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1531-3492
eISSN:
1553-524X
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Discrete & Continuous Dynamical Systems - B
May 2015 , Volume 20 , Issue 3
Special issue on the asymptotic dynamics of non-autonomous systems
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2015, 20(3): i-ii
doi: 10.3934/dcdsb.2015.20.3i
+[Abstract](1861)
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Abstract:
We were very pleased to be given the opportunity by Prof. Peter Kloeden to edit this special issue of Discrete and Continuous Dynamical Systems - Series B on the asymptotic dynamics of non-autonomous systems.
For more information please click the “Full Text” above.
We were very pleased to be given the opportunity by Prof. Peter Kloeden to edit this special issue of Discrete and Continuous Dynamical Systems - Series B on the asymptotic dynamics of non-autonomous systems.
For more information please click the “Full Text” above.
2015, 20(3): 703-747
doi: 10.3934/dcdsb.2015.20.703
+[Abstract](4964)
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Abstract:
This review paper treats the dynamics of non-autonomous dynamical systems. To study the forwards asymptotic behaviour of a non-autonomous differential equation we need to analyse the asymptotic configurations of the non-autonomous terms present in the equations. This fact leads to the definition of concepts such as skew-products and cocycles and their associated global, uniform, and cocycle attractors. All of them are closely related to the study of the pullback asymptotic limits of the dynamical system, from which naturally emerges the concept of a pullback attractor. In the first part of this paper we want to clarify these different dynamical scenarios and the relations between their corresponding attractors.
If the global attractor of an autonomous dynamical system is given as the union of a finite number of unstable manifolds of equilibria, a detailed understanding of the continuity of the local dynamics under perturbation leads to important results on the lower-semicontinuity and topological structural stability for the pullback attractors of evolution processes that arise from small non-autonomous perturbations, with respect to the limit regime. Finally, continuity with respect to global dynamics under non-autonomous perturbation is also studied, for which appropriate concepts for Morse decomposition of attractors and non-autonomous Morse--Smale systems are introduced. All of these results will also be considered for uniform attractors. As a consequence, this paper also makes connections between different approaches to the qualitative theory of non-autonomous differential equations, which are often treated independently.
This review paper treats the dynamics of non-autonomous dynamical systems. To study the forwards asymptotic behaviour of a non-autonomous differential equation we need to analyse the asymptotic configurations of the non-autonomous terms present in the equations. This fact leads to the definition of concepts such as skew-products and cocycles and their associated global, uniform, and cocycle attractors. All of them are closely related to the study of the pullback asymptotic limits of the dynamical system, from which naturally emerges the concept of a pullback attractor. In the first part of this paper we want to clarify these different dynamical scenarios and the relations between their corresponding attractors.
If the global attractor of an autonomous dynamical system is given as the union of a finite number of unstable manifolds of equilibria, a detailed understanding of the continuity of the local dynamics under perturbation leads to important results on the lower-semicontinuity and topological structural stability for the pullback attractors of evolution processes that arise from small non-autonomous perturbations, with respect to the limit regime. Finally, continuity with respect to global dynamics under non-autonomous perturbation is also studied, for which appropriate concepts for Morse decomposition of attractors and non-autonomous Morse--Smale systems are introduced. All of these results will also be considered for uniform attractors. As a consequence, this paper also makes connections between different approaches to the qualitative theory of non-autonomous differential equations, which are often treated independently.
2015, 20(3): 749-779
doi: 10.3934/dcdsb.2015.20.749
+[Abstract](2405)
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Abstract:
We give an abstract framework for studying nonautonomous PDEs, called a generalized evolutionary system. In this setting, we define the notion of a pullback attractor. Moreover, we show that the pullback attractor, in the weak sense, must always exist. We then study the structure of these attractors and the existence of a strong pullback attractor. We then apply our framework to both autonomous and nonautonomous evolutionary systems as they first appeared in earlier works by Cheskidov, Foias, and Lu. In this con- text, we compare the pullback attractor to both the global attractor (in the autonomous case) and the uniform attractor (in the nonautonomous case). Finally, we apply our results to the nonautonomous 3D Navier-Stokes equations on a periodic domain with a translationally bounded force. We show that the Leray-Hopf weak solutions form a generalized evolutionary system and must then have a weak pullback attractor.
