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Equi-attraction and continuity of attractors for skew-product semiflows

Tomás Caraballo, Alexandre N. Carvalho, Henrique B. da Costa, José A. Langa
2016 Discrete and continuous dynamical systems. Series B  
In this paper we prove the equivalence between equi-attraction and continuity of attractors for skew-product semi-flows, and equi-attraction and continuity of uniform and cocycle attractors associated  ...  Results on the upper and lower-semicontinuity of uniform and cocycle attractors are relatively new in the literature, as a deep understanding of the internal structure of these sets is needed, which is  ...  We would like to thank the referees for their helpful and interesting suggestions which allowed us to improve the presentation of the paper.  ... 
doi:10.3934/dcdsb.2016081 fatcat:huvsd2ebhrgjrpcndxinzu5ham

Page 528 of Mathematical Reviews Vol. 56, Issue 2 [page]

1978 Mathematical Reviews  
He does so, however, by introducing the idea of a semiflow associated with a process.  ...  Lysko (Warsaw) Dafermos, Constantine M. 3813 Semiflows associated with and uniform Math. Systems Theory 8 (1974/75), no. 2, 142-149.  ... 

Non-autonomous dynamical systems

Alexandre N. Carvalho, José A. Langa, James C. Robinson
2015 Discrete and continuous dynamical systems. Series B  
This fact leads to the definition of concepts such as skew-products and cocycles and their associated global, uniform, and cocycle attractors.  ...  topological structural stability for the pullback attractors of evolution processes that arise from small non-autonomous perturbations, with respect to the limit regime.  ...  Suppose Σ is compact and A(·) is the cocycle attractor of (ϕ, θ) and that Π(·) is the associated skew-product semiflow.  ... 
doi:10.3934/dcdsb.2015.20.703 fatcat:65bukdbeq5dihcrbeqsqtwsldm

Attractors for impulsive non-autonomous dynamical systems and their relations

E.M. Bonotto, M.C. Bortolan, T. Caraballo, R. Collegari
2017 Journal of Differential Equations  
of asymptotic dynamics, such as the cocycle attractor, the uniform attractor and the global attractor for the impulsive skew-product semiflow.  ...  Lastly, we illustrate our theory by exhibiting an example of a non-classical non-autonomous parabolic equation with subcritical nonlinearity and impulses.  ...  we consider the impulsive skew-product semiflow associated with rpϕ, θq pX,Σq , M, Is, then we obtain the same object as if we first constructed the skew-product semiflow tΠptq : t ě 0u associated with  ... 
doi:10.1016/j.jde.2016.11.036 fatcat:2wvjiwcv5rfdjgbs6r76soig6q

Skew product semiflows and Morse decomposition

M.C. Bortolan, T. Caraballo, A.N. Carvalho, J.A. Langa
2013 Journal of Differential Equations  
In particular, we derive a Morse decomposition for the global attractors of skew product semiflows (and thus for pullback attractors of non-autonomous differential equations) from a Morse decomposition  ...  The description of the asymptotic dynamics of non-autonomous equations lies on dynamical structures of some associated limiting non-autonomous -and autonomous -differential equations (one for each global  ...  We would like to thank the referee for the helpful and interesting suggestions and comments which allowed us to significantly improve the presentation of this paper.  ... 
doi:10.1016/j.jde.2013.06.023 fatcat:444qjec4a5exxmkgtmegdw2wmy

On the uniform exponential stability of linear skew-product semiflows

Ciprian Preda
2006 Journal of Function Spaces and Applications  
The problem of uniform exponential stability of linear skew-product semiflows on locally compact metric space with Banach fibers, is discussed.  ...  It is established a connection between the uniform exponential stability of linear skewproduct semiflows and some admissibility-type condition.  ...  We are setting an connection between the uniform exponential stability of a linear skew-product semiflow on a locally compact metric space and the admissibility of a function space from a large class of  ... 
doi:10.1155/2006/703620 fatcat:dkp4ard33fdudl7tu6uu7oivg4

Local Approximate Forward Attractors of Nonautonomous Dynamical Systems

Ailing Qi, Xuewei Ju
2020 American Journal of Applied Mathematics  
Specifically, let φ be a cocycle semiflow on a Banach space X with driving system θ on a base space P . Suppose that the base space P is compact and φ is uniformly asymptotically compact.  ...  Let A(·) be a local pullback attractor with p∈P A(p) being compact.  ...  Based on the compactness of the base spaces, the pullback attraction of cocycle semiflow and (forward) attraction of the associated autonomous semiflow is shown to be equivalent.  ... 
doi:10.11648/j.ajam.20200805.16 fatcat:eysuipyryjgcxmzsnok75afkgm

