the Response of Cracked Jeffcott Rotor

A cracked rotor is modeled by a piecewise linear system due to thebreath of crack in a rotating shaft. The differential equations ofmotion for the nonsmooth system are derived and solved with thenumerical integration method. From the simulation results, it isobserved that a grazing bifurcation exists in the response. Thegrazing bifurcation can give rise to jumps between periodic motions,quasi-periodic motions from the periodic ones, chaos, and intermittentchaos.     

Phase synchronization on scale-free networks with community structure

In this Letter, we propose a growing network model that can generate scale-free networks with a tunable community strength. The community strength, C, is directly measured by the ratio of the number of external edges to that of the internal ones; a smaller C   corresponds to a stronger community structure. By using the Kuramoto model, we investigated the phase synchronization on this network and found an abnormal region (C?0.002$C?0.002$), in which the network has even worse synchronizability than the unconnected case (C=0$C=0$). On the other hand, the community effect will vanish when C exceeds 0.1. Between these two extreme regions, a stronger community structure will hinder global synchronization.     

A simple global synchronization criterion for coupled chaotic systems

Based on the Lyapunov stabilization theory and Gerschgorin theorem, a simple generic criterion is derived for global synchronization of two coupled chaotic systems with a unidirectional linear error feedback coupling. This simple criterion is applicable to a large class of chaotic systems, where only a few algebraic inequalities are involved. To demonstrate the efficiency of design, the suggested approach is applied to some typical chaotic systems with different types of nonlinearities, such as the original Chua’s circuit, the modified Chua’s circuit with a sine function, and the Rössler chaotic system. It is proved that these synchronizations are ensured by suitably designing the coupling parameters.     

Stabilizing stochastically-forced oscillation generators with hard excitement: a confidence-domain control approach

In this paper, noise-induced destruction of self-sustained oscillations is studied for astochastically-forced generator with hard excitement. The problem is to design a feedbackregulator that can stabilize a limit cycle of the closed-loop system and to provide arequired dispersion of the generated oscillations. The approach is based on the stochasticsensitivity function (SSF) technique and confidence domain method. A theory about thesynthesis of assigned SSF is developed. For the case when this control problem isill-posed, a regularization method is constructed. The effectiveness of the new method ofconfidence domain is demonstrated by stabilizing auto-oscillations in a randomly-forcedgenerator with hard excitement.     

On the invariance of maximal distributional chaos under an annihilation operator

It is known that certain physical systems, which do not generate deterministic chaos under conventional frameworks, may generate such complex behavior in a quantum mechanical setting. In this paper, it is proved that the annihilation operator of an unforced quantum harmonic oscillator admits an invariant distributionally $?$-scrambled set for any $0<?<2$, showing that this operator can exhibit maximal distributional chaos on an uncountable invariant subset.     

Spectral coarse graining of complex clustered networks

Synchronization in complex dynamical networks is in the focus of network science today, where intensive efforts have been devoted to understanding its mechanisms and developing basic theories with applications. However, the sheer sizes of large-scale networks have been the main hurdle in such analysis and applications. Recently, a coarse graining scheme based on network synchronization was proposed to reduce the network size while preserving the synchronizability of the original network. In this research, we investigate the effects of the coarse graining process on synchronizability over complex clustered networks. Numerous experiments demonstrate a close correlation between the degree of clustering of the initial network and the ability of spectral coarse graining in preserving the network synchronizability. It is found that synchronizability can be well preserved after applying the spectral coarse graining if the considered network has a clear cluster structure, whereas this is not so for networks with vague clustering. Since most real-world networks have prominent cluster structures, this research provides new insights into understanding large-scale dynamical networks and analyzing their complex topological characteristics as well as synchronization mechanisms.     

Chaos of time-varying discrete dynamical systems

This paper is concerned with chaos of time-varying (i.e. non-autonomous) discrete systems in metric spaces. Some basic concepts are introduced for general time-varying systems, including periodic point, coupled-expansion for transitive matrix, uniformly topological equiconjugacy, and three definitions of chaos, i.e. chaos in the sense of Devaney and Wiggins, respectively, and in a strong sense of Li–Yorke. An interesting observation is that a finite-dimensional linear time-varying system can be chaotic in the original sense of Li–Yorke, but cannot have chaos in the strong sense of Li–Yorke, nor in the sense of Devaney in a set containing infinitely many points, and nor in the sense of Wiggins in a set starting from which all the orbits are bounded. A criterion of chaos in the original sense of Li–Yorke is established for finite-dimensional linear time-varying systems. Some basic properties of topological conjugacy are discussed. In particular, it is shown that topological conjugacy alone cannot guarantee two topologically conjugate time-varying systems to have the same topological properties in general. In addition, a criterion of chaos induced by strict coupled-expansion for a certain irreducible transitive matrix is established, under which the corresponding nonlinear system is proved chaotic in the strong sense of Li–Yorke. Two illustrative examples are finally provided with computer simulations for illustration.     

A chaotic system with only one stable equilibrium

If you are given a simple three-dimensional autonomous quadratic system that has only one stable equilibrium, what would you predict its dynamics to be, stable or periodic? Will it be surprising if you are shown that such a system is actually chaotic? Although chaos theory for three-dimensional autonomous systems has been intensively and extensively studied since the time of Lorenz in the 1960s, and the theory has become quite mature today, it seems that no one would anticipate a possibility of finding a three-dimensional autonomous quadratic chaotic system with only one stable equilibrium. The discovery of the new system, to be reported in this Letter, is indeed striking because for a three-dimensional autonomous quadratic system with a single stable node-focus equilibrium, one typically would anticipate non-chaotic and even asymptotically converging behaviors. Although the equilibrium is changed from an unstable saddle-focus to a stable node-focus, therefore the familiar Ši’lnikov homoclinic criterion is not applicable, it is demonstrated to be chaotic in the sense of having a positive largest Lyapunov exponent, a fractional dimension, a continuous broad frequency spectrum, and a period-doubling route to chaos.     

Global asymptotical synchronization of chaotic neural networks by output feedback impulsive control: An LMI approach

In this paper, impulsive control for master–slave synchronization schemes consisting of identical chaotic neural networks is studied. Impulsive control laws are derived based on linear static output feedback. A sufficient condition for global asymptotic synchronization of master–slave chaotic neural networks via output feedback impulsive control is established, in which synchronization is proven in terms of the synchronization errors between the full state vectors. An LMI-based approach for designing linear static output feedback impulsive control laws to globally asymptotically synchronize chaotic neural networks is discussed. With the help of LMI solvers, linear output feedback impulsive controllers can be easily obtained along with the bounds of the impulsive intervals for global asymptotic synchronization. The method is finally illustrated by numerical simulations.     

On estimates of Lyapunov exponents of synchronized coupled systems

Lyapunov exponents of a synchronized coupled system consist of those of the underlying individual systems and the transverse systems, based on a mode decomposition along the synchronization manifold. Estimates of bounds on the Lyapunov exponents (including transverse Lyapunov exponents) are derived. Several examples are used to validate the theoretical estimates.