Some results on the precompactness of orbits of dynamical systems

Differential and Integral Equations, 1991

Introduction. In this note we deal with the asymptotic behavior of (eventually) strongly monotone semigroups, S(t), on strictly ordered Banach spaces. In our considerations, the continuity in t does not enter and so our results hold for t in ~d or zt alike; in particular, they hold for discrete semigroups generated by a single map. Moreover, no smoothness assumptions, besides the continuity of the maps S(t) for fixed t, are introduced. The paper addresses the following basic question: under which conditions are all relatively compact orbits convergent? Our answer is very simple and geometrical. For a semigroup S(-) as above, with initial conditions chosen from the order interval [a, b], with a a subsolution, and b a supersolution, all precompact orbits are convergent if there is a continuous, strongly ordered arc r connecting a to b, and which, in general, may consist of two pieces rl and r2, with the lower one made up of subsolutions and the upper one of supersolutions. The most interesting, dynamically, is the case where (part of) r consists of a continuum of equilibria. In that case, f can be split into f1 and f2 in an infinity of ways. We point out that r need not be invariant under S(•), a feature that makes the result flexible in applications. We note that our structure hypotheses do not allow the existence of a non-degenerate unstable equilibrium on r except possibly at the end points. This feature, in general, is in the nature of things for stabilization of all precompact orbits to hold, for otherwise only generic results can be expected

Reports on Mathematical Physics, 1992

2010

The aim of this paper is to give an account of some problems considered in past years in the setting of Dynamical Systems, some new research directions and also state some open problems

Journal of Evolution Equations, 2021

In this paper, we examine regularity and stability issues for two damped abstract elastic systems. The damping involves the average velocity and a fractional power θ, with θ in [−1, 1], of the principal operator. The matrix operator defining the damping mechanism for the coupled system is degenerate. First, we prove that for θ in (1/2, 1], the underlying semigroup is not analytic, but is differentiable for θ in (0, 1); this is in sharp contrast with known results for a single similarly damped elastic system, where the semigroup is analytic for θ in [1/2, 1]; this shows that the degeneracy dominates the dynamics of the interacting systems, preventing analyticity in that range. Next, we show that for θ in (0, 1/2], the semigroup is of certain Gevrey classes. Finally, we show that the semigroup decays exponentially for θ in [0, 1], and polynomially for θ in [−1, 0). To prove our results, we use the frequency domain method, which relies on resolvent estimates. Optimality of our resolvent estimates is also established. Two examples of application are provided. CONTENTS 1. Problem formulation and statements of main results 2. Some technical Lemmas 3. Proof of Theorem 1.1 4. Proof of Theorem 1.2 5. Proof of Theorem 1.3 6. Examples of application 1 2). We assume that V ֒→ H ֒→ V ′ , each injection being dense and compact, where V ′ denotes the topological dual of V. Let a, b, and γ be positive constants. Let θ ∈ [−1, 1], and consider the evolution system (1.1) y tt + aAy + γA θ (y t + z t) = 0 in (0, ∞) z tt + bAz + γA θ (y t + z t) = 0 in (0, ∞) y(0) = y 0 ∈ V, y t (0) = y 1 ∈ H, z(0) = z 0 ∈ V, z t (0) = z 1 ∈ H.

Nonlinear Analysis: Theory, Methods & Applications, 2001

Lettere al Nuovo Cimento, 1980

Nonlinear Analysis: Theory, Methods & Applications, 1985

2023

Journal de Mathématiques Pures et Appliquées, 2013

Under suitable growth and coercivity conditions on the nonlinear damping operator g, we establish boundedness or compactness properties of trajectories to the equation

Journal of Mathematical Analysis and Applications, 1980

Submirred bx V. Lakshmikanrham Explicit criteria for the asymptotic stability (or instability) of bifurcating closed orbits are given for a class of abstract evolution equations. ' For the extensive literature on this subject. see 18. 14. I9 I.

Journal of Mathematical Analysis and Applications, 1987

Ergodic Theory and Dynamical Systems, 2009

This paper is concerned with the lower semicontinuity of attractors for semilinear non-autonomous differential equations in Banach spaces. We require the unperturbed attractor to be given as the union of unstable manifolds of time-dependent hyperbolic solutions, generalizing previous results valid only for gradient-like systems in which the hyperbolic solutions are equilibria. The tools employed are a study of the continuity of the local unstable manifolds of the hyperbolic solutions and results on the continuity of the exponential dichotomy of the linearization around each of these solutions.

We consider dynamical semigroups with unbounded Kossakowski-Lindblad-Davies generators which are related to evolution of an open system with a tuned repeated harmonic perturbation. Our main result is the proof of existence of uniquely determined minimal trace-preserving strongly continuous dynamical semigroups on the space of density matrices. The corresponding dual W *-dynamical system is shown to be unital quasi-free and completely positive automorphisms of the CCR-algebra. We also comment on the action of dynamical semigroups on quasi-free states.

International Journal of Pure and Applied Mathematics

In this paper we study autonomous evolution inclusions in an evolution triple, and satisfying one sided Lipschitzian condition with some negative constant. It is known that the solution set is compact on every bounded interval. Using this fact we prove the existence of a unique strong forward attractor and a unique strong backward attractor when the one sided Lipschitz constant is positive. As a corollary some surjectivity and fixed point results are proved. An example of a parabolic system, satisfying our assumptions is discussed.

2014

The aim of this paper is the study of the problem of global asymptotic stability of trivial solutions of non-autonomous dynamical systems (both with continuous and discrete time). We study this problem in the framework of general non-autonomous dynamical systems (cocycles). In particularly, we present some new results for non-autonomous version of Markus-Yamabe conjecture. Theorem 2.10. [7, Ch.I] For the locally completely (compact) dynamical systems the point, compact and local dissipativity are equivalent.