Converting a general 3-D autonomous quadratic system to an extended Lorenz-type system
Cuncai Hua Guanrong Chen Qunhong Li Juhong Ge
Discrete & Continuous Dynamical Systems - B 2011, 16(2): 475-488 doi: 10.3934/dcdsb.2011.16.475
A problem of reducing a general three-dimensional (3-D) autonomous quadratic system to a Lorenz-type system is studied. Firstly, under some necessary conditions for preserving the basic qualitative properties of the Lorenz system, the general 3-D autonomous quadratic system is converted to an extended Lorenz-type system (ELTS) which contains a large class of existing chaotic dynamical systems. Secondly, some different canonical forms of the ELTS are obtained with the aid of various nonsingular linear transformations and normalization techniques. Thirdly, the conjugate systems of the ELTS are defined and discussed. Finally, a sufficient condition for the nonexistence of chaos in such ELTS is derived.
keywords: Three-dimensional autonomous quadratic system bifurcation analysis. Lorenz-type system
Polynomial maps with hidden complex dynamics
Xu Zhang Guanrong Chen
Discrete & Continuous Dynamical Systems - B 2017, 22(11): 1-14 doi: 10.3934/dcdsb.2018293

The dynamics of a class of one-dimensional polynomial maps are studied, and interesting dynamics are observed under certain conditions: the existence of periodic points with even periods except for one fixed point; the coexistence of two attractors, an attracting fixed point and a hidden attractor; the existence of a double period-doubling bifurcation, which is different from the classical period-doubling bifurcation of the Logistic map; the existence of Li-Yorke chaos. Furthermore, based on this one-dimensional map, the corresponding generalized Hénon map is investigated, and some interesting dynamics are found for certain parameter values: the coexistence of an attracting fixed point and a hidden attractor; the existence of Smale horseshoe for a subshift of finite type and also Li-Yorke chaos.

keywords: Attractor bifurcation Li-Yorke chaos polynomial map Smale horseshoe
Coupled-expanding maps under small perturbations
Xu Zhang Yuming Shi Guanrong Chen
Discrete & Continuous Dynamical Systems - A 2011, 29(3): 1291-1307 doi: 10.3934/dcds.2011.29.1291
This paper studies the $C^1$-perturbation problem of strictly $A$-coupled-expanding maps in finite-dimensional Euclidean spaces, where $A$ is an irreducible transition matrix with one row-sum no less than $2$. It is proved that under certain conditions strictly $A$-coupled-expanding maps are chaotic in the sense of Li-Yorke or Devaney under small $C^1$-perturbations. It is shown that strictly $A$-coupled-expanding maps are $C^1$ structurally stable in their chaotic invariant sets under certain stronger conditions. One illustrative example is provided with computer simulations.
keywords: coupled-expanding structural stability. perturbation Chaos
Consensus of discrete-time linear multi-agent systems with observer-type protocols
Zhongkui Li Zhisheng Duan Guanrong Chen
Discrete & Continuous Dynamical Systems - B 2011, 16(2): 489-505 doi: 10.3934/dcdsb.2011.16.489
This paper concerns the consensus of discrete-time multi-agent systems with linear or linearized dynamics. An observer-type protocol based on the relative outputs of neighboring agents is proposed. The consensus of such a multi-agent system with a directed communication topology can be cast into the stability of a set of matrices with the same low dimension as that of a single agent. The notion of discrete-time consensus region is then introduced and analyzed. For neurally stable agents, it is shown that there exists an observer-type protocol having a bounded consensus region in the form of an open unit disk, provided that each agent is stabilizable and detectable. An algorithm is further presented to construct a protocol to achieve consensus with respect to all the communication topologies containing a spanning tree. Moreover, for the case where the agents have no poles outside the unit circle, an algorithm is proposed to construct a protocol having an origin-centered disk of radius
$\delta$ ($0<\delta<1$) as its consensus region. Finally, the consensus algorithms are applied to solve formation control problems of multi-agent systems.
keywords: discrete-time linear system multi-agent system formation control. consensus region observer-type protocol Consensus
Synchronization of chaotic systems with time-varying coupling delays
Tingwen Huang Guanrong Chen Juergen Kurths
Discrete & Continuous Dynamical Systems - B 2011, 16(4): 1071-1082 doi: 10.3934/dcdsb.2011.16.1071
In this paper, we study the complete synchronization of a class of time-varying delayed coupled chaotic systems using feedback control. In terms of Linear Matrix Inequalities, a sufficient condition is obtained through using a Lyapunov-Krasovskii functional and differential equation inequalities. The conditions can be easily verified and implemented. We present two simulation examples to illustrate the effectiveness of the proposed method.
keywords: Chaotic System Synchronization Time-varying Delay.
Generalized snap-back repeller and semi-conjugacy to shift operators of piecewise continuous transformations
Wei Lin Jianhong Wu Guanrong Chen
Discrete & Continuous Dynamical Systems - A 2007, 19(1): 103-119 doi: 10.3934/dcds.2007.19.103
In this paper, we attempt to clarify an open problem related to a generalization of the snap-back repeller. Constructing a semi-conjugacy from the finite product of a transformation $f:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ on an invariant set $\Lambda$ to a sub-shift of the finite type on a $w$-symbolic space, we show that the corresponding transformation associated with the generalized snap-back repeller on $\mathbb{R}^{n}$ exhibits chaotic dynamics in the sense of having a positive topological entropy. The argument leading to this conclusion also shows that a certain kind of degenerate transformations, admitting a point in the unstable manifold of a repeller mapping back to the repeller, have positive topological entropies on the orbits of their invariant sets. Furthermore, we present two feasible sufficient conditions for obtaining an unstable manifold. Finally, we provide two illustrative examples to show that chaotic degenerate transformations are omnipresent.
keywords: Shift operator Snap-back repeller Chaotic dynamics.
Delay-induced synchronization transition in small-world Hodgkin-Huxley neuronal networks with channel blocking
Qingyun Wang Xia Shi Guanrong Chen
Discrete & Continuous Dynamical Systems - B 2011, 16(2): 607-621 doi: 10.3934/dcdsb.2011.16.607
We study the evolution of spatiotemporal dynamics and synchronization transition on small-world Hodgkin-Huxley (HH) neuronal networks that are characterized with channel noises, ion channel blocking and information transmission delays. In particular, we examine the effects of delay on spatiotemporal dynamics over neuronal networks when channel blocking of potassium or sodium is involved. We show that small delays can detriment synchronization in the network due to a dynamic clustering anti-phase synchronization transition. We also show that regions of irregular and regular wave propagations related to synchronization transitions appear intermittently as the delay increases, and the delay-induced synchronization transitions manifest as well-expressed minima in the measure for spatial synchrony. In addition, we show that the fraction of sodium or potassium channels can play a key role in dynamics of neuronal networks. Furthermore, We found that the fraction of sodium and potassium channels has different impacts on spatiotemporal dynamics of neuronal networks, respectively. Our results thus provide insights that could facilitate the understanding of the joint impact of ion channel blocking and information transmission delays on the dynamical behaviors of realistic neuronal networks.
keywords: synchronization. time delay Neural network

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