The ODE method for some self-interacting diffusions on ℝ d

Aline Kurtzmann
2010 Annales de l'I.H.P. Probabilites et statistiques  
These processes have so far been studied on compact spaces by Benaïm, Ledoux and Raimond, using stochastic approximation methods.  ...  These are solutions to SDEs with a drift term depending on the actual position of the process and its normalized occupation measure µt.  ...  , the deterministic semiflow associated to the self-interacting diffusion and show its local existence. The proofs of the main results are given in Section ??  ... 
doi:10.1214/09-aihp206 fatcat:ivfaxdhqi5frzaw65filecgwwq

Uniform and strict persistence in monotone skew-product semiflows with applications to non-autonomous Nicholson systems

Rafael Obaya, Ana M. Sanz
2016 Journal of Differential Equations  
in terms of the principal spectrums of some associated linear skew-product semiflows which admit a continuous separation.  ...  We determine sufficient conditions for uniform and strict persistence in the case of skew-product semiflows generated by solutions of nonautonomous families of cooperative systems of ODEs or delay FDEs  ...  Uniform and strict persistence in abstract monotone skew-product semiflows with a continuous separation In this section, we analyze the notions of uniform and strict persistence in a very general context  ... 
doi:10.1016/j.jde.2016.06.019 fatcat:v6b5ok2oyjhx7cduhwjnvuarku

Page 7810 of Mathematical Reviews Vol. , Issue 2004j [page]

2004 Mathematical Reviews  
Then they prove that the uniform exponen- tial dichotomy of a linear skew-product semiflow is equivalent to the uniform exponential dichotomy of the associated discrete linear skew-product semiflow.  ...  Based on results of Chow and Leiva and of Latushkin and Schnaubelt, the authors discuss the connections between the uniform exponential dichotomy of a discrete linear skew- product semiflow and the uniform  ... 

Compactifications of Dynamical Systems [article]

Ethan Akin, Joseph Auslander
2010 arXiv   pre-print
In the process we don't want to introduce recurrence which was not there in the original system.  ...  While compactness is an essential assumption for many results in dynamical systems theory, for many applications the state space is only locally compact.  ...  Let φ be a semiflow on X and letX be a proper compactification with inclusion map j : X →X and associated algebra A = j * (B(X)) ⊂ B(X).  ... 
arXiv:1004.0323v1 fatcat:ihnwrom4qrejlp7senlai5dq7i

A generalized Polya's urn with graph based interactions [article]

Michel Benaim, Itai Benjamini, Jun Chen, Yuri Lima
2013 arXiv   pre-print
We also prove that if a<1 then there is a single point v=v(G,a) with nonzero entries such that the proportion converges to v almost surely. A special case is when G is regular and a is at most 1.  ...  We show e.g. that if G is non-bipartite then the proportion of balls in the bins converges to the uniform measure almost surely.  ...  Acknowledgements The authors are thankful to Omer Angel and Omri Sarig for valuable comments and suggestions. I.B. is the incumbent of the Renee and Jay Weiss Professorial Chair.  ... 
arXiv:1211.1247v4 fatcat:j25o24fzdfeflcxwjixsjetpje

Non-autonomous perturbations of a non-classical non-autonomous parabolic equation with subcritical nonlinearity

Matheus C. Bortolan, Felipe Rivero
2017 Applied Mathematics and Nonlinear Sciences  
Dealing with perturbations adds a difficulty to this process, since we must be able to do such study for each ε small and obtain the result with uniformity in this parameter.  ...  A NDS (ϕ, σ ) (X,Σ) has a uniform attractor if and only if there exists a compact uniform attracting set K.  ...  Moreover, for 0 ε ε 1 , we have With this result, we can derive structures for the other types of attractors, as we did for A 0 ; namely, we have that the uniform attractor of (ϕ ε , θ ) (H 1 0 (Ω),Σ ε  ... 
doi:10.21042/amns.2017.1.00004 fatcat:u4yj7vwewfakzlhw7y5b6jbbfq

Minimality properties of set-valued processes and their pullback attractors [article]

Michele Coti Zelati, Piotr Kalita
2014 arXiv   pre-print
The presentation is complemented with examples and counterexamples to test the sharpness of the hypotheses involved, including a reaction-diffusion equation, a discontinuous ordinary differential equation  ...  and an irregular form of the heat equation.  ...  Global attractors for m-semiflows. Observe that an m-semiflow is asymptoti- cally compact if and only if the associated m-process is pullback asymptotically compact.  ... 
arXiv:1407.6135v2 fatcat:w7vd4yhkm5bq3ova32qahqpp5q

On Exponential Stability for Skew-Evolution Semiflows on Banach Spaces [article]

Codruta Stoica
2008 arXiv   pre-print
The paper emphasizes the property of stability for skew-evolution semiflows on Banach spaces, defined by means of evolution semiflows and evolution cocycles and which generalize the concept introduced  ...  A unified treatment in the uniform and in the nonuniform setting is given. The main results are also formulated in discrete time.  ...  Let us consider the set C = C(R, R) of all continuous functions given by x : R → R, endowed with the uniform convergence topology on compact subsets of R.  ... 
arXiv:0804.1479v1 fatcat:6d5dbvvnrrendaflgkgjecnt3e
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