Impulsive stabilization of chaos in fractional-order systems

Nonlinear Dynamics - This paper considers a class of nonlinear impulsive Caputo differential equations of fractional order, which models chaotic systems. Computer-assisted proof of chaos...     

Proximal and syndetical properties in nonautonomous discrete systems

This paper is mainly concerned with a class of nonautonomous discrete systems $(X, f_{1,\infty})$. New definitions of proximity relations and sensitivity in nonautonomous discrete systems are given. Some relations among $P(f_{1,\infty})$, $L(f_{1,\infty})$, $R(f_{1,\infty})$, $S(f_{1,\infty})$ and $P(f_{1,\infty})(x)$ are derived. And some chaotic properties of $f_{1,\infty}$ are proved.     

Degree Sequences Beyond Power Laws in Complex Networks

Many complex networks possess vertex-degree distributions in a power-law form of $ck^{-\gamma}$, where $k$ is the degree variable and $c$ and $\gamma$ are constants. To better understand the mechanism of power-law formation in real-world networks, it is effective to analyze their degree variable sequences. We had shown before that, for a scale-free network of size $N$ ,if its vertex-degree sequence is $k_11$ , then the length $l$ of the vertex-degree sequence is of order $logN$ . In the present paper, we further study complex networks with more general distributions and prove that the same conclusion holds even for non-network type of complex systems. In addition, we support the conclusion by verifying many real-world network and system examples. We finally discuss some potential applications of the new finding in various fields of science, technology and society.     

Distributed adaptive tracking control of multiple flexible spacecraft under various actuator and measurement limitations

This paper investigates the distributed leader–follower tracking problem for a team of flexible spacecraft over an undirected communication network with uncertain parameters subject to various actuator and measurement limitations. Assuming that at least one team member can receive information from the virtual leader, three scenarios are considered: (i) all the states of the flexible spacecraft can be completely measured and driven, (ii) only the rigid part of the flexible spacecraft can be driven with full state feedback and (iii) only the rigid part of the flexible spacecraft can be measured and driven. In the first case, a continuous adaptive control law is designed by building a unified architecture based on the linear-in-parameter property. In the second case, a distributed adaptive control algorithm is developed with a discontinuous parameter update law by treating the team of flexible spacecraft as two cascading subsystems. In the third case, a distributed adaptive control law is established with feedback from the generalized coordinates, generalized velocities and generalized accelerations of the rigid part of the spacecraft. It is theoretically proved that the closed-loop systems under the three designed adaptive control laws are all convergent to the target states. Finally, three numerical examples are presented to illustrate the effectiveness of the three proposed control laws.     

Fractional-order PWC systems without zero Lyapunov exponents

In this paper, it is shown numerically that a class of fractional-order piece-wise continuous systems, which depend on a single real bifurcation parameter, have no zero Lyapunov exponents but can be chaotic or hyperchaotic with hidden attractors. Although not analytically proved, this conjecture is verified on several systems including a fractional-order piece-wise continuous hyperchaotic system, a piece-wise continuous chaotic Chen system, a piece-wise continuous variant of the chaotic Shimizu-Morioka system and a piece-wise continuous chaotic Sprott system. These systems are continuously approximated based on results of differential inclusions and selection theory, and numerically integrated with the Adams-Bashforth-Moulton method for fractional-order differential equations. It is believed that the obtained results are valid for many, if not most, fractional-order PWC systems.     

On a new hyperchaotic system

A new hyperchaotic system is presented, which has two large positive and one small negative Lyapunov exponent over a large range of parameters. As a consequence, system orbits strongly expand in some directions but rapidly shrink in some other directions. These strong expansions and shrinking lead the system orbits to be more disordered and random. Bifurcation and Poincaré-map analyses further show that the system has very rich bifurcations in different directions and extremely complicated dynamics overall. Spectral analysis shows that the system in the hyperchaotic mode has an extremely broad frequency bandwidth of high magnitudes, verifying its unusual random nature and indicating its great potential for some relevant engineering applications such as secure communications.     

On area-preserving non-hyperbolic chaotic maps: A case study

This paper provides a case study of a two-dimensional area-preserving non-hyperbolic chaotic map, revealing to some new phenomena that have not been well discussed in the literature to date.     

Controlling chaos in an economic model

A Cournot duopoly, with a bounded inverse demand function and different constant marginal production costs, can be modeled as a discrete-time dynamical system, which exhibits complex bifurcating and chaotic behaviors. Based on some essential features of the model, we show how bifurcation and chaos can be controlled via the delayed feedback control method. We then propose and evaluate an adaptive parameter-tuning algorithm for control. In addition, we discuss possible economic implications of the chaos control strategies described in the paper.     

Epidemic spreading in lattice-embedded scale-free networks

We study geographical effects on the spread of diseases in lattice-embedded scale-free networks. The geographical structure is represented by the connecting probability of two nodes that is related to the Euclidean distance between them in the lattice. By studying the standard susceptible-infected model, we found that the geographical structure has great influences on the temporal behavior of epidemic outbreaks and the propagation in the underlying network: the more geographically constrained the network is, the more smoothly the epidemic spreads, which is different from the clearly hierarchical dynamics that the infection pervades the networks in a progressive cascade across smaller-degree classes in Barabási–Albert scale-free networks.     

A simple time-delay feedback anticontrol method made rigorous

An effective method of chaotification via time-delay feedback for a simple finite-dimensional continuous-time autonomous system is made rigorous in this paper. Some mathematical conditions are derived under which a nonchaotic system can be controlled to become chaotic, where the chaos so generated is in a rigorous mathematical sense of Li-Yorke in terms of the Marotto theorem. Numerical simulations are given to verify the theoretical analysis.