Equi-Attraction and the Continuous Dependence of Attractors on Parameters

Li, Desheng; Kloeden, Peter E.

doi:10.1017/s0017089503001605

Cited by 27 publications

(4 citation statements)

References 16 publications

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“…Next, we list some definitions that are useful for describing the robustness of dynamical systems. Similar concepts (may be with different names) have been given by many other literatures (see [3,4,5,15] and references therein).…”

mentioning

confidence: 69%

Robustness of exponentially κ-dissipative dynamical systems with perturbations

Zhang¹,

Wang²,

Zhong³

2017

Discrete &Amp; Continuous Dynamical Systems - B

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mentioning

confidence: 69%

Robustness of exponentially κ-dissipative dynamical systems with perturbations

Zhang¹,

Wang²,

Zhong³

2017

Discrete &Amp; Continuous Dynamical Systems - B

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“…There are also fairly general results for the continuity of global attractors of semi-flows [9]. In particular Hoang et.…”

Section: Introductionmentioning

confidence: 95%

“…Let H 0 be the compact filled triangle XZP . The conditions (9) imply that H 0 is contained in the region x 1 > 0, the line segments XZ and ZP belong to the unstable manifold of X, and XP belongs to the local stable manifold of X. It follows that every point on the boundary of the forward invariant set H = ∪ ∞ k=0 L k (H 0 ) belongs to either the unstable manifold of X or the line segment XP .…”

Section: Lozi Mapsmentioning

confidence: 99%

Robust Chaos and the Continuity of Attractors

Glendinning¹,

Simpson²

2019

Preprint

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As the parameters of a map are varied an attractor may vary continuously in the Hausdorff metric. The purpose of this paper is to explore the continuation of chaotic attractors. We argue that this is not a helpful concept for smooth unimodal maps for which periodic windows fill parameter space densely, but that for piecewise-smooth maps it provides a way to delineate structure within parameter regions of robust chaos and form a stronger notion of robustness. We obtain conditions for the continuity of an attractor and demonstrate the results with coupled skew tent maps, the Lozi map, and the border-collision normal form.

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“…Later, the author in [52] established the concept of upper semicontinuity for non-compact random dynamical systems also. But for lower semicontinuity, we require more detailed studies either on the structure of the deterministic attractor or on the equi-attraction of the family of the random attractors of perturbed systems, see [3,10,25,34], etc for more details.…”

mentioning

confidence: 99%

Convergence of random attractors towards deterministic singleton attractor for 2D and 3D convective Brinkman-Forchheimer equations

Kinra

Mohan

2022

EECT

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This work deals with the asymptotic behavior of the two as well as three dimensional convective Brinkman-Forchheimer (CBF) equations in an <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula>-dimensional torus (<inline-formula><tex-math id="M2">\begin{document}$ n = 2, 3 $\end{document}</tex-math></inline-formula>):<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \frac{\partial\boldsymbol{u}}{\partial t}-\mu \Delta\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+\alpha\boldsymbol{u}+\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p = \boldsymbol{f}, \ \nabla\cdot\boldsymbol{u} = 0, $\end{document} </tex-math></disp-formula>where <inline-formula><tex-math id="M3">\begin{document}$ r\geq1 $\end{document}</tex-math></inline-formula>. We prove that the global attractor of the above system is singleton under small forcing intensity (<inline-formula><tex-math id="M4">\begin{document}$ r\geq 1 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M5">\begin{document}$ n = 2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ r\geq 3 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M7">\begin{document}$ n = 3 $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M8">\begin{document}$ 2\beta\mu\geq 1 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M9">\begin{document}$ r = n = 3 $\end{document}</tex-math></inline-formula>). But if one perturbs the above system with an additive or multiplicative white noise, there is no sufficient evidence that the random attractor keeps the singleton structure. We obtain that the random attractor for 2D stochastic CBF equations forced by additive and multiplicative white noise converges towards the deterministic singleton attractor for all <inline-formula><tex-math id="M10">\begin{document}$ 1\leq r<\infty $\end{document}</tex-math></inline-formula>, when the coefficient of random perturbation converges to zero (upper and lower semicontinuity). For the case of 3D stochastic CBF equations perturbed by additive and multiplicative white noise, we are able to establish that the random attractor converges towards the deterministic singleton attractor for <inline-formula><tex-math id="M11">\begin{document}$ 3\leq r<\infty $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M12">\begin{document}$ 2\beta\mu\geq 1 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M13">\begin{document}$ r = 3 $\end{document}</tex-math></inline-formula>), when the coefficient of random perturbation converges to zero.

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Equi-Attraction and the Continuous Dependence of Attractors on Parameters

Cited by 27 publications

References 16 publications

Robustness of exponentially <i>κ</i>-dissipative dynamical systems with perturbations

Robustness of exponentially <i>κ</i>-dissipative dynamical systems with perturbations

Robust Chaos and the Continuity of Attractors

Convergence of random attractors towards deterministic singleton attractor for 2D and 3D convective Brinkman-Forchheimer equations

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