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[en] This article surveys recent progress of results in topology and dynamics based on techniques of closed 1-forms. Our approach lets us draw conclusions about properties of flows by studying homotopical and cohomological features of manifolds. More specifically, a Lusternik-Schnirelmann type theory for closed 1-forms is described, along with the focusing effect for flows and the theory of Lyapunov 1-forms. Also discussed are recent results about cohomological treatment of the invariants cat(X,ξ) and cat1(X,ξ) and their explicit computation in certain examples.
[en] This paper investigates the synchronization problem of general complex networks with fractional-order dynamical nodes. Pinning state feedback controllers have been proved to be effective for synchronization control of fractional-order complex networks. We will show that pinning intermittent controllers are also effective for synchronization control of general fractional-order complex networks. Based on the Lyapunov method and periodically intermittent control method, several low-dimensional criteria are derived for the synchronization of such dynamical networks. Finally, a numerical example is presented to demonstrate the validity and feasibility of the theoretical results.
[en] We consider a two-parameter family of piecewise linear maps in which the moduli of the two slopes take different values. We provide numerical evidence of the existence of some parameter regions in which the Lyapunov exponent and the topological entropy remain constant. Analytical proof of this phenomenon is also given for certain cases. Surprisingly however, the systems with that property are not conjugate as we prove by using kneading theory. (paper)
[en] We present structural improvements of Esseen’s (1969) and Rozovskii’s (1974) estimates for the rate of convergence in the Lindeberg theorem and also compute the appearing absolute constants. We introduce the asymptotically exact constants in the constructed inequalities and obtain upper bounds for them. We analyze the values of Esseen’s, Rozovskii’s, and Lyapunov’s fractions, compare them pairwise, and provide some extremal distributions. As an auxiliary statement, we prove a sharp inequality for the quadratic tails of an arbitrary distribution (with finite second-order moment) and its convolutional symmetrization.
[en] In this paper, the nonlinear model of genetic regulatory networks is described by the Takagi-Sugeno fuzzy model representation with time-varying delays. Due to the highly complicated nonlinear stability and robust stability problems, a fuzzy approximation method is employed to interpolate several linear genetic regulatory networks at different operation points via fuzzy bases to approximate the nonlinear genetic regulatory network. In this context, the methods of the linear matrix inequality (LMI) technique could be employed to simplify the problems related to robust stability of genetic regulatory networks. Further, by involving triple integral terms in Lyapunov-Krasovskii functionals and LMI techniques, the stability criteria for the delayed fuzzy genetic regulatory networks are expressed as a convex combination of LMIs, which can be solved numerically by any LMI solvers. Several numerical examples are given to verify the effectiveness and applicability of the derived approach.
[en] A generalized direct Lyapunov method is put forward for the study of stability and attraction in general time systems of the following types: the classical dynamical system in the sense of Birkhoff, the general system in the sense of Zubov, the general system in the sense of Seibert, the general system with delay, and the general 'input-output' system. For such systems, with the help of generalized Lyapunov functions with respect to two filters, two quasifilters, or two filter bases, necessary and sufficient conditions for stability and attraction are obtained under minimal assumptions about the mathematical structure of the general system
[en] This paper attempts to investigate the stabilization and switching law design for the switched discrete-time systems. A theoretical study of the stabilization and switching law design has been performed using the Lyapunov stability theorem and genetic algorithm. The present results demonstrated that can be applied to cases when all individual subsystems are unstable. Finally, some examples are exploited to illustrate the proposed schemes.
[en] This paper investigates the synchronization of fractional order complex-variable dynamical networks with time-varying coupling. Based on information of the complex network's configuration, an effective adaptive pinning control strategy to adjust simultaneously coupling strength and feedback gain is designed. Besides, we also consider the synchronization in complex networks with time-varying coupling weight. By constructing suitable Lyapunov function and using the presented lemma, some sufficient criteria are derived to achieve the synchronization of fractional order complex-variable dynamical networks under the corresponding update law. The update law is only dependent on the states of the complex dynamical networks, which do not need any other information such as the characteristic of the uncoupled nodes of the networks. Further, the result extends the synchronization condition of the real-variable dynamical networks to the complex-valued field. Finally, the correctness and feasibility of the proposed theoretical results are verified by two examples of fractional complex-variable dynamic networks.
[en] We compute the Lyapunov exponents and the Kolmogorov-Sinai (KS) entropy for a self-bound N-body system that is realized as a convex billiard. This system exhibits truly high-dimensional chaos, and 2N-4 Lyapunov exponents are found to be positive. The KS entropy increases linearly with the numbers of particles. We examine the chaos generating defocusing mechanism and investigate how high-dimensional chaos develops in this system with no dispersing elements. (c) 2000 The American Physical Society
[en] De Rham curves are obtained from a polygonal arc by passing to the limit in repeatedly cutting off the corners: at each step, the segments of the arc are divided into three pieces in the ratio ω : (1-2ω) : ω, where ω element of (0,1/2) is a given parameter. We find explicitly the sharp exponent of regularity of such a curve for any ω. Regularity is understood in the natural parametrization using the arclength as a parameter. We also obtain a formula for the local regularity of a de Rham curve at each point and describe the sets of points with given local regularity. In particular, we characterize the sets of points with the largest and the smallest local regularity. The average regularity, which is attained almost everywhere in the Lebesgue measure, is computed in terms of the Lyapunov exponent of certain linear operators. We obtain an integral formula for the average regularity and derive upper and lower bounds