Class Of Nonlinear Differential Systems Research Articles

New global exponential stability conditions for nonlinear delayed differential systems with three kinds of time-varying delays

For a class of nonlinear differential systems with heterogeneous time-varying delays, including distributed, leakage and transmission time-varying delays, a novel global exponential stability (GES) analysis method was developed. Based on the GES definition, some sufficient conditions and rigorous convergence analysis of nonlinear delayed differential systems are presented directly, which ensure all states to be globally exponentially convergent. The proposed analysis method not only avoids the construction of the Lyapunov–Krasovskii functional, but also uses some simple integral reduction techniques to determine the global exponential convergence rate. Furthermore, the main advantages and low calculation complexity are demonstrated through a theoretical comparison. Finally, three numerical examples are provided to verify the effectiveness of the theoretical results.

Existence of periodic solutions for a class of $ (\phi_{1}, \phi_{2}) $-Laplacian discrete Hamiltonian systems

<abstract><p>In this paper, we consider the existence of periodic solutions for a class of nonlinear difference systems involving classical $ (\phi_{1}, \phi_{2}) $-Laplacian. By using the least action principle, we obtain that the system with classical $ (\phi_{1}, \phi_{2}) $-Laplacian has at least one periodic solution when potential function is $ (p, q) $-sublinear growth condition, subconvex condition. The results obtained generalize and extend some known works.</p></abstract>

Fixed points and stability of a class of nonlinear differential systems with several delays of feedback control

We offer existence criteria and sufficient conditions, so that the trivial solution of the differential system with several delays of feedback control is asymptotically stable. Here the fixed point technique is a practical method for this purpose. When these results are applied to some special delay mathematics models, some new results are obtained, and many known results are improved. Lastly, we provide an example that illustrates our results.

Existence result of solutions for a class of nonlinear differential systems

In this paper, we will discuss the existence of bounded positive solutions for a class of nonlinear differential systems. The objective will be achieved by applying some results and techniques of functional analysis such as Schauder's fixed point theorem and potential theory tools.

一类非线性时滞微分系统的稳定性

一类非线性时滞微分系统的稳定性

Existence of Solutions to Boundary Value Problems for a Class of Nonlinear Difference Systems

Abstract This paper is devoted to investigate a question of the existence of solutions to boundary value problems for a class of nonlinear difference systems. The proof is based on the notable mountain pass lemma in combination with variational technique. By using the critical point theory, some new existence criteria are obtained.

Delay-independent stability conditions for a class of nonlinear difference systems

The paper addresses the asymptotic stability problem for a class of difference systems with nonlinearities of a sector type and time-delay. A new approach to Lyapunov–Krasovskii functionals constructing for considered systems is proposed. On the basis of the approach, delay-independent asymptotic stability conditions and estimates of the convergence rate of solutions are derived. In addition, stability of perturbed systems is investigated in the case where nonstationary perturbations admit zero mean values. Some examples are given to illustrate the obtained results.

On stability of solutions for a class of nonlinear difference systems with switching

The stability of the trivial solution for a class of difference systems with switching and sector-type nonlinearities is studied. Different approaches to common Lyapunov function design for the family of subsystems corresponding to the considered switched system are proposed. Sufficient conditions making the trivial solution asymptotically stable for any switching law are determined. In the case when common Lyapunov function design fails, multiple Lyapunov functions are used to obtain the restrictions on the switching law guaranteeing the asymptotic stability of the trivial solution.

Homoclinic solutions for a class of nonlinear difference systems with classical ( ϕ 1 , ϕ 2 ) -Laplacian

In this paper, we consider the existence of homoclinic solutions for a class of nonlinear difference systems involving classical $(\phi_{1}, \phi_{2})$ -Laplacian. First, we improve some inequalities in known literature. Then, by using the variational method, some new existence results are obtained. Finally, some examples are given to verify our results.

Multiple periodic solutions for a class of nonlinear difference systems with classical or bounded ( ϕ 1 , ϕ 2 ) -Laplacian

In this paper, we consider the multiplicity of periodic solutions for a class of difference systems involving the -Laplacian in the cases when the gradient of the nonlinearity has a sublinear growth. By using the variational method, some existence results are obtained. Our results generalize some recent results in (Mawhin in Discrete Contin. Dyn. Syst. 6:1065-1076, 2013).

