周期激励频率对蔡氏振子的动力学行为的影响

蔡氏振子在不同的参数条件下可以表现为两个共存的对称单涡卷周期振荡以及单个对称的双涡卷周期行为．两个对称的单涡卷周期振荡在周期外激励作用下，随着频率的变化，分别演化为两个对称的混沌吸引子，在混沌区域中存在着不同形式的周期窗口以及倍周期分岔过程．同时在低频激励下，两个对称的混沌吸引子会相互作用，形成扩大了的混沌吸引子，其相应的轨迹交替在两个混沌子结构上来回长时间逗留．而双涡卷周期行为在周期激励下，可以导致共存的两个对称的概周期振荡．它们之间经相互作用形成单个概周期解，进而演化为双涡卷混沌吸引子．     

一类脉冲动力系统的状态反馈控制

对一类具有状态反馈控制的脉冲动力系统的动力学性质进行了研究．由周期解的扰动解得到了一个Poincare映射，利用Poincare映射讨论了系统周期解的分岔，并得到了半平凡周期解和正周期-1解存在和稳定的充分条件．定性分析和数学模拟表明，半平凡周期解通过fold分岔分钻出正周期-1解，正周期-1解通过flip分岔分岔出正周期-2解，再通过一系列flip分岔通向混沌，此外，讨论了脉冲状态反馈控制的效果．     

基于捕食者间扩散项的多时滞捕食模型研究

研究了一类基于捕食者间扩散项的多时滞非自治捕食模型,利用微分方程比较原理得到了系统持久生存的充分条件;通过构造一个合理的Lyapunov函数,得到了该系统存在唯一的全局吸引的正的概周期解的充分条件;运用matlab数学软件进行数值模拟验证了理论分析的可行性。     

广义系统的周期解

本文研究了几类广义系统，得出了一类广义线性非齐次系统存在周期解的充要条件，一类广义线性时变系统存在唯一周期解的充分条件；一类广义非线性系统存在周期解的充分条件。     

一类经济模型的分岔周期解

讨论了一类经济模型的稳定性及Hopf分岔．根据特征根给出系统失稳的条件后，利用伪振子法和迭代法得到了Hopf分岔的方向以及周期解的振幅估计，计算式较简洁，最后的数值算例很好地验证了方法的准确性．特别地，当周期解振幅较大时，迭代法的估算更准确．尽管系统失稳后，产生分岔周期解，但适当调整参数大小，仍然可以保证周期解稳定，也就意味着经济的良性发展．     

非线性增益下半导体激光方程的动力学行为分析

分析了在非线性增益下各种因素对Lang-Kobayashi模型动力学行为的影响．由于时滞反馈的作用,系统中存在着不同的ECM解,Hopf分岔是这些ECM解失稳的主要原因,进而演化为各种形式的混沌解,不同吸引子之间的相互作用会引发混沌结构的突变,表现为混沌吸引子在空间尺寸上的明显变化,随着时滞量的变化,这些演化模式会重复出现．值得指出的是,在一定条件下,不同频率的两个ECM共存,其中之一会由Hopf分岔失稳,并由倍周期分岔进入混沌,最终通过混沌危机回到另一个稳定的ECM上．另外,随着非线性增益系数的变化,在极坐标下系统的概周期运动的两个频率相差很大,激光器呈现出明显的快慢效应．     

一类超混沌系统的动力学研究及其仿真

混沌系统的全局指数吸引集在混沌系统的控制与同步之中起着非常重要的作用。借助一个适当的Lyapunov函数和一元函数极值理论研究了一个新超混沌系统的全局指数吸引集,得到了该系统的全局指数吸引集表达式Ω。可以断定在全局指数吸引集Ω之外混沌系统的平衡位置、周期解、概周期解、游荡回复解和其他任何混沌吸引子都不复存在,这大大简化了对该系统的分析工作。确定轨线从吸引集外走向吸引集的速度是指数速率。同时得到的全局指数吸引集表达式Ω为该系统的控制和同步提供了理论依据。通过计算机进行了模拟,数值模拟与理论计算的结果相吻合。     

四边简支矩形薄板的双Hopf分岔分析

基于奇异性理论,研究了主参数共振-1∶3内共振情形下参数激励与外激励联合作用下四边简支矩形薄板的双Hopf分岔问题.考虑弱阻尼和弱激励的情形,得到了四边简支矩形薄板的分岔方程,给出了四边简支矩形薄板在参数平面μ-σ1上的分岔图.对参数激励与外激励联合作用下四边简支矩形薄板的阻尼系数、外激励、参数激励以及调谐参数进行不同的取值,通过数值模拟得到了四边简支矩形薄板平衡解将发生Hopf分岔,并分岔出周期解,薄板系统的非线性振动形式为周期运动.当四边简支矩形薄板的参数满足给定条件时,我们得到薄板的1∶3共振双Hopf分岔.随后,四边简支矩形薄板将会呈现概周期振动.     

