Nonlinear Nonautonomous Differential Equations Research Articles

Asymptotic Representations for the Solutions of Third-Order Nonlinear Differential Equations

We study the asymptotic behavior of solutions for a certain class of third-order nonlinear nonautonomous differential equations.

Stability of Caputo fractional differential equations by Lyapunov functions

The stability of the zero solution of a nonlinear nonautonomous Caputo fractional differential equation is studied using Lyapunov-like functions. The novelty of this paper is based on the new definition of the derivative of a Lyapunov-like function along the given fractional equation. Comparison results using this definition for scalar fractional differential equations are presented. Several sufficient conditions for stability, uniform stability and asymptotic uniform stability, based on the new definition of the derivative of Lyapunov functions and the new comparison result, are established.

Synchronization in a nonisochronous nonautonomous system

We study a model system of nonautonomous nonlinear differential equations arising in magnetodynamics theory. We find constraints on the parameters such that Lyapunov-stable solutions with a stabilized phase exist. These solutions describe the synchronization phenomenon in a nonisochronous system with slowly varying parameters.

Method of limiting equations for the stability analysis of equations with infinite delay in the Carathéodory conditions: I

We consider systems of nonautonomous nonlinear differential equations with infinite delay. We introduce Caratheodory type conditions for the right-hand side in an equation, which permit one, on the one hand, to cover a fairly broad class of systems and, on the other hand, include the right-hand side in a compact function space and construct the so-called limiting equations. In the investigation, we use the construction of admissible spaces with fading memory, which permits one to obtain constructive results for the class of equations under study.

Generalization of the simplest equation method for nonlinear non-autonomous differential equations

It is known that the simplest equation method is applied for finding exact solutions of autonomous nonlinear differential equations. In this paper we extend this method for finding exact solutions of non-autonomous nonlinear differential equations (DEs). We applied the generalized approach to look for exact special solutions of three Painlevé equations. As ODE of lower order than Painlevé equations the Riccati equation is taken. The obtained exact special solutions are expressed in terms of the special functions defined by linear ODEs of the second order.

Dynamic Analysis of Nonlinear Impulsive Neutral Nonautonomous Differential Equations with Delays

A class of neural networks described by nonlinear impulsive neutral nonautonomous differential equations with delays is considered. By means of Lyapunov functionals and differential inequality technique, criteria on global exponential stability of this model are derived. Many adjustable parameters are introduced in criteria to provide flexibility for the design and analysis of the system. The results of this paper are new and they supplement previously known results. An example is given to illustrate the results.

Local energy decay for the wave equation with a nonlinear time-dependent damping

This article addresses a wave equation on a exterior domain in ℝ d (d odd) with nonlinear time-dependent dissipation. Under a microlocal geometric condition we prove that the decay rates of the local energy functional are obtained by solving a nonlinear non-autonomous differential equation

ON THE FUNDAMENTAL THEORY OF IMPULSIVE DIFFERENTIAL EQUATIONS WITH VARIABLE STRUCTURE

A special class of nonlinear non-autonomous systems ordinary differential equations with variable structure and impulses is studied in the paper. The main research goal is to identify the reasons for which such impulsive systems have solutions which are not continuable up to infinity. The case of non continuability of the solutions (or as it is accepted to say “death” of the solutions) due to the impulsive effects is investigated in the paper.

Stability of autoresonance models

We consider systems of two nonlinear nonautonomous differential equations on the real line which arise when averaging rapid nonlinear vibrations. We study the Lyapunov stability of solutions with infinitely increasing amplitude. Such solutions are related to the description of the initial stage of autoresonance or resonance trapping in oscillating nonlinear systems with a small perturbation. We obtain conditions on the coefficients of the equations under which the increasing solutions are stable or unstable. The problem is reduced to the analysis of an equilibrium. The stability of the equilibrium is studied by the Lyapunov second method. The construction of Lyapunov functions is based the presence of dissipative terms with coefficients moderately decaying in time.

On asymptotic completeness of scattering in the nonlinear Lamb system, II

We establish the asymptotic completeness in the nonlinear Lamb system for hyperbolic stationary states. For the proof we construct a trajectory of a reduced equation (which is a nonlinear nonautonomous ordinary differential equation) converging to a hyperbolic stationary point using the inverse function theorem in a Banach space. We give counterexamples which show nonexistence of such trajectories for nonhyperbolic stationary points.

An analog to Kamenkov’s critical stability case for nonstationary systems of impulsive differential equations

A new approach to the investigation of the stability of nonlinear nonautonomous differential equations with impulse effects in critical cases is proposed. The approach is based on the direct method of Lyapunov with the use of piecewise differentiable functions. The sufficient conditions of the asymptotic stability of the critical position of equilibrium in one case are obtained. The case is analogous to Kamenkov’s critical case.

