Asymptotic Stability Of Periodic Solution Research Articles

Asymptotic Stability of Periodic Solutions for a Nonlinear Neutral First-order Differential Equation with Functional Delay

We consider a nonlinear neutral differential equation and use the contraction mapping principle to show the asymptotic stability of the zero solution provided that 𝑄(𝑡, 0) = 𝑓(0) = 0.

A remark about asymptotic stability in Duffing equations: lateral stability in Comb-drive finger MEMS

In this short paper we tackle two subjects. First, we provide a lower bound for the first eigenvalue of the antiperiodic problem for a Hill’s equation based on L^{p}-conditions, and as a consequence, we introduce an adjusted statement of the main result about the asymptotic stability of periodic solutions for the general Duffing equation in (Torres in Mediterr. J. Math. 1(4):479–486, 2004) (Theorem 4). This appropriate version of the result arises because of one subtlety in the proof provided in (Torres in Mediterr. J. Math. 1(4):479–486, 2004). More precisely, the lower bound of the first antiperiodic eigenvalue associated with Hill’s equations of potential a(t) employed there may be negative, thus the conclusion is not completely attained. Hence, the adjustments considered here provide a mathematically correct result. On the other hand, we apply this result to obtain a lateral asymptotic stable periodic oscillation in the Comb-drive finger MEMS model with a cubic nonlinear stiffness term and linear damping. This fact is not typical in Comb-drive finger devices, thus our results could provide a new possibility; a new design principle for stabilization in Comb-drive finger MEMS.

Asymptotic stability of a periodic GA-predation system with infinite distributed lags on time scales

The Gilpin-Ayala (GA) ecosystem is one of the most important biological mathematical models. The exploration of various kinetics behaviours of GA-ecological model has attracted the attention of many mathematicians. This paper focuses on a nonlinear periodic GA-predation ecosystem with infinite distributed lags on time scales. In the sense of time scale, our model unifies the difference model and differential model of GA-predation ecosystem. We first discuss a class of auxiliary functions with only two real roots. By using these auxiliary functions, coincidence degree and some inequality techniques, we next obtain some existence conditions of periodic solution. Furthermore, we explore the global asymptotic stability of periodic solution on account of Lyapunov stability theory. An example and its simulation are provided to inspect the correctness and availability of our main results at last.

Asymptotic stability of periodic solutions for a nonlinear neutral first-order differential equation with functional delay

We consider the nonlinear neutral differential equation 
 
 
 and use the contraction mapping principle to show the asymptotic stability of the zero solution provided that

Stabilization of periodic sweeping processes and asymptotic average velocity for soft locomotors with dry friction

<p style='text-indent:20px;'>We study the asymptotic stability of periodic solutions for sweeping processes defined by a polyhedron with translationally moving faces. Previous results are improved by obtaining a stronger <inline-formula><tex-math id="M1">\begin{document}$ W^{1,2} $\end{document}</tex-math></inline-formula> convergence. Then we present an application to a model of crawling locomotion. Our stronger convergence allows us to prove the stabilization of the system to a running-periodic (or derivo-periodic, or relative-periodic) solution and the well-posedness of an average asymptotic velocity depending only on the gait adopted by the crawler. Finally, we discuss some examples of finite-time versus asymptotic-only convergence.</p>

Periodic solutions for feedback control coupled systems on networks

This paper is concerned with the existence of periodic solutions for feedback control coupled systems on networks (FCCSNs) by a novel approach, which is made up of the continuation theorem of coincidence degree theory, graph theory, Lyapunov method, and some novel analysis skills. As an application of our approach, the existence and global asymptotic stability of periodic solutions for feedback control coupled oscillators on networks are investigated. Finally, numerical simulations are given to illustrate the effectiveness and feasibility of our results.