一类自治脉冲微分方程的动力学研究

对一类自治脉冲微分方程的动力学性质进行了研究,给出了半平凡周期解的存在与稳定的充分条件,建立Poincare映射将周期解问题转化为不动点问题．理论分析及数值模拟表明,半平凡周期解通过跨临界分岔获得稳定的正周期-1解．数值模拟显示,随着控制参数的变化,正周期-1解通过倍周期分岔出正周期-2解,再通过一系列倍周期分岔通向混沌．     

神经元周期放电模式的分岔

利用一种可以计算自治非线性系统周期解及周期的改进打靶法,求解了神经元电活动Rose—Hind-marsh（R-H）模型自发放电的周期解和周期;计算了周期放电的Floquet乘子并分析了周期解的分岔,如倍周期分岔,鞍-结分岔．研究结果有助于进一步理解神经放电模式转迁的动力学和生物学意义．     

Existence and global exponential stability of almost periodic solution for cellular neural networks with variable coefficients and time-varying delays

In this paper, we study cellular neural networks with almost periodic variable coefficients and time-varying delays. By using the existence theorem of almost periodic solution for general functional differential equations, introducing many real parameters and applying the Lyapunov functional method and the technique of Young inequality, we obtain some sufficient conditions to ensure the existence, uniqueness, and global exponential stability of almost periodic solution. The results obtained in this paper are new, useful, and extend and improve the existing ones in previous literature.     

Almost periodic dynamics of a delayed three-level food-chain model

This paper is concerned with the dynamics of a three-level food-chain model with time delays. By means of Mawhin's continuation theorem of coincidence degree theory, we investigate the existence of positive almost periodic solutions in a three-level food-chain model with time delays. A set of sufficient conditions for the existence of at least one positive almost periodic solution of the model is obtained. In addition, the global asymptotical stability of the almost periodic solution of the above model is also studied. Finally, two examples and simulations are also given to illustrate the main results in this paper.     

Permanence and almost periodic solution of two-species delayed Lotka–Volterra cooperative systems with impulsive perturbations

By applying the comparison theorem and the Lyapunov method of the impulsive differential equations, this paper gives some new sufficient conditions for the permanence and existence of a unique uniformly asymptotically stable positive almost periodic solution in a class of two-species Lotka–Volterra cooperative systems with impulses. The method used in this paper provides a possible method to study the permanence and existence of a unique uniformly asymptotically stable positive almost periodic solution of the models with impulsive perturbations in biological populations. Finally, an example and numerical simulations are given to illustrate that our results are feasible.     

Almost periodic dynamics of a class of delayed neural networks with discontinuous activations

We use the concept of the Filippov solution to study the dynamics of a class of delayed dynamical systems with discontinuous right-hand side, which contains the widely studied delayed neural network models with almost periodic self-inhibitions, interconnection weights, and external inputs. We prove that diagonal-dominant conditions can guarantee the existence and uniqueness of an almost periodic solution, as well as its global exponential stability. As special cases, we derive a series of results on the dynamics of delayed dynamical systems with discontinuous activations and periodic coefficients or constant coefficients, respectively. From the proof of the existence and uniqueness of the solution, we prove that the solution of a delayed dynamical system with high-slope activations approximates to the Filippov solution of the dynamical system with discontinuous activations.     

Positive almost periodic solution for a class of Lasota–Wazewska model with multiple time-varying delays

In this paper, we study the existence and global exponential convergence of positive almost periodic solutions for the generalized Lasota–Wazewska model with multiple time-varying delays. Under proper conditions, we employ a novel proof to establish some criteria to ensure that all solutions of this model converge exponentially to a positive almost periodic solution. Moreover, we give an example to illustrate our main results.     

Almost periodic cellular neural networks with neutral-type proportional delays

This paper presents a new result on the existence, uniqueness and generalised exponential stability of almost periodic solutions for cellular neural networks with neutral-type proportional delays and D operator. Based on some novel differential inequality techniques, a testable condition is derived to ensure that all the state trajectories of the system converge to an almost periodic solution with a positive exponential convergence rate. The effectiveness of the obtained result is illustrated by a numerical example.     

The almost periodic solution of Lotka-Volterra recurrent neural networks with delays

By the fixed-point theorem subject to different polyhedrons and some inequalities (e.g., the inequality resulted from quadratic programming), we obtain three theorems for the Lotka-Volterra recurrent neural networks having almost periodic coefficients and delays. One of the three theorems can only ensure the existence of an almost periodic solution, whose existence and uniqueness the other two theorems are about. By using Lyapunov function, the sufficient condition guaranteeing the global stability of the solution is presented. Furthermore, two numerical examples are employed to illustrate the feasibility and validity of the obtained criteria. Compared with known results, the networks model is novel, and the results are extended and improved.     

Properties of the solutions of rational matrix difference equations

We prove a comparison theorem for the solutions of a rational matrix difference equation, generalizing the Riccati difference equation, and existence and convergence results for the solutions of this equation. Moreover, we present conditions ensuring that the corresponding algebraic matrix equation has a stabilizing or almost stabilizing solution.     

Existence and Global Exponential Stability of Almost Periodic Solution for High-Order BAM Neural Networks with Delays on Time Scales

In this paper, by using a fixed point theorem and by constructing a suitable Lyapunov functional, we study the existence and global exponential stability of almost periodic solution for high-order bidirectional associative memory neural networks with delays on time scales. An examples shows the feasibility of our main results.     

A note on explosion suppression for nonlinear delay differential systems by polynomial noise

Liu and Shen discussed the role of stochastic suppression on the explosive solution by a polynomial noise for a deterministic differential system satisfying a general polynomial growth condition. They further showed that the global solution of the corresponding perturbed system grows at most polynomially. However, the estimation of the asymptotic property of polynomial growth is rough, and we see the necessity to develop a more accurate estimation which is the main motivation of the present paper. As to the existence of time delays, we aim to discuss the stochastic roles of the polynomial noise for a deterministic delay differential system with the general polynomial growth condition. We show that a properly chosen polynomial stochastic noise not only can guarantee the existence and uniqueness of the global solution of the stochastically perturbed delay differential system, but also can make almost every sample path of the global solution grow at most with polynomial rate and even decay to the zero solution exponentially.