Simple polynomial classes of chaotic jerky dynamics

Third-order explicit autonomous differential equations, commonly called jerky dynamics, constitute a powerful approach to understand the properties of functionally very simple but nonlinear three-dimensional dynamical systems that can exhibit chaotic long-time behavior. In this paper, we investigate the dynamics that can be generated by the two simplest polynomial jerky dynamics that, up to these days, are known to show chaotic behavior in some parameter range. After deriving several analytical properties of these systems, we systematically determine the dependence of the long-time dynamical behavior on the system parameters by numerical evaluation of Lyapunov spectra. Some features of the systems that are related to the dependence on initial conditions are also addressed. The observed dynamical complexity of the two systems is discussed in connection with the existence of homoclinic orbits.     

Robust stabilization for a class of uncertain dynamical systems with time delay

In this paper, a new and simple approach whereby we derive several sufficient conditions on robust stabilizability for a class of uncertain dynamical systems with time delay is presented. Some analytical methods and the Bellman-Gronwall inequality are employed to investigate these sufficient conditions. The notable features of the results obtained are their simplicity in testing the stability of uncertain dynamical systems with time delay and their clarity in giving insight into system analysis. Finally, several numerical examples are given to demonstrate the utilization of the results.The authors would like to acknowledge the many helpful comments provided by the reviewer. Particularly, in the light of these comments, the proof of Theorem 3.1 has been considerably shortened.     

Statistical analysis of Lyapunov exponents from time series: A Jacobian approach

In this paper, we provide a statistical analysis for the Lyapunov exponents estimated from time series. Through the Jacobian estimation approach, the asymptotic distributions of the estimated Lyapunov exponents of discrete-time dynamical systems are studied and characterized based on the time series. Some new results under weak conditions are obtained. The theoretical results presented in the paper are illustrated by numerical simulations.     

Robust stabilization of uncertain linear dynamical systems with time-varying delay

In this paper, the problem of the robust stabilization for a class of uncertain linear dynamical systems with time-varying delay is considered. By making use of an algebraic Riccati equation, we derive some sufficient conditions for robust stability of time-varying delay dynamical systems with unstructured or structured uncertainties. In our approach, the only restriction on the delay functionh(t) is the knowledge of its upper boundh ^–. Some analytical methods are employed to investigate these stability conditions. Since these conditions are independent of the delay, our results are also applicable to systems with perturbed time delay. Finally, a numerical example is given to illustrate the use of the sufficient conditions developed in this paper.     

On the uniqueness of a cycle in an asymmetric three-dimensional model of a molecular repressilator

Some sufficient conditions are established for the nonlinear dynamical systems of dimension 3 that simulate the functioning of a molecular repressilator to have exactly one cycle. Some constructive method is described for finding an invariant surface that contains this cycle.     

Feigenbaum刚性和界于简单与混沌之间的动力系统

本文主要综述了关于一类光滑动力系统族周期倍分支现象中通有比例规则的最新研究进展，而这一规则是由Ｆｅｉｇｅｎｂａｕｍ在美国、Ｃｏｕｌｌｅｔ与Ｔｒｅｓｓｅｒ在法国分别独立发现的，文中还阐述了一类界于简单与混沌之间的动力系统的几何性质。     

On the uniqueness of solutions to a class of discontinuous dynamical systems

The study of uniqueness of solutions of discontinuous dynamical systems has an important implication: multiple solutions to the initial value problem could not be found in real dynamical systems; also the (attracting or repulsive) sliding mode is inherently linked to the uniqueness of solutions. In this paper a strengthened Lipschitz-like condition for differential inclusions and a geometrical approach for the uniqueness of solutions for a class of Filippov dynamical systems are introduced as tools for uniqueness. Several theoretical and practical examples are discussed.     

An approach of partial control design for system control and synchronization

In this paper, a general approach of partial control design for system control and synchronization is proposed. It turns control problems into simpler ones by reducing their control variables. This is realized by utilizing the dynamical relations between variables, which are described by the dynamical relation matrix and the dependence–influence matrix. By adopting partial control theory, the presented approach provides a simple and general way to stabilize systems to their partial or whole equilibriums, or to synchronize systems with their partial or whole states. Further, based on this approach, the controllers can be simplified. Two examples of synchronizing chaotic systems are given to illustrate its effectiveness.     

BIFURCATIONS OF TRAVELLING WAVE SOLUTIONS FOR THE GENERALIZED DODD-BULLOUGH-MIKHAILOV EQUATION

In this paper, the generalized Dodd-Bullough-Mikhailov equation is studied. The existence of periodic wave and unbounded wave solutions is proved by using the method of bifurcation theory of dynamical systems. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given.Some exact explicit parametric representations of the above travelling solutions are obtained.     

Sample-Path Stability of Non-Stationary Dynamic Economic Systems

The goal of this paper is to introduce and illustrate a new approach to the stability analysis of sample-paths of non-linear stochastic economic models with non-stationary components. We place our study within the mathematical theory of random dynamical systems and apply the concept of a random fixed point which is tailor-made for the study of the long-term behavior of sample-paths in stochastic systems. The main tool for the application of this approach is a Banach-type fixed point theorem for non-stationary random dynamical systems which is proved here. The concept and the theorem are thoroughly explained and illustrated by examples from stochastic growth theory.     