We give an abstract framework for studying nonautonomous PDEs, called a generalized evolutionary system. In this setting, we define the notion of a pullback attractor. Moreover, we show that the pullback attractor, in the weak sense, must always exist. We then study the structure of these attractors and the existence of a strong pullback attractor. We then apply our framework to both autonomous and nonautonomous evolutionary systems as they first appeared in earlier works by Cheskidov, Foias, and Lu. In this con- text, we compare the pullback attractor to both the global attractor (in the autonomous case) and the uniform attractor (in the nonautonomous case). Finally, we apply our results to the nonautonomous 3D Navier-Stokes equations on a periodic domain with a translationally bounded force. We show that the Leray-Hopf weak solutions form a generalized evolutionary system and must then have a weak pullback attractor.
2015, 20(3): 781-810
doi: 10.3934/dcdsb.2015.20.781
+[Abstract](2606)
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We give a comprehensive study of strong uniform attractors of nonautonomous dissipative systems for the case where the external forces are not translation compact. We introduce several new classes of external forces that are not translation compact, but nevertheless allow the attraction in a strong topology of the phase space to be verified and discuss in a more detailed way the class of so-called normal external forces introduced before. We also develop a unified approach to verify the asymptotic compactness for such systems based on the energy method and apply it to a number of equations of mathematical physics including the Navier-Stokes equations, damped wave equations and reaction-diffusing equations in unbounded domains.
We give a comprehensive study of strong uniform attractors of nonautonomous dissipative systems for the case where the external forces are not translation compact. We introduce several new classes of external forces that are not translation compact, but nevertheless allow the attraction in a strong topology of the phase space to be verified and discuss in a more detailed way the class of so-called normal external forces introduced before. We also develop a unified approach to verify the asymptotic compactness for such systems based on the energy method and apply it to a number of equations of mathematical physics including the Navier-Stokes equations, damped wave equations and reaction-diffusing equations in unbounded domains.
2015, 20(3): 811-832
doi: 10.3934/dcdsb.2015.20.811
+[Abstract](2526)
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We construct the trajectory attractor $\mathfrak{A}_{\Sigma }$ for the non-autonomous dissipative 2d Euler systems with periodic boundary conditions that contain time dependent dissipation terms $-r(t)u$ such that $0<\alpha \le r(t)\le \beta$, for $t\ge 0$. External forces $g(x,t),x\in \mathbb{T}^{2},t\ge 0,$ also depend on time. The corresponding non-autonomous dissipative 2d Navier--Stokes systems with the same terms $-r(t)u$ and $g(x,t)$ and with viscosity $\nu >0$ also have the trajectory attractor $\mathfrak{A}_{\Sigma }^{\nu }.$ Such systems model large-scale geophysical processes in atmosphere and ocean. We prove that $\mathfrak{A}_{\Sigma }^{\nu }\rightarrow \mathfrak{A}_{\Sigma }$ as viscosity $\nu \rightarrow 0+$ in the corresponding metric space. Moreover, we establish the existence of the minimal limit $\mathfrak{A}_{\Sigma }^{\min }\subseteq \mathfrak{A}_{\Sigma }$ of the trajectory attractors $\mathfrak{A}_{\Sigma }^{\nu }$ as $\nu \rightarrow 0+.$ Every set $\mathfrak{A}_{\Sigma }^{\nu }$ is connected. We prove that $\mathfrak{A}_{\Sigma }^{\min }$ is a connected invariant subset of $\mathfrak{A}_{\Sigma }.$ The problem of the connectedness of the trajectory attractor $\mathfrak{A}_{\Sigma }$ itself remains open.
