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Register a new userEstimate of the attractive velocity of attractors for some dynamical systems
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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.
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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.
We study the period doubling renormalization operator for dynamics which present two coupled laminar regimes with two weakly expanding fixed points. We focus our analysis on the potential point of view, meaning we want to solve $$V=mathcal{R} (V):=Vcirc fcirc h+V circ h,$$ where $f$ and $h$ are naturally defined. Under certain hypothesis we show the existence of a explicit ``attracting fixed point $V^*$ for $mathcal{R} $. We call $mathcal{R}$ the renormalization operator which acts on potentials $V$. The log of the derivative of the main branch of the Manneville-Pomeau map appears as a special ``attracting fixed point for the local doubling period renormalization operator. We also consider an analogous definition for the one-sided 2-full shift $S$ (and also for the two-sided shift) and we obtain a similar result. Then, we consider global properties and we prove two rigidity results. Up to some weak assumptions, we get the uniqueness for the renormalization operator in the shift. In the last section we show (via a certain continuous fraction expansion) a natural relation of the two settings: shift acting on the Bernoulli space ${0,1}^mathbb{N}$ and Manneville-Pomeau-like map acting on an interval.
Attractors of cooperative dynamical systems are particularly simple; for example, a nontrivial periodic orbit cannot be an attractor. This paper provides characterizations of attractors for the wider class of coherent systems, defined by the property that no directed feedback loops are negative. Several new results for cooperative systems are obtained in the process.
This paper shows that the celebrated Embedding Theorem of Takens is a particular case of a much more general statement according to which, randomly generated linear state-space representations of generic observations of an invertible dynamical system carry in their wake an embedding of the phase space dynamics into the chosen Euclidean state space. This embedding coincides with a natural generalized synchronization that arises in this setup and that yields a topological conjugacy between the state-space dynamics driven by the generic observations of the dynamical system and the dynamical system itself. This result provides additional tools for the representation, learning, and analysis of chaotic attractors and sheds additional light on the reservoir computing phenomenon that appears in the context of recurrent neural networks.
We study the topological properties of attractors of Iterated Function Systems (I.F.S.) on the real line, consisting of affine maps of homogeneous contraction ratio. These maps define what we call a second generation I.F.S.: they are uncountably many and the set of their fixed points is a Cantor set. We prove that when this latter either is the attractor of a finite, non-singular, hyperbolic, I.F.S. (of first generation), or it possesses a particular dissection property, the attractor of the second generation I.F.S. consists of finitely many closed intervals.
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