Universal theory of dynamical chaos in nonlinear dissipative systems of differential equations

The article presents a new universal theory of dynamical chaos in nonlinear dissipative systems of differential equations, including autonomous and nonautonomous ordinary differential equations (ODE), partial differential equations, and delay differential equations. The theory relies on four remarkable results: Feigenbaum’s period doubling theory for cycles of one-dimensional unimodal maps, Sharkovskii’s theory of birth of cycles of arbitrary period up to cycle of period three in one-dimensional unimodal maps, Magnitskii’s theory of rotor singular point in two-dimensional nonautonomous ODE systems, acting as a bridge between one-dimensional maps and differential equations, and Magnitskii’s theory of homoclinic bifurcation cascade that follows the Sharkovskii cascade. All the theoretical propositions are rigorously proved and illustrated with numerous analytical examples and numerical computations, which are presented for all classical chaotic nonlinear dissipative systems of differential equations.     

Chaotic attractors in nonlinear dissipative systems of ordinary differential equations

A mechanism is proposed describing the formation of irregular attractors in a wide class of three-dimensional nonlinear autonomous dissipative systems of ordinary differential equations with singular cycles. The attractors of such systems, called singular attractors, lie on two-dimensional surfaces in the phase space and have no positive Lyapunov exponents. In all systems of this class the onset of chaos follows the same universal mechanism: a cascade of Feigenbaum’s period doubling bifurcations, a subharmonic cascade of Sharkovskii’s bifurcations, and eventually a homoclinic cascade. All classical chaotic systems, including Lorenz, Rössler, and Chua systems, satisfy these conditions.     

A scenario for the creation of chaotic attractors in Chua’s system

We investigate a scenario for the creation of irregular chaotic attractors in Chua’s system. We show that irregular attractors in Chua’s system are created by those and only those mechanisms that characterize Lorenz, Rössler, and other dissipative nonlinear systems described by ordinary differential equations. These mechanisms include cascades of Feigenbaum period doubling bifurcations, subharmonic cascades of cycle bifurcations in Sharkovskii’s order, and homoclinic cascades of bifurcations.     

Bifurcations in non-autonomous scalar equations

In a previous paper we introduced various definitions of stability and instability for non-autonomous differential equations, and applied these to investigate the bifurcations in some simple models. In this paper we present a more systematic theory of local bifurcations in scalar non-autonomous equations.     

奇异耦合分数阶微分方程组Dirichlet型边值问题的正解

研究了一非线性奇异非自治耦合半正分数阶微分方程组Dirichlet型边值问题.利用Schauder不动点定理,获得了该非自治耦合分数阶微分方程组Dirichlet型边值问题的正解.     

Effective approaches to explore rich dynamics of delay-differential equations

Discrete models are proposed to delve into the rich dynamics of nonlinear delayed systems under Euler discretization, such as backwards bifurcations, stable limit cycles, multiple limit-cycle bifurcations and chaotic behavior. The effect of breaking the special symmetry of the system is to create a wide complex operating conditions which would not otherwise be seen. These include multiple steady states, complex periodic oscillations, chaos by period doubling bifurcations. Effective computation of multiple bifurcations, stable limit cycles, symmetrical breaking bifurcations and chaotic behavior in nonlinear delayed equations is developed.     

Integrable Duffing’s maps and solutions of the Duffing equation

In the numerical integration of nonlinear differential equations, discretization of the nonlinear terms poses extra ambiguity in reducing the differential equation to a discrete difference equation. As for the cubic nonlinear Schrodinger equation, it was well known that there exists the corresponding discrete soliton system. Here, representing the discrete systems by the mappings, we explore structure of the integrable mappings. We introduce the first kind and the second kind of Duffing’s map, and investigate temporal evolution of the orbits. Although the smooth periodic orbits are in accord with the solutions of the Duffing equation, the integrable Duffing’s maps provide much wider variety of orbits.     

New theory of spatio-temporal chaos in nonlinear systems of partial differential equations

It is shown that there exists one universal scenario of transition to chaos in three considered systems of nonlinear partial differential equations of diffusion type. It is the subharmonic cascade of bifurcations of two-dimensional tori along one or both frequencies. So, it is shown that the new universal Feigenbaum-Sharkovskii-Magnitskii theory of dynamical chaos in nonlinear differential equations is applicable also for nonlinear partial differential equations of diffusion type. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Space–time chaos in a system of reaction–diffusion equations

We find conditions for the bifurcation of periodic spatially homogeneous and spatially inhomogeneous solutions of a three-dimensional system of nonlinear partial differential equations describing a soil aggregate model. We show that the transition to diffusion chaos in this model occurs via a subharmonic cascade of bifurcations of stable limit cycles in accordance with the universal Feigenbaum–Sharkovskii–Magnitskii bifurcation theory.     

Approximate controllability of the non-autonomous impulsive evolution equation with state-dependent delay in Banach spaces

In this paper, we consider the non-autonomous semilinear impulsive differential equations with state-dependent delay. The approximate controllability results of the first-order systems are obtained in a separable reflexive Banach space, which has a uniformly convex dual. In order to establish sufficient conditions of the approximate controllability of such a system, we have used the theory of linear evolution systems, properties of the resolvent operator and Schauder’s fixed point theorem. Finally, we provide two concrete examples to validate our results.     