Periodic solution for inertial neural networks with variable parameters

<abstract><p>We discuss periodic solution problems and asymptotic stability for inertial neural networks with $ D- $operator and variable parameters. Based on Mawhin's continuation theorem and Lyapunov functional method, some new sufficient conditions on the existence and asymptotic stability of periodic solutions are established. Finally, a numerical example verifies the effectiveness of the obtained results.</p></abstract>

Periodic solutions of discrete-time Quaternion-valued BAM neural networks

The existence and global asymptotic stability of periodic solutions for a class of discrete-time Quaternion-valued BAM neural networks are considered. Without using the estimate priori approach of periodic solutions, the fixed point theorem and the approach of combining a continuation theorem of coincidence degree theory with a Lyapunov sequence, without decomposing the discrete-time Quaternion-valued BAM neural networks into eight real-valued differential equations, by using a continuation theorem and constructing two Lyapunov sequences, the criterion to ensure the existence of periodic solutions for the considered networks is put forward. Next, by constructing two Lyapunov sequences, a criterion assuring the global asymptotic stability of periodic solutions of the quaternion-valued neural networks is achieved. Compared with the skills and results in the existing literature, our skills and results are completely new.

Existence and asymptotic stability of periodic solutions for neutral evolution equations with delay

In this paper, we are devoted to consider the periodic problem for the neutral evolution equation with delay in Banach space. By using operator semigroups theory and fixed point theorem, we establish some new existence theorems of periodic mild solutions for the equation. In addition, with the aid of a new integral inequality with delay, we present essential conditions on the nonlinear function to guarantee that the equation has an asymptotically stable periodic mild solution.

Input-to-State Stability of Periodic Orbits of Systems With Impulse Effects via Poincaré Analysis

In this paper, we investigate the relation between robustness of periodic orbits exhibited by systems with impulse effects and robustness of their corresponding Poincare maps. In particular, we prove that input-to-state stability (ISS) of a periodic orbit under external excitation in both continuous and discrete time is equivalent to ISS of the corresponding zero-input fixed point of the associated forced Poincare map. This result extends the classical Poincare analysis for asymptotic stability of periodic solutions to establish orbital ISS of such solutions under external excitation. In our proof, we define the forced Poincare map, and use it to construct ISS estimates for the periodic orbit in terms of ISS estimates of this map under mild assumptions on the input signals. As a consequence of the availability of these estimates, the equivalence between exponential stability (ES) of the fixed point of the zero-input (unforced) Poincare map and the ES of the corresponding orbit is recovered. The results can be applied naturally to study the robustness of periodic orbits of continuous-time systems as well. Although our motivation for extending classical Poincare analysis to address ISS stems from the need to design robust controllers for limit-cycle walking and running robots, the results are applicable to a much broader class of systems that exhibit periodic solutions.

Global Asymptotic Stability of Periodic Solutions for Neutral-Type BAM Neural Networks with Delays

In this paper, we study the neutral-type BAM neural networks with time-varying delays. By applying the continuation theorem and some analysis techniques, some sufficient conditions to guarantee the neutral-type BAM neural networks have at least one periodic solution are proposed. Moreover, we also consider the asymptotic behaviours of periodic solutions by Lyapunov function and inequality $$2ab\le a^{2}+b^{2}$$. At last, an example is given to illustrate the effectiveness and feasibility of the obtain results.

Existence and asymptotic stability of periodic solutions for impulsive delay evolution equations

This paper we devote to considering the periodic problem for the impulsive evolution equations with delay in Banach space. By using operator semigroup theory and the fixed point theorem, we establish some new existence theorems of periodic mild solutions for the equations. In addition, with the aid of an integral inequality with impulsive and delay, we present essential conditions on the nonlinear and impulse functions to guarantee that the equations have an asymptotically stable ω-periodic mild solution.

The existence and asymptotic stability of periodic solutions with an interior layer of Burgers type equations with modular advection

We consider a new class of singularly perturbed parabolic periodic boundary value problems for reaction-advection-diffusion equations: Burgers type equations with modular advection. We construct the interior layer type formal asymptotics and propose a modified procedure to get asymptotic lower and upper solutions. By using sufficiently precise lower and upper solutions, we prove the existence of a periodic solution with an interior layer and estimate the accuracy of its asymptotics. The asymptotic stability of this solution is also established.

