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This paper is devoted to the quantitative study of the attractive velocity of generalized attractors for infinite-dimensional dynamical systems. We introduce the notion of~$varphi$-attractor whose attractive speed is characterized by a general non-negative decay function~$varphi$, and prove that~$varphi$-decay with respect to noncompactness measure is a sufficient condition for a dissipitive system to have a~$varphi$-attractor. Furthermore, several criteria for~$varphi$-decay with respect to noncompactness measure are provided. Finally, as an application, we establish the existence of a generalized exponential attractor and the specific estimate of its attractive velocity for a semilinear wave equation with a critical nonlinearity.
In this paper, we first prove an abstract theorem on the existence of polynomial attractors and the concrete estimate of their attractive velocity for infinite-dimensional dynamical systems, then apply this theorem to a class of wave equations with nonlocal weak damping and anti-damping in case that the nonlinear term~$f$~is of subcritical growth.
We study the attractor of Iterated Function Systems composed of infinitely many affine, homogeneous maps. In the special case of second generation IFS, defined herein, we conjecture that the attractor consists of a finite number of non-overlapping intervals. Numerical techniques are described to test this conjecture, and a partial rigorous result in this direction is proven.
This paper is concerned with the dynamics of an infinite-dimensional gradient system under small almost periodic perturbations. Under the assumption that the original autonomous system has a global attractor given as the union of unstable manifolds of a finite number of hyperbolic equilibrium solutions, we prove that the perturbed non-autonomous system has exactly the same number of almost periodic solutions. As a consequence, the pullback attractor of the perturbed system is given by the union of unstable manifolds of these finitely many almost periodic solutions. An application of the result to the Chafee-Infante equation is discussed.
Certain dynamical models of competition have a unique invariant hypersurface to whichevery nonzero tractory is asymptotic, having simple geometry and topology.
We consider an independently identically distributed random dynamical system generated by finitely many, non-uniformly expanding Markov interval maps with a finite number of branches. Assuming a topologically mixing condition and the uniqueness of the equilibrium state of product form, we establish an almost-sure weighted equidistribution of cycles with respect to a natural stationary measure, as the periods of the cycles tend to infinity. This result is an analogue of Bowens theorem on periodic orbits of topologically mixing Axiom A diffeomorphisms in random setup. We also prove averaging results over all samples, as well as another samplewise result. We apply our result to the random $beta$-expansion of real numbers, and obtain a new formula for the mean relative frequencies of digits in the series expansion.