Asymptotic behavior of one class of solutions of nonlinear nonautonomous second-order differential equations

For a second-order ordinary differential equation whose right-hand side contains a sum of terms with nonlinearities regularly varying with respect to the unknown function and its derivatives, we establish necessary and sufficient conditions for the existence of a broad class of monotone solutions and obtain exact asymptotic representations of solutions of this class in a neighborhood of a singular point.

STABILITY OF THE DIFFERENTIAL EQUATIONS WITH VARIABLE STRUCTURE AND NON FIXED IMPULSIVE MOMENTS USING SEQUENCES OF LYAPUNOV'S FUNCTIONS

A specific class of non-linear non-autonomous systems ordinary differential equations with variable structure and impulses are studied in the paper. The change of the system right side and impulsive effects of the solution are realized at the moments, at which the so-called switching functions, defined in the system phase space, are canceled. Sufficient conditions for stability, uniform stability and uniform asymptotically stability of zero solution for the systems investigated are obtained. The results are received using a modification of the Lyapunov's Second Method. For this purpose, there are introduced sequences of the scalar piecewise - continuous functions of the Lyapunov class. The consecutive change (activation) of the Lyapunov's functions from the sequence are synchronized with the changing of the structure of the system investigated. Note that, it is allowed each one of the Lyapunov's functions of the sequence to be piecewise - continuous. The points of discontinuity coincide with the points of the set of switching of the corresponding right side of the system.

Bifurcation analyses of nonlinear dynamical systems: From theory to numerical computations

In this paper, we explain how to compute bifurcation parameter values of periodic solutions for non-autonomous nonlinear differential equations. Although various approaches and tools are available for solving this problem nowadays, we have devised a very simple method composed only of basic computational algorithms appearing in textbooks for beginner's, i.e., Newton's method and the Runge-Kutta method. We formulate the bifurcation problem as a boundary value problem and use Newton's method as a solver consistently. All derivatives required in each iteration are obtained by solving variational equations about the state and the parameter. Thanks to the quadratic convergence ability of Newton's method, accurate results can be quickly and effectively obtained without using any sophisticated mathematical library or software. If a discontinuous periodic force is applied to the system, we can use the same strategy to solve the bifurcation problem. The key point of this method is deriving a differentiable composite map from the various information about the problem such as the location of sections, the periodicity, the Poincaré mapping, etc.

Ultimate boundedness of the solutions of certain third order nonlinear non-autonomous differential equations

In this paper, we shall establish sufficient conditions for the uniform ultimate boundedness of solutions of a certain third order nonlinear non-autonomous differential equation, by using a Lyapunov function as basic tool. In doing so we extend some existing results. Examples are given to illustrate our results.

Asymptotic representations of solutions of one class of nonlinear differential equations of the second order

We establish asymptotic properties of some types of solutions of one class of essentially nonlinear nonautonomous differential equations of the second order and find necessary and sufficient conditions for the existence of these solutions.

Conditions for the existence of solutions of real nonautonomous systems of quasilinear differential equations vanishing at a singular point

We establish conditions for the existence of solutions vanishing at a singular point for various classes of systems of quasilinear differential equations appearing in the investigation of the asymptotic behavior of solutions of essentially nonlinear nonautonomous differential equations of higher orders.

Parametric resonance in non-linear viscoelasticity: solids of differential type

We show that finite-amplitude shearing motions superimposed on an 'unsteady' simple extension are admissible for incompressible isotropic viscoelastic materials of differential kind. The amplitude of these motions is determined by solving a non-linear non-autonomous differential equation for which we show that limit cycles are possible.

Comparison results and existence of bounded solutions to strongly nonlinear second order differential equations

We investigate the existence of bounded solutions on the whole real line of the following strongly non-linear non-autonomous differential equation $$ (a(x(t))x'(t))'= f(t,x(t),x'(t)) \quad \text{a.e } t\in \mathbb R \tag \text{\rm E} $$ where $a(x)$ is a generic continuous positive function, $f$ is a Carathéodory right-hand side. We get existence results by combining the upper and lower-solutions method to fixed-point techniques. We also provide operative comparison criteria ensuring the well-ordering of pairs of upper and lower-solutions.

Dynamical analysis of the transmission of seasonal diseases using the differential transformation method

The aim of this paper is to apply the differential transformation method ( D T M ) to solve systems of nonautonomous nonlinear differential equations that describe several epidemic models where the solutions exhibit periodic behavior due to the seasonal transmission rate. These models describe the dynamics of the different classes of the populations. Here the concept of D T M is introduced and then it is employed to derive a set of difference equations for this kind of epidemic models. The D T M is used here as an algorithm for approximating the solutions of the epidemic models in a sequence of time intervals. In order to show the efficiency of the method, the obtained numerical results are compared with the fourth-order Runge–Kutta method solutions. Numerical comparisons show that the D T M is accurate, easy to apply and the calculated solutions preserve the properties of the continuous models, such as the periodic behavior. Furthermore, it is showed that the D T M avoids large computational work and symbolic computation.