Attractors for infinite-dimensional non-autonomous dynamical systems

Carvalho, Alexandre N.; Langa, José A.; Robinson, James C.

doi:10.1007/978-1-4614-4581-4

Cited by 420 publications

(538 citation statements)

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“…The limit t / 2' is the pullback limit and it provides the natural generalization of the forward asymptotics associated with autonomous systems (Arnold 1998;Rasmussen 2007;Kloeden and Rasmussen 2011;Carvalho et al 2012;Ghil et al 2008;Chekroun et al 2011); see the appendix herein. In practical terms, though, a question arises: How large should jtj be for A(t, t) to be virtually independent of t?…”

Section: Convergence To Pbasmentioning

confidence: 99%

“…, W 4 ) for such an existence to be guaranteed. General results are known from the specialized literature on the existence of pullback attractors or related invariant manifolds (Kloeden and Rasmussen 2011;Carvalho et al 2012;Chekroun et al 2015a,b). For the sake of conciseness and clarity, however, we provide below the main elements of such an existence theory, while emphasizing the energy estimates involved; see also Kondrashov et al (2015, Theorem 3.1 and Corollary 3.2).…”

Section: Existence Of a Global Pullback Attractormentioning

confidence: 99%

“…(A12) We have thus proved that U(t, t)D (t) & B [0, R(t)] for t # t 0 (t), ensuring in turn the desired pullback dissipation. The existence and uniqueness proceed then from this dissipation property and the theory of pullback attractors [e.g., Carvalho et al (2012)]. ∎…”

Section: Theoremmentioning

confidence: 99%

“…A global PBA is defined in the mathematical literature as a time-dependent set A(t) in the system's phase space X that is invariant under its governing equations-together with the equally time-dependent, invariant measure m(t) supported on this set-and to which all trajectories in X starting in the distant past converge (Arnold 1998;Rasmussen 2007;Kloeden and Rasmussen 2011;Carvalho et al 2012). As we shall see later, such a global PBA might include two or more local attractors, which only attract trajectories from certain subsets of X.…”

Section: Introductionmentioning

confidence: 99%

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Exploring the Pullback Attractors of a Low-Order Quasigeostrophic Ocean Model: The Deterministic Case

2016

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A low-order quasigeostrophic double-gyre ocean model is subjected to an aperiodic forcing that mimics time dependence dominated by interdecadal variability. This model is used as a prototype of an unstable and nonlinear dynamical system with time-dependent forcing to explore basic features of climate change in the presence of natural variability. The study relies on the theoretical framework of nonautonomous dynamical systems and of their pullback attractors (PBAs), that is, of the time-dependent invariant sets attracting all trajectories initialized in the remote past. The existence of a global PBA is rigorously demonstrated for this weakly dissipative nonlinear model. Ensemble simulations are carried out and the convergence to PBAs is assessed by computing the probability density function (PDF) of localization of the trajectories. A sensitivity analysis with respect to forcing amplitude shows that the PBAs experience large modifications if the underlying autonomous system is dominated by small-amplitude limit cycles, while less dramatic changes occur in a regime characterized by large-amplitude relaxation oscillations. The dependence of the attracting sets on the choice of the ensemble of initial states is then analyzed. Two types of basins of attraction coexist for certain parameter ranges; they contain chaotic and nonchaotic trajectories, respectively. The statistics of the former does not depend on the initial states whereas the trajectories in the latter converge to small portions of the global PBA. This complex scenario requires separate PDFs for chaotic and nonchaotic trajectories. General implications for climate predictability are finally discussed.

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Section: Convergence To Pbasmentioning

confidence: 99%

Section: Existence Of a Global Pullback Attractormentioning

confidence: 99%

Section: Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Exploring the Pullback Attractors of a Low-Order Quasigeostrophic Ocean Model: The Deterministic Case

2016

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“…Moreover, similar properties are used to prove that a gradient system with a finite number of hyperbolic equilibria (see [18,19]) can be completely characterized by the internal dynamics between equilibria: every global solution connects two different equilibria and there are no homoclinic structures connecting equilibria (see [11,13]). …”

Section: Structural Stability: Robustness Under Perturbationmentioning

confidence: 99%

Attracting Complex Networks

Guerrero

Langa

Suárez

2016

Lecture Notes in Economics and Mathematical Systems

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Real phenomena from different areas of Life Sciences can be described by complex networks, whose structure is usually determining their intrinsic dynamics. On the other hand, Dynamical Systems Theory is a powerful tool for the study of evolution processes in real situations. The concept of global attractor is the central one in this theory. In the last decades there has been an intensive research in the geometrical characterization of global attractors. However, there still exists a weak connection between the asymptotic dynamics of a complex network and the structure of associated global attractors. In this paper we show that, in order to analyze the long-time behavior of the dynamics on a complex network, it is the topological and geometrical structure of the attractor the subject to take into account. In fact, given a complex network, a global attractor can be understood as the new attracting complex network which is really describing and determining the forwards dynamics of the phenomena. We illustrate our discussion with models of differential equations related to mutualistic complex networks in Economy and Ecology.

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Extremes and Recurrence in Dynamical Systems

Lucarini¹,

Faranda²,

Freitas³

et al. 2016

174

249

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ii 4.1.2 The New Conditions 44 4.1.3 The Existence of EVL for General Stationary Stochastic Processes under Weaker Hypotheses 46 4.1.4 Proofs of Theorem 4.1.4 and Corollary 4.1.5 48 4.2 Extreme Values for Dynamically Defined Stochastic Processes 53 4.2.1 Observables and Corresponding Extreme Value Laws 55 4.2.2 Extreme Value Laws for Uniformly Expanding Systems 59 4.2.3 Example 4.2.1 revisited 61 4.2.4 Proof of the Dichotomy for Uniformly Expanding Maps 63 4.3 Point Processes of Rare Events 64 4.3.1 Absence of Clustering 64 4.3.2 Presence of Clustering 65 4.3.3 Dichotomy for Uniformly Expanding Systems for Point Processes 67 4.4 Conditions Д q (u n ), D 3 (u n ), D p (u n ) * and Decay of Correlations 68 4.5 Specific Dynamical Systems where the Dichotomy Applies 71 4.5.1 Rychlik Systems 72 4.5.2 Piecewise Expanding Maps in Higher Dimensions 73 4.6 Extreme Value Laws for Physical Observables 74

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Attractors for infinite-dimensional non-autonomous dynamical systems

Cited by 420 publications

References 0 publications

Exploring the Pullback Attractors of a Low-Order Quasigeostrophic Ocean Model: The Deterministic Case

Exploring the Pullback Attractors of a Low-Order Quasigeostrophic Ocean Model: The Deterministic Case

Attracting Complex Networks

Extremes and Recurrence in Dynamical Systems

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