Nonautonomous Differential Systems Research Articles

On the stability of nonautonomous systems

In Kalitine (RAIRO Automatique/Systems Anal. Control 16(3) (1982) 275) the use of semi-definite Lyapunov functions for exploring the local stability of autonomous dynamical systems has been introduced. In this paper, we give an extension of the results of Kalitine (1982) that allows to study the local stability of nonautonomous differential systems. We give an application to the algebraic Riccati equation.

Applications of a Zp Index Theory to Periodic Solutions for a Class of Functional Differential Equations

By means of a new geometrical index with Zp group actions, multiplicity results for a certain class of nonautonomous time periodic functional differential systems are obtained.

On the functional polystability of certain essentially nonlinear nonautonomous differential systems

For essentially nonlinear differential systems with the limit matrix of coefficients of the first-approximation system, we establish sufficient conditions for functional polystability, which generalizes the notion of exponential polystability.

On the “freezing” method for nonlinear nonautonomous systems with delay

Nonlinear nonautonomous differential systems with delaying argument are considered. Explicit conditions for absolute stability are derived. The proposed approach is based on the generalization of the “freezing” method for ordinary differential equations.

Functional polystability of some nonautonomous quasilinear differential systems

For quasilinear differential systems with a boundary matrix of coefficients of the system of the first approximation, we obtain sufficient conditions of functional polystability, which generalizes the notion of exponential polystability.

Nonresonance Conditions for Asymptotically Positively Homogeneous Differential Systems: The Fučik Spectrum and Its Generalization

In this article we propose how to define the Fučik spectra for differential systems in Rnwith jumping nonlinearities. The PropertyP, a generalization of the non-Fučik spectra to positively homogeneous nonautonomous differential systems, is also defined in the most general setting. A quantitative estimate result concerning the parameterized completely continuous, positively homogeneous perturbations of the identity in Banach spaces is given in this article. Such a result is a generalization of one property in the spectrum theory for completely continuous linear operators. Using this estimate result, we will give some significantly improved nonresonance results from the Fučik spectra and PropertyPfor asymptotically positively homogeneous differential systems.

Permanence in Kolmogorov-Type Systems of Nonautonomous Functional Differential Equations

In this paper we establish some general results on the permanence of then-species Kolmogorov-type nonautonomous functional differential system[formula],i=1,2,…,n. These results are applied to then-species Lotka–Volterra-type systems with distributed delays, that is, whenfi(t,xt) are linear functions ofxt. Our method consists of constructingnLiapunov-like functions, which can be regarded as boundary functions of some compact region inside the positive cone inRn.

On the asymptotic stability for a two-dimensional linear nonautonomous differential system

On the asymptotic stability for a two-dimensional linear nonautonomous differential system

On stability of the trivial solution of a nonautonomous quasilinear system whose characteristic equation has multiple roots

We obtain sufficient conditions for the Lyapunov stability of the trivial solution of a nonautonomous differential system of a special form ast ↑ ω, ω ≤ + ∞. For this system, the coefficient matrix of a differential system of the first approximation has almost Jordan form with triangular blocks. We indicate methods that enable one to reduce certain classes of differential systems of the general form to special differential systems.

Exponential Dichotomy and Rotation Number for Linear Hamiltonian Systems

Let x′ = H( t) x be a non-autonomous Hamiltonian linear differential system. We show that if the rotation number α(λ) for the associated family of systems x′ = ( H( t) + λ Jγ( t)) x (where J = ( 0 In − In 0) and γ( t) = γ*( t) ≥ 0 for all t) is constant on an interval containing λ = 0, then x′ = H( t) x has an exponential dichotomy.

Uniform Persistence in Nonautonomous Delay Differential Kolmogorov-Type Population Models

In this paper we establish sufficient conditions for uniform persistence in nonautonomous Kolmogorov-type delayed population models. The method involves the construction of a set of proper autonomous ordinary differential systems whose solutions can serve as lower or upper bounds for the delayed system in certain regions. The results are new even for nonautonomous ordinary differential systems.

On asymptotic stability for nonautonomous differential systems

On asymptotic stability for nonautonomous differential systems

Boundary-value problems for systems of ordinary differential equations

This article contains an exposition of fundamental results of the theory of boundary-value problems for systems of linear and nonlinear ordinary differential equations. In particular, criteria are given for problems with functional, many-point, and two-point boundary conditions to be solvable and well-posed, as well as methods of finding approximate solutions. We also examine questions of existence, uniqueness, and stability of periodic and bounded solutions of nonautonomous differential systems.

Total Stability of Sets for Nonautonomous Differential Systems

Total Stability of Sets for Nonautonomous Differential Systems

Total stability of sets for nonautonomous differential systems

The principal purpose of this paper is to present sufficient conditions for total stability, or stability under constantly acting perturbations, of sets of a sufficiently general kind for nonautonomous ordinary differential equations. To do this, two Liapunov-like functions with specific properties are used. The obtained results include and considerably improve the classical results on total stability of isolated equilibrium points. Applications are presented to study the stability of nonautonomous Lurie-type nonlinear equations.

Stability and Asymptotic Estimates in Nonautonomous Linear Differential Systems

A new theory is presented, in which a generalized kinematic similarity transformation is used to diagonalize linear differential systems. No matrices of Jordan form are needed. The relation to Lyapunov’s classical stability theory is explored, and asymptotic estimates of fundamental solutions are given. Finally, some possible numerical applications of the presented theory are suggested.

Nonautonomous differential systems: several functions in the stability problem

Results of asymptotic stability and instability for nonautonomous differential systems with unbounded right hand side are obtained by using several auxiliary functions and topological properties of the Ω-limit set of solutions. The theorem established is applied to prove the asymptotic stability of the stationary solution of the Lienard equation with unbounded dampings.

Hyperbolic almost periodic solutions and toroidal limit sets

We consider an almost periodic solution phi(t) of an autonomous differential equation x' = f(x). Let k denote the topological dimension of the hull H(phi). By considering the linearized equation x' = A(t)x alone [where A(t) is the Jacobian matrix of f evaluated along phi(t)], one can derive a sufficient condition in order that phi(t) be quasi-periodic and that the hull H(phi) be diffeomorphic to a k-dimensional torus T(k). The proof is based upon an extension of the center manifold theorem to nonautonomous nonperiodic differential systems.

The Lyapunov theorem on the behavior of solutions of a nonautonomous differential system in a neighborhood of a periodic solution with corresponding multivalued variational equation

The Lyapunov theorem on the behavior of solutions of a nonautonomous differential system in a neighborhood of a periodic solution with corresponding multivalued variational equation

Solution of certain nonautonomous differential systems by series of exponential functions

Solution of certain nonautonomous differential systems by series of exponential functions