Chaotification of discrete dynamical systems via impulsive control

This Letter is concerned with chaotification of discrete dynamical systems in finite-dimensional real spaces, via impulsive control techniques. Chaotification theorems for one-dimensional discrete dynamical systems and general higher-dimensional discrete dynamical systems are derived, respectively, whether the original systems are stable or not. Finally, the effectiveness of the theoretical results is illustrated by some numerical examples.     

二维抛物线离散映射的动力学研究

由两个一维抛物线离散映射作推广并非线性耦合,实现了一个新的二维抛物线离散映射.利用不动点稳定性分析和映射分岔分析,研究了所提出的二维离散映射的复杂动力学行为及其吸引子的演变过程,阐述了它所特有的共存分岔模式和快慢周期振荡效应等动力学特性.研究结果表明:二维抛物线离散映射具有动力学特性调节和动态幅度调节的两个功能不同的控制参数,存在Hopf分岔、分岔模式共存、锁频和周期振荡快慢效应等非线性物理现象.并基于微控制器实现的数字电路验证了相应的理论分析和数值仿真结果. 关键词： 二维离散映射 分岔 吸引子 参数     

A “No-Go” theorem for the existence of a discrete action principle

In this paper, we study the problem of the existence of a least-action principle for invertible, second-order dynamical systems, discrete in time and space. We show that, when the configuration space is finite and arbitrary state transitions are allowed, a least-action principle does not exist for such systems. We dichotomize discrete dynamical systems with infinite configuration spaces into those of finite type for which this theorem continues to hold, and those not of finite type for which it is possible to construct a least-action principle. We also show how to recover an action, by restriction of the phase space of certain second-order discrete dynamical systems. We provide numerous examples to illustrate each of these results.     

Symmetrical dynamical characteristic of peak and valley current-mode controlled single-phase H-bridge inverter

Complex dynamical phenomenon was studied in the single phase H-bridge inverter which was controlled by either a peak current or a valley current. The state functions and the discrete iterative map equations were established to analyze the dynamical phenomenon in the single phase H-bridge inverter. The dynamical characteristics of the single phase H- bridge inverter, such as time domain waveform diagram, bifurcation diagram, and folding map, were obtained by using the numerical calculation when the circuit parameters varied in specific range. Moreover, the simulation results were obtained by using the OrCAD-PSpice software to validate the numerical calculation. Both the numerical calculation and the circuit simulation show that the symmetrical dynamical phenomenon occurs in the single phase H-bridge inverter controlled by the peak current or the valley current.     

忆阻混沌振荡器的动力学分析

忆阻器(memristor)是一种有记忆功能的非线性电阻器,它是除电阻器、电容器和电感器之外的第四种基本电路元件.采用一个具有光滑磁控特性曲线的忆阻器和一个负电导替换蔡氏振荡器中的蔡氏二极管,导出了一个基于忆阻器的振荡器电路.采用常规的动力学分析手段研究了电路参数和初始条件变化时该光滑忆阻振荡器的动力学特性.研究结果表明,光滑忆阻振荡器与一般的混沌系统完全不同,它的动力学行为除了与电路参数有关外,还极端依赖于电路的初始条件,存在瞬态混沌和状态转移等奇异的非线性物理现象.     

Nonlinear nonautonomous discrete dynamical systems from a general discrete isomonodromy problem

Nonlinear nonautonomous discrete dynamical systems (DDS) whose continuum limits are the well-known Painlevé equations, have recently arisen in models of quantum gravity. The Painlevé equations are believed integrable because each is the isomonodromy condition for an associated linear differential equation. However, not every DDS with an integrable continuum limit is necessarily integrable. Which of the many discrete versions of the Painlevé equations inherit their integrability is not known. How to derive all their integrable discrete versions is also not known. We provide a systematic method of attacking these questions by giving a general discrete isomonodromy problem. Discrete versions of the first and second Painlevé equations are deduced from this general problem.     