Effect of leakage time-varying delay on stability of nonlinear differential systems

In this paper, a class of nonlinear differential systems with leakage time-varying delay is considered. Sufficient conditions for the existence-uniqueness and global asymptotic stability of the equilibrium point are derived by using fixed point theorems, Lyapunov–Krasovskii functional and model transformation technique. The stability criterion that depends on the upper bounds of the leakage time-varying delay and its derivative is given in terms of a linear matrix inequality (LMI), which can be efficiently solved via standard numerical software. Even for the case of leakage constant delay, the criterion is shown to be less conservative than a recent publication. Finally, two numerical examples are provided to demonstrate the less conservatism and effectiveness of the proposed results.

A REFINEMENT OF LYAPUNOV-TYPE INEQUALITY FOR A CLASS OF NONLINEAR SYSTEMS

Some new Lyapunov-type inequalities for a class of nonlinear differential systems, which are natural refinements and generalizations of the well-known Lyapunov inequality for linear second order differential equations, are given. The results of this paper cover some previous results on this topic.

Stability and stabilization of the family of pseudolinear differential systems

New methods for the analysis of the robust stability of equilibrium states are developed for some classes of nonlinear differential systems. We formulate sufficient conditions for the stability of the trivial solutions of the families of pseudolinear controlled systems with undetermined coefficient matrices and feedback from the measured output. A method is developed for the analysis of stability according to the first approximation to the family of nonlinear systems. The application of the obtained results is reduced to the solution of systems of linear differential matrix inequalities. We present an example of a system of stabilization of a double inverted pendulum.

Stability Results for a Class of Difference Systems with Delay

Considering the linear delay difference system , where , is a real matrix, and is a positive integer, the stability domain of the null solution is completely characterized in terms of the eigenvalues of the matrix . It is also shown that the stability domain becomes smaller as the delay increases. These results may be successfully applied in the stability analysis of a large class of nonlinear difference systems, including discrete-time Hopfield neural networks.

Existence and uniqueness results for a class of nonlinear differential systems with second‐order differences

AbstractIn a real Hilbert space we investigate the existence, uniqueness and asymptotic properties of the strong and weak solutions to an infinite nonlinear differential system with second–order differences, subject to an extreme condition and initial data. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A symbolic algorithm for the automatic computation of multitone-input harmonic balance equations for nonlinear systems

A symbolic algorithm is developed for the automatic generation of harmonic balance equations for multitone input for a class of nonlinear differential systems with polynomial nonlinearities. Generalized expressions are derived for the construction of balance equations for a defined multitone signal form. Procedures are described for determining combinations for a given output frequency from the desired set obtained from box truncated spectra and their permutations to automate symbolic algorithm. An application of method is demonstrated using the well-known Duffing–Van der Pol equation. Then the obtained analytical results are compared with numerical simulations to show the accuracy of the approach. The computation times for both the numerical solutions of equations versus the number of frequency components and the symbolic generation of the equations versus the power of nonlinearity are also investigated.

Permanence for a class of nonlinear difference systems

A class of nonlinear difference systems is considered in this paper. By exploring the relationship between this system and a correspondent first-order difference system, some permanence results are obtained.

Global attractor of coupled difference equations and applications to Lotka-Volterra systems

This paper is concerned with a coupled system of nonlinear difference equations which is a discrete approximation of a class of nonlinear differential systems with time delays. The aim of the paper is to show the existence and uniqueness of a positive solution and to investigate the asymptotic behavior of the positive solution. Sufficient conditions are given to ensure that a unique positive equilibrium solution exists and is a global attractor of the difference system. Applications are given to three basic types of Lotka-Volterra systems with time delays where some easily verifiable conditions on the reaction rate constants are obtained for ensuring the global attraction of a positive equilibrium solution.

On Stability of Solutions to One Class of Nonlinear Difference Systems

We consider some class of systems of nonlinear ordinary differential equations. We adjust the difference schemes corresponding to the equations under study in order to guarantee agreement between differential and difference systems in the sense of stability of the zero solution. We obtain conditions under which perturbations do not violate the asymptotic stability of solutions to difference systems.

Qualitative behavior of stage-structured populations: application to structural validation

The transient behavior of a class of nonlinear differential systems representing stage-structured populations is studied. The qualitative dynamics are described in terms of succession of extrema for the state variables, or for the integrated difference between two trajectories. The rules giving the possibilities of extrema are derived, they characterize the classical stage-structured models. These rules can be compared with experiments to validate the structure of the model. An explanation for the disagreement of this transition scheme with some experiments could be an unexpected interaction with another variable. A new model taking the interaction into account thus engenders new transition rules, which are to be compared with experiments. These results are illustrated with experiments on copepods, showing how the qualitative experimental features can help the construction and the validation of the models.