Existence Of Time-periodic Solutions Research Articles

Stability and attractivity of periodic solutions of parabolic systems with time delays

This paper is concerned with the existence, stability, and global attractivity of time-periodic solutions for a class of coupled parabolic equations in a bounded domain. The problem under consideration includes coupled system of parabolic and ordinary differential equations, and time delays may appear in the nonlinear reaction functions. Our approach to the problem is by the method of upper and lower solutions and its associated monotone iterations. The existence of time-periodic solutions is for a class of locally Lipschitz continuous reaction functions without any quasimonotone requirement using Schauder fixed point theorem, while the stability and attractivity analysis is for quasimonotone nondecreasing and mixed quasimonotone reaction functions using the monotone iterative scheme. The results for the general system are applied to the standard parabolic equations without time delay and to the corresponding ordinary differential system. Applications are also given to three Lotka–Volterra reaction diffusion model problems, and in each problem a sufficient condition on the reaction rates is obtained to ensure the stability and global attractivity of positive periodic solutions.

Solutions périodiques en temps des équations de Vlasov–Maxwell

We study here the existence of time periodic solution for the Vlasov–Maxwell equations in a three dimensional bounded domain. We assume that the boundary of the domain is strictly star-shaped. We give a priori estimates for the kinetic and electro-magnetic energy, and also for the normal and tangential traces of the electro-magnetic field. This method allows us to treat both classical and relativistic cases. To cite this article: M. Bostan, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

On the time periodic thermistor problem

In this paper we study a nonlocal parabolic/elliptic system which models thermistor behaviour in cases where heat losses to the surrounding gas play a significant role. The existence of time periodic solutions for the system is established through Faedo-Galerkin approximations and the Leray–Schauder degree theory. We show that for the small gas pressure case, the temperature of the time periodic solutions is positive. Moreover we consider the long time behaviour of the system and prove the existence of a uniform attractor. Finally, the finite dimensionality of the attractor is discussed.

Effects on global coupling of an interface problem in an activator–inhibitor system

In this paper, we shall explore the dynamics of interfaces in the problem with global coupling in order to investigate the existence of time periodic solutions as the global coupling intensity and the bifurcation parameter vary.

Time-periodic solutions of the time-dependent Ginzburg-Landau equations of superconductivity

The problem of existence of time-periodic solutions for the time-dependent Ginzburg-Landau equations of superconductivity is discussed. It is assumed that the applied magnetic field H is τ-periodic in time, and so is the associated dynamical process. We prove the existence of τ-periodic solutions in time, which are exactly the fixed points of the associated period mapping.

Existence of time periodic solutions for a damped generalized coupled nonlinear wave equations

The time periodic solution problem of damped generalized coupled nonlinear wave equations with periodic boundary condition was studied. By using the Galerkin method to construct the approximating sequence of time periodic solutions, a priori estimate and Laray-Schauder fixed point theorem to prove the convergence of the approximate solutions, the existence of time periodic solutions for a damped generalized coupled nonlinear wave equations can be obtained.

On approximations of first integrals for strongly nonlinear oscillators

In this paper strongly nonlinear oscillator equations will be studied.It will be shown that the recently developed perturbation method based onintegrating factors can be used to approximate first integrals. Not onlyapproximations of first integrals will be given, butit will also be shown how in a rather efficient way the existence and stability oftime-periodic solutions can be obtained from these approximations. In particularthe generalized Rayleigh oscillator equation \(\ddot X + 9X + \mu X^2 + {\lambda }X^3 = \varepsilon (\dot X - \dot X^3 )\) will be studied in detail, and it will beshown that at least five limit cycles can occur.

Free Vibrations of Beams on Bearings with Nonlinear Elastic Responses

Existence of time periodic solutions is investigated for certain beam partial differential equations.

Existence of Time Periodic Solutions for the Ginzburg-Landau Equations of Superconductivity

This paper deals with the time dependent Ginzburg-Landau equations of superconductivity with an applied magnetic field. The existence of time periodic solutions is established when the applied magnetic field is time periodic.

The Existence of Periodic Solutions to Reaction-Diffusion Systems with Periodic Data

The existence of time-periodic solutions is proven for a large class of reaction-diffusion systems in which Dirichlet boundary data, diffusivities, and reaction rates are periodic with common period.

Non-collision solutions for a second order singular Hamiltonian system with weak force

Under a weak force type condition, we consider the existence of time periodic solutions of singular Hamiltonian systems: (HS)q¨+Vq(q,t)=0q(t+T)=q(t).}We assume V (q, t) < 0 for all q, t and V (q, t), Vq(q, t) → 0 as |q| → ∞. Moreover we assume V (q, t) is of a form: V(q,t)=−1|q|α+U(q,t)where 0 < α <2 and U(q, t) ∈ C2 ((RN\\{0}) × R, R) is a T-periodic funetion in t such that |q|α U (q, t), |q|α + 1 Uq(q, t), |q|α+2 Uqq, (q, t), |q|α Ut, (q, t) → 0 as |q| → 0.For α ∈ (1, 2], we prove the existence of a non-collision solution of (HS). For α ∈ (0, 1], we prove that the generalized solution of (HS), which is introduced in [BR], enters the singularity 0 at most one time in its period. Our argument depends on a minimax argument due to [BR] and an estimate of Morse index of corresponding functional, which will be obtained via re-scaling argument.

Weakly damped quasilinear wave equation: existence of time-periodic solutions

AMONG many questions concerning global behaviour of hyperbolic equations, the existence of time-periodic solutions seems to be of great interest both from the mathematical and physical points of view. The aim of the present paper is to add to the vast literature on that subject a result related to the weakly damped quasilinear wave equation (see Section 1, theorem 1). With a function U = U(X, t) of x E (0, 1), t E R’ consider the equation

The existence of periodic solutions of nonlinear wave equations on Sn

We are concerned with the existence of time periodic solutions of nonlinear wave equations on n-dimensional spheres. The existence of nontrivial periodic solution in case 0 is a solution is proved. the existence of multiple, especially infinitely many, time periodic solutions is established for several classes f nonlinear terms which satisfy some symmetry properties such as time translation invariance or oddness. Furhermore, this paper also studies the effect of perturbations which are not small and which destroy the symmetry, and shows how multiple solutions persist despite the nonsymmetric perturbations if the growth of nonlinear term at infinity is suitable controlled.

A Hopf bifurcation for a reaction-diffusion equation with concentrated chemical kinetics

A reaction-diffusion equation related to some mathematical models of gasless combustion of solid fuel is studied. A formal bifurcation analysis by B. J. Matkowsky and G. I. Sivashinsky ( SIAM J. Appl. Math. 35 (1978), 465–478) shows that solutions demonstrate behavior typical for the Hopf bifurcation. A rigorous treatment of this phenomenon is developed. The problem is considered as an evolution equation in a Banach space. To circumvent difficulties involving a possible resonance with the continuous spectrum, appropriate weighted norms are introduced. A suitable version of the Hopf bifurcation theorem is developed and the existence of time periodic solutions is proved for values of the parameter near some critical value.

Critical point theorems for indefinite functionals

A variational principle of a minimax nature is developed and used to prove the existence of critical points for certain variational problems which are indefinite. The proofs are carried out directly in an infinite dimensional Hilbert space. Special cases of these problems previously had been tractable only by an elaborate finite dimensional approximation procedure. The main applications given here are to Hamiltonian systems of ordinary differential equations where the existence of time periodic solutions is established for several classes of Hamiltonians.