Gradient-like Dynamical Systems Research Articles

Equilibrium-Like Solutions of Asymptotically Autonomous Differential Equations

We analyze the chain recurrent set of skew product semiflows obtained from nonautonomous differential equations—ordinary differential equations or semilinear parabolic differential equations. For many gradient-like dynamical systems, Morse–Smale dynamical systems e.g., the chain recurrent set contains only isolated equilibria. The structure in the asymptotically autonomous setting is richer but still close to the structure of a Morse–Smale dynamical system. The main tool used in this paper is a nonautonomous flavour of Conley index theory developed by the author. We will see that for a class of good equations, the Conley index can be understood in terms of equilibria (in a generalized meaning) and their connections. This allows us to find specific solutions of asymptotically autonomous equations and generalizes properties of Morse–Smale dynamical systems to the asymptotically autonomous setting.

The Structure of Global Attractors for Non-autonomous Perturbations of Gradient-Like Dynamical Systems

In this paper we give the complete description of the structure of compact global (forward) attractors for non-autonomous perturbations of autonomous gradient-like dynamical systems under the assumption that the original autonomous system has a finite number of hyperbolic stationary solutions. We prove that the perturbed non-autonomous (in particular $$\tau $$ -periodic, quasi-periodic, Bohr almost periodic, almost automorphic, recurrent in the sense of Birkhoff) system has exactly the same number of invariant sections (in particular the perturbed systems has the same number of $$\tau $$ -periodic, quasi-periodic, Bohr almost periodic, almost automorphic, recurrent in the sense of Birkhoff solutions). It is shown the compact global (forward) attractor of non-autonomous perturbed system coincides with the union of unstable manifolds of this finite number of invariant sections.

Gradient Dynamics: Motion Near a Manifold of Quasi-Equilibria

In a general Banach space we consider gradient-like dynamical systems with the property that there is a manifold along which solutions move slowly compared to attraction in the transverse direction...

Asymptotic behavior of gradient-like dynamical systems involving inertia and multiscale aspects

In a Hilbert space H, we study the asymptotic behavior, as time variable t goes to +∞, of nonautonomous gradient-like dynamical systems involving inertia and multiscale features. Given Φ:H→R and Ψ:H→R two convex differentiable functions, γ a positive damping parameter, and ϵ(t) a function of t which tends to zero as t goes to +∞, we consider the second-order differential equationx¨(t)+γx˙(t)+∇Φ(x(t))+ϵ(t)∇Ψ(x(t))=0. This system models the emergence of various collective behaviors in game theory, as well as the asymptotic control of coupled nonlinear oscillators. Assuming that ϵ(t) tends to zero moderately slowly as t goes to infinity, we show that the trajectories converge weakly in H. The limiting equilibria are solutions of the hierarchical minimization problem which consists in minimizing Ψ over the set C of minimizers of Φ. As key assumptions, we suppose that ∫0+∞ϵ(t)dt=+∞ and that, for every p belonging to a convex cone C depending on the data Φ and Ψ∫0+∞[Φ⁎(ϵ(t)p)−σC(ϵ(t)p)]dt<+∞ where Φ⁎ is the Fenchel conjugate of Φ, and σC is the support function of C. An application is given to coupled oscillators.

The structure of global attractors for non-autonomous perturbations of discrete gradient-like dynamical systems

In this paper we give the complete description of the structure of compact global (forward) attractors for non-autonomous perturbations of discrete autonomous gradient-like dynamical systems under the assumption that the original discrete autonomous system has a finite number of hyperbolic stationary solutions. We prove that the perturbed non-autonomous (in particular -periodic, quasi-periodic, Bohr almost periodic, almost automorphic, recurrent in the sense of Birkhoff) system has exactly the same number of invariant sections (in particular the perturbed systems has the same number of -periodic, quasi-periodic, Bohr almost periodic, almost automorphic, recurrent in the sense of Birkhoff solutions). It is shown that the compact global (forward) attractor of non-autonomous perturbed system coincides with the union of unstable manifolds of this finite number of invariant sections.

A dynamic gradient approach to Pareto optimization with nonsmooth convex objective functions

In a general Hilbert framework, we consider continuous gradient-like dynamical systems for constrained multiobjective optimization involving nonsmooth convex objective functions. Based on the Yosida regularization of the subdifferential operators involved in the system, we obtain the existence of strong global trajectories. We prove a descent property for each objective function, and the convergence of trajectories to weak Pareto minima. This approach provides a dynamical endogenous weighting of the objective functions, a key property for applications in cooperative games, inverse problems, and numerical multiobjective optimization.