Sub-manifold and traveling wave solutions of Ito's 5th-order mKdV equation

In this paper, we study Ito''s 5th-order mKdV equation with the aid of symbolic computation system and by qualitative analysis of planar dynamical systems. We show that the corresponding higher-order ordinary differential equation of Ito''s 5th-order mKdV equation, for some particular values of the parameter, possesses some sub-manifolds defined by planar dynamical systems. Some solitary wave solutions, kink and periodic wave solutions of the Ito''s 5th-order mKdV equation for these particular values of the parameter are obtained by studying the bifurcation and solutions of the corresponding planar dynamical systems.     

具有时滞和非时滞耦合的复杂动态网络的自适应同步

对于同时具有时滞耦合和非时滞耦合的复杂动态网络,以时滞动力系统的稳定性理论为基础,设计了全新的自适应控制器,同时给出了一些简单而又一般的网络同步化准则.对由三个节点组成的动态网络的同步化进行了数值模拟,使得理论分析与数值模拟得到了相互验证.     

Types of convergence for sequences of probability densities and attractors

Some necessary and sufficient conditions on densities convergence for the existence of an ergoidc mixing, or exact dynamical system on a probability space given in [8] are extended to a measure space to obtain an ergodic, mixing or exact dynamical system on this measure space. More general sufficient conditions are given for the existence of those kinds of dynamical systems; as a consequence of these conditions it is obtained an ergodic, exact attractor for the orbits of almost every phase state. This orbits’ behaviour recall the thermodynamical evolution of systems from nonequilibrium to equilibrium states     

HOPF BIFURCATION ANALYSIS OF A 3D CHAOTIC SYSTEM

In this paper,a 3D chaotic system with multi-parameters is introduced. The dynamical systems of the original ADVP circuit and the modified ADVP model are regarded as special examples to the system.Some basic dynamical behaviors such as the stability of equilibria,the existence of Hopf bifurcation are investigated.We analyse the Hopf bifurcation of the system comprehensively using the first Lyapunov coefficient by precise symbolic computation.As a result,in fact we have studied the further dynamical behaviors.     

Multicriteria optimization of convex dynamical systems

We suggest an approach to the solution of multicriteria optimization problems for dynamical systems described by differential inclusions. The investigation is restricted to dynamical systems with concave differential inclusion, for which the trajectory tube is convex. Such systems are typical of economic models. We assume that the criteria for the choice of the solution depend on the system state at a given terminal time and are related to it by sufficiently arbitrary functions. The approach is based on the interactive visualization of the Pareto frontier, which is carried out by approximating the reachable set of the dynamical system and the Edgeworth-Pareto set of feasible criteria vectors.     

On the Lagrangian form of the variational equations of Lagrangian dynamical systems

An extremal principle for obtaining the variational equations of a Lagrangian system is reviewed and formalized. Formalization is accomplished by relating the new Lagrangian function γ needed in such scheme to a prolongation of the original Lagrangian L. This formalization may be regarded as a necessary step before using the approach for stablishing nonintegrability of dynamical systems, or before applying it to analyse chaos-producing perturbations of integrable Lagrangian systems. The configuration manifold in which γ is defined is the double tangent bundle T(TQ) of the original configuration manifold Q modulo a flip mapping in such manifold. Our main result establishes that both the Euler–Lagrange equations and the corresponding variational equations of the original system can be viewed as the Lagrangian vector field associated with the composition of the first prolongation of the original Lagrangian with a flip mapping. Some applications of the approach to chaos and integrability issues are discussed.     

Constrained controllability of nonlinear systems

In the paper, infinite-dimensional dynamical systems describedby nonlinear abstract differential equations are considered.Using a generalized open-mapping theorem sufficient conditionsfor constrained exact local controllability are formulated andproved. It is generally assumed that the values of controlsare in a convex and closed cone with the vertex at zero. Asan illustrative example. the constrained exact local controllabilityproblem for nonlinear delayed dynamical system is solved indetail. Some remarks and comments on the existing results forcontrollability of nonlinear dynamical systems are also presented.     

Stability and robustness of quasi-linear impulsive hybrid systems

Both hybrid dynamical systems and impulsive dynamical systems are studied extensively in the literature. However, impulsive hybrid systems are not yet well studied. Nonetheless, many physical systems exhibit both system switching and impulsive jump phenomena. This paper investigates stability and robust stability of a class of quasi-linear impulsive hybrid systems by using the methods of Lyapunov functions and Riccati inequalities. Sufficient conditions for stability and robust stability of those systems are established. Some examples are given to illustrate the applicability of our results.     

On numerical approaches to the analysis of topology of the phase space for dynamical integrability

In this paper we consider the possibility to use numerical simulations for a computer assisted qualitative analysis of dynamical systems. We formulate a rather general method of recovering the obstructions to dynamical integrability for the systems that after reduction have a small number of degrees of freedom. We generalize this method using the results of KAM theory and stochastic approaches to the families of parameter depending systems. This permits to localize possible integrability regions in the parameter space. We give some examples of application of this approach to dynamical systems having a mechanical origin.     

On Dynamics Related to a Class of Numeration Systems

 In this paper, we investigate the class of numeration systems and we study the associated dynamical systems, called odometers. It is shown that these odometers are measure-theoretically isomorphic to rank one transformations on the unit interval, constructed by a cutting-stacking method. Furthermore, a symbolic coding leads to isomorphic shift systems arising from substitutions. Some skew products of the odometers by cocycles related to the sum of digits are shown to be ergodic.