脉冲序列控制双断续导电模式BIFRED变换器的动力学建模与多周期行为

根据BIFRED变换器四种不同的工作模式, 建立了脉冲序列(PT)控制双断续导电模式(DCM-DCM) BIFRED变换器的分段光滑线性时变模型, 数值仿真得到了负载不同时的时域波形和相轨图. 通过分析在一个开关周期内输出电容的电荷变化量, 导出了PT控制DCM-DCM BIFRED变换器的一维近似离散映射模型, 研究了随负载电阻和输入电压变化的多周期行为, 并进行了稳定性分析和负载取阻范围估算. 研究发现, 一维近似离散映射模型刻画的动力学行为与分段光滑线性时变模型描述的完全一致, 均较好地揭示了PT控制DCM-DCM BIFRED变换器所存在的复杂多周期行为. PSIM电路仿真和物理电路实验测量获得了与理论分析结果所一致的波形, 有效验证了两种动力学模型的正确性.     

Discrete variational principle and first integrals for Lagrange--Maxwell mechanico-electrical systems

This paper presents a discrete variational principle and a method to build first-integrals for finite dimensional Lagrange--Maxwell mechanico-electrical systems with nonconservative forces and a dissipation function. The discrete variational principle and the corresponding Euler--Lagrange equations are derived from a discrete action associated to these systems. The first-integrals are obtained by introducing the infinitesimal transformation with respect to the generalized coordinates and electric quantities of the systems. This work also extends discrete Noether symmetries to mechanico-electrical dynamical systems. A practical example is presented to illustrate the results.     

Differential System＇s Nonlinear Behaviour of Real Nonlinear Dynamical Systems

Chaos attractor behaviour is usually preserved if the four basic arithmetic operations, i.e. addition, subtraction, multiplication, division, or their compound, are applied. First-order differential systems of one-dimensional real discrete dynamical systems and nonautonomous real continuous-time dynamical systems are also dynamical systems and their Lyapunov exponents are kept, if they are twice differentiable. These two conclusions are shown here by the definitions of dynamical system and Lyapunov exponent. Numerical simulations support our analytical results. The conclusions can apply to higher order differential systems if their corresponding order differentials exist.     

Numerical computation of orbits and rigorous verification of existence of snapback repellers

In this paper we show how analysis from numerical computation of orbits can be applied to prove the existence of snapback repellers in discrete dynamical systems. That is, we present a computer-assisted method to prove the existence of a snapback repeller of a specific map. The existence of a snapback repeller of a dynamical system implies that it has chaotic behavior [F. R. Marotto, J. Math. Anal. Appl. 63, 199 (1978)]. The method is applied to the logistic map and the discrete predator-prey system.     

Mapping equivalent approach to analysis and realization of memristor-based dynamical circuit

A novel mapping equivalent approach is proposed in this paper, which can be used for analyzing and realizing a memristor-based dynamical circuit equivalently by a nonlinear dynamical circuit with the same topologies and circuit parameters. A memristor-based chaotic circuit and the corresponding Chua’s chaotic circuit with two output differentiators are taken as examples to illustrate this approach. Equivalent dynamical analysis and realization of the memristor-based chaotic circuit are performed by using Chua’s chaotic circuit. The results indicate that the outputs of memristor-based chaotic circuit and the corresponding outputs of Chua’s chaotic circuit have identical dynamics. The proposed approach verified by numerical simulations and experimental observations is useful in designing and analyzing memristor-based dynamical circuits.     

The principle of maximum entropy and Boltzmann type kinetic equations

The problem of the paper is the possibility of a dynamical justification of the principle of maximum entropy in the sense of a dynamical semigroup of open systems. It has been shown that, under the assumption of a convex dynamical semigroup defined on discrete and finite probability distributions (a finite sample space), this principle cannot be realized. This is possible, however, for non-linear dynamical semigroups for some random variables called p-collision-type variables in analogy to the Boltzmann 2-collision problem.     

Non-Gaussian Fluctuations of Local Lyapunov Exponents at Intermittency

In intermittent dynamical systems, the distributions of local Lyapunov exponents are markedly non-Gaussian and tend to be asymmetric and fat-tailed. A comparative analysis of the different time-scales in intermittency provides a heuristic explanation for the origin of the exponential tails, for which we also obtain an analytic expression deriving from a more quantitative theory. Application is made to several examples of discrete dynamical systems displaying intermittent dynamics.     

