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A general explicit formula is derived for controlling bifurcations using nonlinear state feedback. This method does not increase the dimension of the system, and can be used to either delay (or eliminate) existing bifurcations or change the stability of bifurcation solutions. The method is then employed for Hopf bifurcation control. The Lorenz equation and Rössler system are used to illustrate the application of the approach. It is shown that a simple control can be obtained to simultaneously stabilize two symmetrical equilibria of the Lorenz system, and keep the symmetry of Hopf bifurcations from the equilibria. For the Rössler system, a control is also obtained to simultaneously stabilize two nonsymmetric equilibria and meanwhile stabilize possible Hopf bifurcations from the equilibria. Computer simulation results are presented to confirm the analytical predictions.
This paper addresses the fundamental problem of complex network synchronizability from a graph-theoretic approach. First, the existing results are briefly reviewed. Then, the relationships between the network synchronizability and network structural parameters (e.g., average distance, degree distribution, and node betweenness centrality) are discussed. The effects of the complementary graph of a given network and some graph operations on the network synchronizability are discussed. A basic theory based on subgraphs and complementary graphs for estimating the network synchronizability is established. Several examples are given to show that adding new edges to a network can either increase or decrease the network synchronizability. To that end, some new results on the estimations of the synchronizability of coalescences are reported. Moreover, a necessary and sufficient condition for a network and its complementary network to have the same synchronizability is derived. Finally, some examples on Chua circuit networks are presented for illustration.
This article is concerned with the cluster consensus control problem for multiagent linear systems with a directed communication topology, where only relative output measurements of neighboring agents are available to each agent. Motivated by the pinning control technique, both static and dynamic intermittent output control strategies are proposed. Using Lyapunov functions, sufficient conditions are developed to ensure cluster consensus with existence-guaranteed control parameters. Both periodic and nonperiodic operations of intermittent controllers are investigated. Finally, the effectiveness of the theoretical results is demonstrated by a simulation example.
Small-world networks are highly clustered networks with small average distance among the vertices. There are many natural and technological networks that present this kind of connections. We study the phase synchronization of small-world chaotic oscillator networks in this paper. We find that for Rössler oscillator networks, in the synchronous regime, the oscillation phases are locked, while the amplitudes vary chaotically. We further show the dependence of phase synchronization on the network coupling strength, the product of the shortcuts-adding probability and the number of chaotic oscillators, as well as the maximal frequency mismatch.
Symmetric block encryption schemes, designed on invertible two-dimensional chaotic maps on a torus or a square, prove feasible and secure for real-time image encryption according to the commonly used criteria given in the literature. In this paper, a typical map of this kind, namely, the baker map, is further extended to be three-dimensional and then used to speed up image encryption while retaining its high degree of security. The proposed algorithm is described in detail, along with its security analysis and implementation. Experimental results show that this three-dimensional baker map is 2–3 times faster than the two-dimensional one, showing its great potential in real-time image encryption applications.
Being a pair of dual concepts, the normalized distance and similarity measures are very important tools for decision-making and pattern recognition under intuitionistic fuzzy sets framework. To be more effective for decision-making and pattern recognition applications, a good normalized distance measure should ensure that its dual similarity measure satisfies the axiomatic definition. In this paper, we first construct some examples to illustrate that the dual similarity measures of two nonlinear distance measures introduced in [A distance measure for intuitionistic fuzzy sets and its application to pattern classification problems, \emph{IEEE Trans. Syst., Man, Cybern., Syst.}, vol.~51, no.~6, pp. 3980--3992, 2021] and [Intuitionistic fuzzy sets: spherical representation and distances, \emph{Int. J. Intell. Syst.}, vol.~24, no.~4, pp. 399--420, 2009] do not meet the axiomatic definition of intuitionistic fuzzy similarity measure. We show that (1) they cannot effectively distinguish some intuitionistic fuzzy values (IFVs) with obvious size relationship; (2) except for the endpoints, there exist infinitely many pairs of IFVs, where the maximum distance 1 can be achieved under these two distances; leading to counter-intuitive results. To overcome these drawbacks, we introduce the concepts of strict intuitionistic fuzzy distance measure (SIFDisM) and strict intuitionistic fuzzy similarity measure (SIFSimM), and propose an improved intuitionistic fuzzy distance measure based on Jensen-Shannon divergence. We prove that (1) it is a SIFDisM; (2) its dual similarity measure is a SIFSimM; (3) its induced entropy is an intuitionistic fuzzy entropy. Comparative analysis and numerical examples demonstrate that our proposed distance measure is completely superior to the existing ones.
A feedback control design method was proposed in Chen and Lai (1996, Internat. J. Bifur, Chaos 6, 1341–1349) towards anti-control of chaos by means of making an arbitrary discrete-time dynamical system behave chaotically, where the feedback control can force all the Lyapunov exponents of the controlled system become strictly positive. Since a control sequence so designed is not uniquely determined, random mechanism can be used to select a suitable control-gain sequence. The behavior (distribution) of the controlled Lyapunov exponents generated by the proposed method is important and hence calls for further statistical analysis. Using Monte Carlo simulations, we examine the distribution empirically in this paper.
In this paper, a new strategy is developed for controlling chaos and hyperchaos in a nonlinear system via switching modulation of systems parameters. The control method is illustrated by examples of the third-order chaotic Chua's circuit and a sixth-order coupled hyperchaotic Chua's circuit. Numerical simulations show that this method is very effective.
This paper is concerned with chaotification of discrete dynamical systems in finite-dimensional real spaces, via feedback control techniques. A chaotification theorem for one-dimensional discrete dynamical systems and a chaotification theorem for general higher-dimensional discrete dynamical systems are established, respectively. The controlled systems are proved to be chaotic in the sense of Devaney. In particular, the maps corresponding to the original systems and designed controllers are only required to satisfy some mild assumptions on two very small disjoint closed subsets in the domains of interest. This condition is weaker than those in the existing relevant literature.
The existence of breaking wave solutions of the second class of singular nonlinear wave equations is proved by methods from the dynamical systems theory. For the second class of singular nonlinear traveling wave equations, dynamical behaviors of the traveling wave solutions are completely classified and thoroughly discussed. Corresponding to some bounded orbits of the traveling systems, exact parametric representations of traveling wave solutions are derived within different parameter regions of the parameter space.
Anticontrol of chaos by making a nonchaotic system chaotic has led to the discovery of some new chaotic systems, particularly the continuous-time three-dimensional autonomous Chen's equation with only two quadratic terms. This paper further investigates some basic dynamical properties and various bifurcations of Chen's equation, thereby revealing its different features from some other chaotic models such as its origin, the Lorenz system.