Global Asymptotic Stability of Periodic Solutions for Neutral-Type Delayed BAM Neural Networks by Combining an Abstract Theorem of k-Set Contractive Operator with LMI Method

The paper considers the existence and global asymptotic stability of periodic solutions for a class of neutral-type BAM neural networks with time delays. By combining an abstract theorem of k-set contractive operator with LMI method as well as inequality techniques, a sufficient condition to guarantee the existence of periodic solutions for the above neutral-type BAM neural networks with time delays is established. Then by combining LMI method with inequality techniques, a sufficient condition of the global asymptotic stability of periodic solutions for the above neutral-type BAM neural networks is obtained. Our method and results on periodic solutions for the above neural networks are new and complementary to the existing papers.

Global Asymptotic Stability of Periodic Solutions for Discrete Time Delayed BAM Neural Networks by Combining Coincidence Degree Theory with LMI Method

In this paper, we are concerned with the existence and global asymptotic stability of periodic solutions for a class of delayed discrete-time BAM neural networks. Instead of using the method of the priori estimate of periodic solutions in existing papers to study periodic solutions of neural networks, by combining Mawhin’s continuation theorem of coincidence degree theory with linear matrix inequality (LMI) method as well as inequality techniques, some novel LMI-based sufficient conditions to guarantee the existence and global asymptotic stability of periodic solutions for the neural networks are established. Our results which are both dependent on time delay and external inputs of the neural networks are new and complementary to the existing papers.

Global asymptotic stability of periodic solutions for delayed complex-valued Cohen–Grossberg neural networks by combining coincidence degree theory with LMI method

The paper is concerned with the existence and global asymptotic stability of periodic solutions for a class of delayed complex-valued Cohen–Grossberg neural networks. Without using the method of the a priori estimate of periodic solutions, by combining Mawhin’s continuation theorem of coincidence degree theory with LMI method and using inequality techniques, a novel LMI-based sufficient condition on the existence of periodic solutions is established for the complex-valued Cohen–Grossberg neural networks. Then by using inequality techniques, a novel sufficient condition on the global asymptotic stability of periodic solutions for the above complex-valued neural networks is established. Our results and method are new and complementary to the existing papers on the study of periodic solutions of neural networks.

Existence and Asymptotic Stability of Periodic Solutions of the Reaction–Diffusion Equations in the Case of a Rapid Reaction

A singularly perturbed periodic in time problem for a parabolic reaction-diffusion equation in a two-dimensional domain is studied. The case of existence of an internal transition layer under the conditions of balanced and unbalanced rapid reaction is considered. An asymptotic expansion of a solution is constructed. To justify the asymptotic expansion thus constructed, the asymptotic method of differential inequalities is used. The Lyapunov asymptotic stability of a periodic solution is investigated.

Periodicity and stability in neutral nonlinear differential equations by Krasnoselskii’s fixed point theorem

The nonlinear neutral functional differential equation with variable delay is investigated. By using Krasnoselskii’s fixed point theorem we obtain the existence and the asymptotic stability of periodic solutions. Sufficient conditions are established for the existence and the stability of the above equation. Our results extend some results obtained in the work [19].

Global asymptotic stability of periodic solutions for inertial delayed BAM neural networks via novel computing method of degree and inequality techniques

Firstly, by combining Mawhin’s continuation theorem of coincidence degree theory with Lyapunov functional method and by using inequality techniques, a sufficient condition on the existence of periodic solutions for inertial BAM neural networks is obtained. Secondly, a novel sufficient condition which can ensure the global asymptotic stability of periodic solutions of the system is obtained by using Lyapunov functional method and by using inequality techniques. In our paper, the assumption for boundedness on the activation functions in existing paper is removed, the conditions in inequality form in existing papers are replaced with novel conditions, and the prior estimate method of periodic solutions is replaced with Lyapunov functional method. Hence, our result on global asymptotic stability of periodic solutions for above system is less conservative than those obtained in existing paper and more novel than those obtained in existing papers.

About periodicity of impulsive evolution equations through fixed point theory

By processing the problem through fixed point theory and propagator theory, we investigate the periodicity of solutions to a class of impulsive evolution equations in Hilbert spaces and establish some existence theorems for periodic solutions. Moreover, the asymptotic stability of periodic solutions is obtained under suitable conditions. As one will see, the concept of an impulsive propagator is introduced for the first time in the paper.