Synchronization of impulsively coupled complex networks

We investigate the synchronization of complex networks,which are impulsively coupled only at discrete instants.Based on the comparison theory of impulsive differential systems,a distributed impulsive control scheme is proposed for complex dynamical networks to achieve synchronization.The proposed scheme not only takes into account the influence of all nodes to network synchronization,which depends on the weight of each node in the network,but also provides us with a flexible method to select the synchronized state of the network.In addition,it is unnecessary for the impulsive coupling matrix to be symmetrical.Finally,the proposed control scheme is applied to a chaotic Lorenz network and Chua’s circuit network.Numerical simulations are used to illustrate the validity of this control scheme.     

AVERAGING IN WEAKLY COUPLED DISCRETE DYNAMICAL SYSTEMS

In [Y. Kifer, Averaging in difference equations driven by dynamical systems, Asterisque 287 (2003) 103–123] a general averaging principle for slow-fast discrete dynamical systems was presented. In this paper we extend this method to weakly coupled slow-fast systems. For this setting we obtain sharper estimates than in the mentioned paper.     

Localization of elastic waves in randomly disordered multi-coupled multi-span beams

The wave localization in randomly disordered periodic multi-span continuous beams is studied. The transfer matrix method is used to deduce transfer matrices of two kinds of multi-span beams. To calculate the Lyapunov exponents in discrete dynamical systems, the algorithm for determining all the Lyapunov exponents in continuous dynamical systems presented by Wolf et al is employed. The smallest positive Lyapunov exponent of the corresponding discrete dynamical system is called the localization factor, which characterizes the average exponential rates of growth or decay of wave amplitudes along the randomly mistuned multi-span beams. For two kinds of disordered periodic multi-span beams, numerical results of localization factors are given. The effects of the disorder of span-length, the non-dimensional torsional spring stiffness and the non-dimensional linear spring stiffness on the wave localization are analysed and discussed. It can be observed that the localization factors increase with the increase of the coefficient of variation of random span-length and the degree of localization for wave amplitudes increases as the torsional spring stiffness and the linear spring stiffness increase.     

Rotation Numbers of Linear Hamiltonian Systems with Phase Transitions over Almost Periodic Lattices

This paper deals with the dynamics of linear Hamiltonian systems which have almost periodic Hamiltonians and symplectic phase transitions over almost periodic lattices. By introducing some discrete skew-product dynamical systems based on certain joint hulls of Hamiltonians and lattices, it will be proved that such a system admits a well-defined rotation number, which gives a global, topological understanding on the motion of these systems in symplectic groups and manifolds of Lagrangian planes.     

Algebraic Entropy of Birational Maps with Invariant Curves

The relevance of the algebraic entropy in the study of birational discrete time dynamical systems highlights the need to relate it to other characteristics of these systems. In this Letter, two complementary proofs are given that the foliation of the space by invariant curves implies that the algebraic entropy is zero.     

Variational principle and dynamical equations of discrete nonconservative holonomic systems

By analogue with the methods and processes in continuous mechanics, a Lagrangian formulation and a Hamiltonian formulation of discrete mechanics are obtained. The dynamical equations including Euler--Lagrange equations and Hamilton's canonical equations of the discrete nonconservative holonomic systems are derived on a discrete variational principle. Some illustrative examples are also given.     

Birkhoff系统的离散最优控制及其在航天器交会对接中的应用

由于非线性,最优控制问题通常依赖于数值求解,即通过离散目标泛函和受控运动方程转化为一有限维的非线性最优化问题.最优控制问题中的受控运动方程在表示为受控Birkhoff方程的形式之后,可以利用受控Birkhoff方程的离散变分差分格式进行离散.与按照传统差分格式近似受控运动方程相比,此途径可以诱导更加真实可靠的非线性最优化问题,进而也会诱导更加精确有效的离散最优控制.应用于航天器交会对接问题,该种数值求解最优控制问题的方法在较大时间步长的情况下仍然求得了一个有效实现交会对接的离散最优控制.模拟结果验证了该方法的有效性.