Dynamics of a class of ODEs more general than almost periodic

Kloeden, Peter E.; Rodrigues, Hildebrando M.

doi:10.1016/j.na.2010.12.025

Cited by 21 publications

(45 citation statements)

References 22 publications

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“…Then H( f ) is a compact metric space provided f satisfies appropriate hypotheses (e.g., f is periodic or almost periodic in t, or it belongs to a class of functions more general than almost periodic ones, as introduced by Kloeden and Rodrigues in [46]) and the closure in (3.9) is taken in the uniform convergence topology (see e.g., [51]). Define the shift mapping θ t :…”

Section: Skew Product Flow Formulationmentioning

confidence: 99%

“…The nonautonomous terms F(·) and G(·) can be periodic, almost periodic or even belong to a type of functions more general than almost periodic, as in [46]. To illustrate the theory of attractors for the process formalism, we will consider very general nonautonomous terms.…”

Section: Nonautonomous Lorenz-84 Modelmentioning

confidence: 99%

“…A more detailed analysis concerning the structure of the pullback attractor for the nonautonomous Lorenz-84 model can be found in [1]. More precisely, one can see how the properties of the nonautonomous terms (periodicity, almost periodicity or a property called more general than almost periodicity by Kloeden and Rodrigues in [46]) in the model are inherited by the pullback attractor. Also an estimate of the Hausdorff dimension of the pullback attractor is constructed in [1], for which the skew product flow formalism has to be used.…”

Section: Nonautonomous Lorenz-84 Modelmentioning

confidence: 99%

“…This assumption results in a system of random differential equations: 46) where ν, β, γ are positive constants which represent the death rate, the infection rate and the removal rate, respectively, and N(t) = S (t)+ I(t)+R(t) is the total population.…”

Section: Random Sir With Real Noisementioning

confidence: 99%

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Applied Nonautonomous and Random Dynamical Systems

Caraballo¹,

Han²

2016

SpringerBriefs in Mathematics

114

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PrefaceDuring the past two decades, the theory of nonautonomous dynamical systems and random dynamical systems has made substantial progress in studying the long term dynamics of open systems subject to time-dependent or random forcing. However, most of the existing pertinent literatures are fairly technical and almost impenetrable for a general audience, except the most dedicated specialists. Moreover, the concepts and methods of nonautonomous and random dynamical systems, though well-established, are extremely nontrivial to apply to real-world problems. The aim of this work is to provide an accessible and broad introduction to the theory of nonautonomous and random dynamical systems, with an emphasis on applications of the theory to problems arising in the applied sciences and engineering.The book starts with basic concepts in the theory of autonomous dynamical systems which are easier to understand, and used as the motivation for the study of non-autonomous and random dynamical systems. Then the framework of nonautonomous dynamical systems is set up, including various approaches to analyze the long time behavior of non-autonomous problems. The major emphasis is given to the novel theory of pullback attractors, as it can be regarded as a natural extension of the autonomous theory and allows a larger variety of time-dependence forcing than other alternatives such as skew-product flows or cocycles. In the end the theory of random dynamical systems and random attractors is introduced and shown to fairly informative to the study of long term behavior of stochastic systems with random forcing.Each set of theory is illustrated by being applied to three different models, the chemostat model, the SIR epidemic model, and the Lorenz-84 model, in their autonomous, nonautonomous, and stochastic formulations, respectively. The techniques and methods adopted can be applied to the study of the long term behavior of a wide range of applications arising in applied sciences and engineering.

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Section: Skew Product Flow Formulationmentioning

confidence: 99%

Section: Nonautonomous Lorenz-84 Modelmentioning

confidence: 99%

Section: Nonautonomous Lorenz-84 Modelmentioning

confidence: 99%

Section: Random Sir With Real Noisementioning

confidence: 99%

See 2 more Smart Citations

Applied Nonautonomous and Random Dynamical Systems

Caraballo¹,

Han²

2016

SpringerBriefs in Mathematics

114

View full text Add to dashboard Cite

show abstract

“…Our first goal is to show that a compact subset A ⊂ P × R d is a Lyapunov attractor if and only if it is a pullback attractor [4,17,18,5,6]. We give a definition of the latter concept which seems appropriate in the context of skew-product local flows.…”

Section: Attractorsmentioning

confidence: 99%