Forward attraction in nonautonomous difference equations

Two types of attractors consisting of families of sets that are mapped into each other under the dynamics have been defined for nonautonomous difference equations, one using pullback convergence with information about the system in the past and the other using forward convergence with information about the system in the future. In both cases, the component sets are constructed using a pullback argument within a positively invariant family of sets. The forward attractor so constructed also uses information about the past, which is very restrictive and not essential for determining future behaviour. Here an alternative is investigated, essentially the omega-limit set of the system, which Chepyzhov and Vishik called the uniform attractor. It is shown here that this set is asymptotically positively invariant, thus providing it with an hitherto missing form of invariance, if in somewhat weaker than usual, that one expects an attractor to possess. As a consequence this set provides useful information about the behaviour in current time during the approach to the limit.     

A definition of spectrum for differential equations on finite time

Hyperbolicity of an autonomous rest point is characterised by its linearization not having eigenvalues on the imaginary axis. More generally, hyperbolicity of any solution which exists for all times can be defined by means of Lyapunov exponents or exponential dichotomies. We go one step further and introduce a meaningful notion of hyperbolicity for linear systems which are defined for finite time only, i.e. on a compact time interval. Hyperbolicity now describes the transient dynamics on that interval. In this framework, we provide a definition of finite-time spectrum, study its relations with classical concepts, and prove an analogue of the Sacker-Sell spectral theorem: For a d-dimensional system the spectrum is non-empty and consists of at most d disjoint (and often compact) intervals. An example illustrates that the corresponding spectral manifolds may not be unique, which in turn leads to several challenging questions.     

Limiting equations in asymptotic stability problems for nonautonomous differential equations

Summary By limiting equations, we prove some asymptotic stability theorems for the origin ofR ⁿ with respect to the solutions of a differential equation , also when the functionf is not defined forx=0. Further we examine similar problems concerning the asymptotic stability of a setM ofR ⁿ that can be unbounded.

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Riassunto Mediante le equazioni limiti, si dimostrano alcuni teoremi di stabilità asintotica per l'origine diR ⁿ rispetto alle soluzioni di un'equazione differenziale , anche quando la funzionef non è definita perx=0. Vengono inoltre esaminati analoghi problemi relativi alla stabilità asintotica di un insiemeM diR ⁿ anche non limitato.

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Work performed under the auspices of the Italian Council of Research (G.N.F.M. del C.N.R.).     

Moving finite element methods for time fractional partial differential equations

With the aim of simulating the blow-up solutions, a moving finite element method, based on nonuniform meshes both in time and in space, is proposed in this paper to solve time fractional partial differential equations (FPDEs). The unconditional stability and convergence rates of 2-α for time and r for space are proved when the method is used for the linear time FPDEs with α-th order time derivatives. Numerical examples are provided to support the theoretical findings, and the blow-up solutions for the nonlinear FPDEs are simulated by the method.     

Periodic solutions of nonautonomous ordinary differential equations

For higher order ordinary differential equations, new sufficient conditions on the existence and uniqueness of periodic solutions are established. Results obtained cover the case when the right-hand side of the equation is not of a constant sign with respect to an independent variable.     

Uniformly attracting solutions of nonautonomous differential equations

Understanding the structure of attractors is fundamental in nonautonomous stability and bifurcation theory. By means of clarifying theorems and carefully designed examples we highlight the potential complexity of attractors for nonautonomous differential equations that are as close to autonomous equations as possible. We introduce and study bounded uniform attractors and repellors for nonautonomous scalar differential equations, in particular for asymptotically autonomous, polynomial, and periodic equations. Our results suggest that uniformly attracting or repelling solutions are the true analogues of attracting or repelling fixed points of autonomous systems. We provide sharp conditions for the autonomous structure to break up and give way to a bewildering diversity of nonautonomous bifurcations.     

Method of limit differential equations for nonautonomous discontinuous systems

For nonautonomous differential equations with discontinuous right-hand sides solvable in the sense of Filippov, an analogue of LaSalle’s invariance principle is proved by using Lyapunov functions with derivatives of constant sign. The specifics of the construction of the corresponding limit differential inclusions is taken into account.     

Stability of nonautonomous second-order partial differential equations

Existence, uniqueness, and three sufficient stability criteria will be established for a class of differential equations of the form: $\text{d}{}^2\text{u(t)}\text{dt}{}^2\text{+ C(t)}\text{du(t)}\text{dt}\text{+ K(t) u(t) = 0}$.The stability criteria emphasize the situations where the nonautonomous portion of the linear operators C(t) and K(t): (a) have small magnitudes, (b) have small derivatives (slowly varying), and (c) have small integrals (rapidly varying). Although this paper directs attention only to the second-order equation, corresponding stability results for the first-order equation could be obtained utilizing similar arguments.     

Periodic solutions of nonautonomous ordinary differential equations

For higher order ordinary differential equations, new sufficient conditions on the existence and uniqueness of periodic solutions are established. Results obtained cover the case when the right-hand side of the equation is not of a constant sign with respect to an independent variable.     

Stability conditions for systems of linear nonautonomous delay differential equations

Systems of linear nonautonomous delay differential equations are considered which are of the form y′_i(t) = ∑_{k = 1}ⁿ ∝₀^T b_ik(t, s) y_k(t − s) dη_ik(s) − c_i(t) y_i(t), where I = 1,…, n. Sufficient conditions are derived for both the asymptotic stability and the instability of the zero solution. The main result is found by a monotone technique using elementary methods only. Moreover, additional criteria are obtained by using the method of Lyapunov functionals.     

Stability of nonautonomous differential equations in Hilbert spaces

We introduce a large class of nonautonomous linear differential equations v^′=A(t)v in Hilbert spaces, for which the asymptotic stability of the zero solution, with all Lyapunov exponents of the linear equation negative, persists in v^′=A(t)v+f(t,v) under sufficiently small perturbations f. This class of equations, which we call Lyapunov regular, is introduced here inspired in the classical regularity theory of Lyapunov developed for finite-dimensional spaces, that is nowadays apparently overlooked in the theory of differential equations. Our study is based on a detailed analysis of the Lyapunov exponents. Essentially, the equation v^′=A(t)v is Lyapunov regular if for every k the limit of Γ(t)^1/t as t→∞ exists, where Γ(t) is any k-volume defined by solutions v₁(t),…,v_k(t). We note that the class of Lyapunov regular linear equations is much larger than the class of uniformly asymptotically stable equations.     

Classical solutions for partial functional differential equations with nonautonomous past

We study partial functional differential equations with infinite delay where the history function is modified by a backward evolution family. Under appropriate assumptions and using semigroup techniques we prove the existence of a unique classical solution. Die endgültige Fassung ging am 20. 6. 2001 einein     

On a resonance periodic problem for nonautonomous higher-order differential equations

For nonlinear nonautonomous higher-order ordinary differential equations, we prove in a sense optimal criteria for the solvability and unique solvability of a resonance periodic problem.     

Periodic nonautonomous differential equations with noninstantaneous impulsive effects

In this paper, we study a new class of periodic nonautonomous differential equations with periodic noninstantaneous impulsive effects. A concept of noninstantaneous impulsive Cauchy matrix is introduced, and some basic properties are considered. We give the representation of solutions to the homogeneous problem and nonhomogeneous problem by using noninstantaneous impulsive Cauchy matrix, and the variation of constants method, adjoint systems, and periodicity of solutions is verified under standard periodicity conditions. Further, we show the existence and uniqueness of solutions of semilinear problem and establish existence result for periodic solutions via Brouwer fixed point theorem and uniqueness and global asymptotic stability via Banach fixed point theorem.