The fundamental existence theorem on invariant fiber bundles

For discrete dynamical systems the theory of invariant manifolds is well known to be of vital importance. In terms of difference equations this theory is basically concerned with autonomous equations. However, the crucial and currently most difficult questions in this field are related to non-periodic, in particular chaotic motions. Since this topic - even in the autonomous context is an intrinsically time-variant matter. There is and urgent need for a non-autonomous version of invariant manifold theory. In this paper we present we present a very general version of the classical result on stable and unstable manifolds for hyperbolic fixed points of diffeomorphisms. In fact, we drop the assumption of invertibility of the mapping, we consider non-autonomous difference equations rather than mappings In effect, we generalize the notion of invariant manifold to the concept of invariant fiber bundle.     

Stability analysis for a class of functional differential equations and applications

The problem of Lyapunov stability for functional differential equations in Hilbert spaces is studied. The system to be considered is non-autonomous and the delay is time-varying. Known results on this problem are based on the Gronwall inequality yielding relative conservative bounds on nonlinear perturbations. In this paper, using more general Lyapunov-Krasovskii functional, neither model variable transformation nor bounding restriction on nonlinear perturbations is required to obtain improved conditions for the global exponential stability of the system. The conditions given in terms of the solution of standard Riccati differential equations allow to compute simultaneously the two bounds that characterize the stability rate of the solution. The proposed method can be easily applied to some control problems of nonlinear non-autonomous control time-delay systems.     

Border-collision bifurcations on a two-dimensional torus

This paper studies resonance phenomena in a piecewise-smooth dynamical system with external periodic action and examines transitions to chaos via border-collision bifurcations of cycles on a two-dimensional torus. As an example we consider a control system with pulse-width modulation described by a three-dimensional set of piecewise-linear non-autonomous equations. It is shown that the domains of synchronization of quasiperiodic oscillations for piecewise-smooth dynamical systems differ in an essential way from the classical Arnol'd tongues. The difference lies in the inner structure and bifurcational transitions. There are two different kinds of synchronization domains, one of which contains regions of bistability. The structure of border-collision bifurcation boundaries of synchronization tongues and transitions to chaos via border-collision bifurcations of cycles on a two-dimensional torus are described in detail.     

Local Hopf bifurcation and global existence of periodic solutions in a kind of physiological system

The dynamics of a physiological control systems described by a first-order nonlinear delay differential equations are investigated. we proved that a sequence of Hopf bifurcations occur at the positive equilibrium as the delay increases. Explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived, using the theory of normal form and center manifold. Global existence of periodic solutions are established using a global Hopf bifurcation result due to Wu [Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc. 350 (1998) 4799–4838].     

时标上一类非自治捕食系统的周期解(英文)

本文在时标上运用迭合度理论中的Gaines和Mawhin连续性定理研究了一类非自治捕食系统的周期解的存在性,得到了此模型周期解存在的充分条件,该方法可将证明连续和离散微分方程的周期解的存在性统一起来.     

ON EXPONENTIAL STABILITY OF NON-AUTONOMOUS STOCHASTIC SEMILINEAR EVOLUTION EQUATIONS

11MroductlonThe purpose ofthls paper Is to Investigate eWone尬lal stability of*theity mild solutions forcenain Hilbert space－Mued stochastlc evoMlon eqll砒ions,Roughy spe出0ng;we cons讪r山efollowing equation:I 伏I＝*x,＋风Il加L十从L,剧dWn,c〔瓜＋咖。(””””“”(11)D 人n 二x．Where A Is the Infinlteslmalgener砒or ofa certain几semigroup S(t),t＞0;on H and F(t;、)and B(t;·)are In general nonlinear mappings from H to H and H to L(x,H),the family ofall bounded linear operators from …     

Classification of solutions of a system of nonlinear diffusion equations in a neighborhood of a bifurcation point

The theory of reaction-diffusion systems in a neighborhood of a bifurcation point is considered. The basic types of space-time ordering, diffusion chaos in such systems, and sequences of bifurcations leading to complication of solutions are studied. A detailed discussion is given of a hierarchy of simplified models (one- and two-dimensional mappings, systems of ordinary differential equations, and others) which make it possible to carry out a qualitative analysis of the problem studied in the case of small regions. A number of generalizations of the equations studied and the simplest types of ordering in the two-dimensional case are described.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 28, 207–313, 1986.     

On the reflecting function and the qualitative behavior of solutions of some non-autonomous differential equations

In this article, we use the Mironenko''s method to discuss the qualitative behavior of some non-autonomous differential equations. We study the structure of the reflecting functions of the simplest differential equations, and obtain some sufficient conditions under which these equations have the rational reflecting functions. We apply the obtained results to discuss the numbers of periodic solutions of the non-autonomous differential systems and derive some sufficient conditions for a critical point of theirs to be a center.     

Nonlinear Boundary Value Problems Describing Mobile Carrier Transport in Semiconductor Devices

The present paper describes mobile carrier transport in semiconductor devices with constant densities of ionized impurities. For this purpose we use one-dimensional partial differential equations. The work gives the proofs of global existence of solutions of systems of such kind, their bifurcations and their stability under the corresponding assumptions.     

非锥映射的不动点指数计算及其应用

本文应用锥理论获得了一类不是锥映射的非线性算子不动点指数计算的若干结果,并将所得结果应用于非线性椭圆型偏微分方程边值问题.