We construct the trajectory attractor $\mathfrak{A}_{\Sigma }$ for the non-autonomous dissipative 2d Euler systems with periodic boundary conditions that contain time dependent dissipation terms $-r(t)u$ such that $0<\alpha \le r(t)\le \beta$, for $t\ge 0$. External forces $g(x,t),x\in \mathbb{T}^{2},t\ge 0,$ also depend on time. The corresponding non-autonomous dissipative 2d Navier--Stokes systems with the same terms $-r(t)u$ and $g(x,t)$ and with viscosity $\nu >0$ also have the trajectory attractor $\mathfrak{A}_{\Sigma }^{\nu }.$ Such systems model large-scale geophysical processes in atmosphere and ocean. We prove that $\mathfrak{A}_{\Sigma }^{\nu }\rightarrow \mathfrak{A}_{\Sigma }$ as viscosity $\nu \rightarrow 0+$ in the corresponding metric space. Moreover, we establish the existence of the minimal limit $\mathfrak{A}_{\Sigma }^{\min }\subseteq \mathfrak{A}_{\Sigma }$ of the trajectory attractors $\mathfrak{A}_{\Sigma }^{\nu }$ as $\nu \rightarrow 0+.$ Every set $\mathfrak{A}_{\Sigma }^{\nu }$ is connected. We prove that $\mathfrak{A}_{\Sigma }^{\min }$ is a connected invariant subset of $\mathfrak{A}_{\Sigma }.$ The problem of the connectedness of the trajectory attractor $\mathfrak{A}_{\Sigma }$ itself remains open.
2015, 20(3): 833-852
doi: 10.3934/dcdsb.2015.20.833
+[Abstract](2233)
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Abstract:
We consider a stochastically perturbed coupled system consisting of linearized 3D Navier--Stokes equations in a bounded domain and the classical (nonlinear) elastic plate equation for in-plane motions on a flexible flat part of the boundary. This kind of models arises in the study of blood flows in large arteries. Our main result states the existence of a random pullback attractor of finite fractal dimension. Our argument is based on some modification of the method of quasi-stability estimates recently developed for deterministic systems.
We consider a stochastically perturbed coupled system consisting of linearized 3D Navier--Stokes equations in a bounded domain and the classical (nonlinear) elastic plate equation for in-plane motions on a flexible flat part of the boundary. This kind of models arises in the study of blood flows in large arteries. Our main result states the existence of a random pullback attractor of finite fractal dimension. Our argument is based on some modification of the method of quasi-stability estimates recently developed for deterministic systems.
2015, 20(3): 853-859
doi: 10.3934/dcdsb.2015.20.853
+[Abstract](2176)
+[PDF](341.8KB)
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As a direct consequence of well-established proof techniques, we establish that the invariant projectors of exponential dichotomies for parameter-dependent nonautonomous difference equations are as smooth as their right-hand sides. For instance, this guarantees that the saddle-point structure in the vicinity of hyperbolic solutions inherits its differentiability properties from the particular given equation.
As a direct consequence of well-established proof techniques, we establish that the invariant projectors of exponential dichotomies for parameter-dependent nonautonomous difference equations are as smooth as their right-hand sides. For instance, this guarantees that the saddle-point structure in the vicinity of hyperbolic solutions inherits its differentiability properties from the particular given equation.
2015, 20(3): 861-874
doi: 10.3934/dcdsb.2015.20.861
+[Abstract](3344)
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The multiplicative ergodic theorem by Oseledets on Lyapunov spectrum and Oseledets subspaces is extended to linear random difference equations with random delay. In contrast to the general multiplicative ergodic theorem by Lian and Lu, we can prove that a random dynamical system generated by a difference equation with random delay cannot have infinitely many Lyapunov exponents.
