It is well known that a finite-dimensional linear system cannot be chaotic. In this article, by introducing a weak topology into a two-dimensional Euclidean space, it shows that Li–Yorke chaos can be generated by a linear map, where the weak topology is induced by a linear functional. Some examples of linear systems are presented, some are chaotic while some others regular. Consequently, several open problems are posted.
This paper investigates the global synchronization problem of complex dynamical networks consisting of a drive network and a response network. Using the decentralized and variable structure control techniques, a control law is derived which guarantees the global exponential synchronization of the drive–response network even with the presence of input nonlinearity. The proposed controller is applicable to complex networks with general nonlinear dynamical nodes. Chaotic networks are used as illustrative examples to demonstrate the effectiveness of the proposed control scheme.
We propose and study a novel evolving network model with the new concept of local-world connectivity, which exists in many physical complex networks. The local-world evolving network model represents a transition between power-law and exponential scaling, while the Barabási–Albert scale-free model is only one of its special (limiting) cases. We found that this local-world evolving network model can maintain the robustness of scale-free networks and can improve the network reliance against intentional attacks, which is the inherent fragility of most scale-free networks.
In this paper, the problem of approximating hidden chaotic attractors of a general class of nonlinear systems is investigated. The parameter switching (PS) algorithm is utilized, which switches the control parameter within a given set of values with the initial value problem numerically solved. The PS-generated attractor approximates the attractor obtained by averaging the control parameter with the switched values, which represents the hidden chaotic attractor. The hidden chaotic attractors of a generalized Lorenz system and the Rabinovich-Fabrikant system are simulated for illustration.
This paper studies synchronization of a dynamical complex network consisting of nodes being generalized Lorenz chaotic systems and connections created with transmitted synchronizing signals. The focus is on the robustness of the network synchronization with respect to its topology. The robustness is analyzed theoretically for the case of two nodes with two-sided (bidirectional) connections, and numerically for various cases with large numbers of nodes. It is shown that, unless a certain minimal coherent topology is present in the network, synchronization is always preserved. While for a minimal network where synchronization is global, the resulting synchrony reduces to semi-global if redundant connections are added.
This note points out that the assertions of (Chen's attractor exists if Lorenz repulsor exists: The Chen system is a special case of the Lorenz system, CHAOS 23, 033108 (2013)) are groundless and incorrect. The failure of that criticism actually supports the strong standing of the Chen system.
This paper derives some sufficient conditions for asymptotic stability of neural networks with constant or time-varying delays. The Lyapunov-Krasovskii stability theory for functional differential equations and the linear matrix inequality (LMI) approach are employed to investigate the problem. It shows how some well-known results can be refined and generalized in a straightforward manner. For the case of constant time delays, the stability criteria are delay-independent; for the case of time-varying delays, the stability criteria are delay-dependent. The results obtained in this paper are less conservative than the ones reported so far in the literature and provides one more set of criteria for determining the stability of delayed neural networks.
By using the approach of dynamical systems, the bifurcations of phase portraits for the traveling system of the Kudryashov–Sinelshchikov equation with ν = δ = 0 are studied, in different parametric regions of (α, c)-parametric plane. Corresponding to different phase orbits of the traveling system, more than 26 exact explicit traveling wave solutions are derived. The dynamics of singular nonlinear traveling system is completely determined.
Complex networks are wide spread in the real world, arising in fields as disparate as sociology, physics and biology. The information spreading through a complex network is often associated with time delays due to the finite speeds of signal transmission over a distance. Hence, complex networks with coupling delays have gained increasing attention in various fields of science and engineering today. In this paper, based on the theory of asymptotic stability of linear time-delay systems, synchronization stability in complex dynamical networks with coupling delays is investigated, and we derive novel criteria of synchronization state for both delay-independent and delay-dependent stabilities. As illustrative examples, we use the networks with coupling delays and a given coupling scheme to test the theoretical results.
No abstract is provided for this article.
This paper addresses the global consensus problem of multi-agent systems with a leader-follower communication topology consisting of Lur'e type of node dynamics. A decomposition approach is proposed to convert the consensus of a high-dimensional Lur'e network into the test of a set of matrix inequalities whose dimensions are the same as a single node system. The notion of global consensus region is then introduced and analyzed. A necessary and sufficient condition is derived for the existence of a consensus protocol to guarantee a desirable unbounded consensus region. A multi-step design procedure is given for constructing such a consensus protocol, which maintains a favorable decoupling property. The effectiveness of the theoretical results is demonstrated through a network of Chua's circuits.
No abstract is provided for this article.
No abstract is provided for this article.
This article develops an event‐based control strategy to achieve finite‐time consensus for a chain of nonholonomic multi‐agent systems subject to unknown disturbances. First, the controller switching time is determined for the state coupling of the systems in the chain. Then, adaptive event‐based integral sliding‐mode control protocols with novel triggering functions are proposed, which can avoid the inherent singularity problem. To this end, adaptive laws are designed to deal with the disturbances without knowing exactly their upper bounds, which can adjust the gains of the controllers automatically. Moreover, an algorithm is developed to compute the sliding‐mode coefficients, adaptive parameters as well as the controller switching time. It is shown that all states of the systems in the chain can achieve consensus in finite time against disturbances. Furthermore, the Zeno behavior is proved to be excluded by showing that the inter‐execution time is lower‐bounded. Finally, a numerical example is presented to demonstrate the effectiveness of the proposed framework and methodology.
