DCDS-B

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.

DCDS-B

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.

DCDS

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.

DCDS-B

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.

DCDS-B

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.

DCDS

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.

DCDS-B

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.