A Continuous Gradient-like Dynamical Approach to Pareto-Optimization in Hilbert Spaces

In a Hilbert space setting, we consider new continuous gradient-like dynamical systems for constrained multiobjective optimization. This type of dynamics was first investigated by Cl. Henry, and B. Cornet, as a model of allocation of resources in economics. Based on the Yosida regularization of the discontinuous part of the vector field which governs the system, we obtain the existence of strong global trajectories. We prove a descent property for each objective function, and in the quasi-convex case, convergence of the trajectories to Pareto critical points. We give an interpretation of the dynamic in terms of Pareto equilibration for cooperative games. By time discretization, we make a link to recent studies of Svaiter et al. on the algorithm of steepest descent for multiobjective optimization.

An estimate on the fractal dimension of attractors of gradient-like dynamical systems

This paper is dedicated to estimate the fractal dimension of exponential global attractors of some generalized gradient-like semigroups in a general Banach space in terms of the maximum of the dimension of the local unstable manifolds of the isolated invariant sets, Lipschitz properties of the semigroup and the rate of exponential attraction. We also generalize this result for some special evolution processes, introducing a concept of Morse decomposition with pullback attractivity. Under suitable assumptions, if (A,A∗) is an attractor–repeller pair for the attractor A of a semigroup {T(t):t≥0}, then the fractal dimension of A can be estimated in terms of the fractal dimension of the local unstable manifold of A∗, the fractal dimension of A, the Lipschitz properties of the semigroup and the rate of the exponential attraction. The ingredients of the proof are the notion of generalized gradient-like semigroups and their regular attractors, Morse decomposition and a fine analysis of the structure of the attractors. As we said previously, we generalize this result for some evolution processes using the same basic ideas.

Barrier Operators and Associated Gradient-Like Dynamical Systems for Constrained Minimization Problems

We study some continuous dynamical systems associated with constrained optimization problems. For that purpose, we introduce the concept of elliptic barrier operators and develop a unified framework to derive and analyze the associated class of gradient-like dynamical systems, called A-driven descent method (A-DM). Prominent methods belonging to this class include several continuous descent methods studied earlier in the literature such as steepest descent method, continuous gradient projection methods and Newton-type methods as well as continuous interior descent methods such as Lotka-Volterra-type differential equations and Riemannian gradient methods. Related discrete iterative methods such as proximal interior point algorithms based on Bregman functions and second order homogeneous kernels can also be recovered within our framework and allow for deriving some new and interesting dynamics. We prove global existence and strong viability results of the corresponding trajectories of (A-DM) for a smooth objective function. When the objective function is convex, we analyze the asymptotic behavior at infinity of the trajectory produced by the proposed class of dynamical systems (A-DM). In particular, we derive a general criterion ensuring the global convergence of the trajectory of (A-DM) to a minimizer of a convex function over a closed convex set. This result is then applied to several dynamics built upon specific elliptic barrier operators. Throughout the paper, our results are illustrated with many examples.

CONVERGENCE TO EQUILIBRIUM FOR SEMILINEAR PARABOLIC PROBLEMS IN

ABSTRACT We consider the semilinear parabolic equation on , assuming that f is an arbitrary C 1 function satisfying and . We prove that any bounded positive solution that decays to zero at spatial infinity, uniformly with respect to t, converges to a (single) stationary solution as . Our proof combines energy and comparison techniques with dynamical system arguments. We first establish an asymptotic symmetrization result: as , approaches a set of steady states that are radially symmetric about a common origin in . To this aim we introduce a new tool that we call first moments of energy. Having established the symmetrization, we apply a general convergence result for gradient-like dynamical systems. This amounts to showing that the dimension of the kernel of the linearized operator around an equilibrium w matches the dimension of a manifold of equilibria passing through w.

An analysis of maximum clique formulations and saturated linear dynamical network

Several formulations and methods used in solving an NP-hard discrete optimization problem, maximum clique, are considered in a dynamical system perspective proposing continuous methods to the problem. A compact form for a saturated linear dynamical network, recently developed for obtaining approximations to maximum clique, is given so its relation to the classical gradient projection method of constrained optimization becomes more visible. Using this form, gradient-like dynamical systems as continuous methods for finding the maximum clique are discussed. To show the one to one correspondence between the stable equilibria of the saturated linear dynamical network and the minima of objective function related to the optimization problem, La Salle’s invariance principle has been extended to the systems with a discontinuous right-hand side. In order to show the efficiency of the continuous methods simulation results are given comparing saturated the linear dynamical network, the continuous Hopfield network, the cellular neural networks and relaxation labelling networks. It is concluded that the quadratic programming formulation of the maximum clique problem provides a framework suitable to be incorporated with the continuous relaxation of binary optimization variables and hence allowing the use of gradient-like continuous systems which have been observed to be quite efficient for minimizing quadratic costs.