The multiplicative ergodic theorem by Oseledets on Lyapunov spectrum and Oseledets subspaces is extended to linear random difference equations with random delay. In contrast to the general multiplicative ergodic theorem by Lian and Lu, we can prove that a random dynamical system generated by a difference equation with random delay cannot have infinitely many Lyapunov exponents.
2015, 20(3): 875-887
doi: 10.3934/dcdsb.2015.20.875
+[Abstract](2765)
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The dichotomy spectrum is introduced for linear mean-square random dynamical systems, and it is shown that for finite-dimensional mean-field stochastic differential equations, the dichotomy spectrum consists of finitely many compact intervals. It is then demonstrated that a change in the sign of the dichotomy spectrum is associated with a bifurcation from a trivial to a non-trivial mean-square random attractor.
The dichotomy spectrum is introduced for linear mean-square random dynamical systems, and it is shown that for finite-dimensional mean-field stochastic differential equations, the dichotomy spectrum consists of finitely many compact intervals. It is then demonstrated that a change in the sign of the dichotomy spectrum is associated with a bifurcation from a trivial to a non-trivial mean-square random attractor.
2015, 20(3): 889-914
doi: 10.3934/dcdsb.2015.20.889
+[Abstract](2778)
+[PDF](503.6KB)
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We study the concept of dissipativity in the sense of Willems for nonautonomous linear-quadratic (LQ) control systems. A nonautonomous system of Hamiltonian ODEs is associated with such an LQ system by way of the Pontryagin Maximum Principle. We relate the concepts of exponential dichotomy and weak disconjugacy for this Hamiltonian ODE to that of dissipativity for the LQ system.
We study the concept of dissipativity in the sense of Willems for nonautonomous linear-quadratic (LQ) control systems. A nonautonomous system of Hamiltonian ODEs is associated with such an LQ system by way of the Pontryagin Maximum Principle. We relate the concepts of exponential dichotomy and weak disconjugacy for this Hamiltonian ODE to that of dissipativity for the LQ system.
2015, 20(3): 915-944
doi: 10.3934/dcdsb.2015.20.915
+[Abstract](2310)
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This paper investigates relevant dynamical properties of nonautonomous linear cooperative families of ODEs and FDEs based on the existence of a continuous separation. It provides numerical algorithms for the computation of the dominant one-dimensional subbundle of the continuous separation and the upper Lyapunov exponent of the semiflow. The extension of the theory to general linear cooperative families of ODEs and FDEs without strong monotonicity is also given. Finally these methods and results are applied in the study of nonlinear families of neural networks of Hopfield type with sigmoidal activation function.
This paper investigates relevant dynamical properties of nonautonomous linear cooperative families of ODEs and FDEs based on the existence of a continuous separation. It provides numerical algorithms for the computation of the dominant one-dimensional subbundle of the continuous separation and the upper Lyapunov exponent of the semiflow. The extension of the theory to general linear cooperative families of ODEs and FDEs without strong monotonicity is also given. Finally these methods and results are applied in the study of nonlinear families of neural networks of Hopfield type with sigmoidal activation function.
2015, 20(3): 945-959
doi: 10.3934/dcdsb.2015.20.945
+[Abstract](1884)
+[PDF](392.7KB)
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The aim of this paper is study the problem of global asymptotic stability of solutions for $\mathbb C$-analytical dynamical systems (both with continuous and discrete time). In particular we present some new results for the $C$-analytical version of Belitskii--Lyubich conjecture. Some applications of these results for periodic $\mathbb C$-analytical differential/difference equations and the equations with impulse are given.
The aim of this paper is study the problem of global asymptotic stability of solutions for $\mathbb C$-analytical dynamical systems (both with continuous and discrete time). In particular we present some new results for the $C$-analytical version of Belitskii--Lyubich conjecture. Some applications of these results for periodic $\mathbb C$-analytical differential/difference equations and the equations with impulse are given.
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