ISSN:
1531-3492
eISSN:
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Discrete & Continuous Dynamical Systems - B
May 2008 , Volume 9 , Issue 3&4
Special Issue on Nonautonomous Dynamics
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2008, 9(3&4, May): i-ii
doi: 10.3934/dcdsb.2008.9.3i
+[Abstract](1949)
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Abstract:
In recent years, the area of nonautonomous dynamical systems has matured into a field with recognizable contours together with well-defined themes and methods. Its development has been strongly stimulated by various problems of applied mathematics, and it has in its turn influenced such areas of applied and pure mathematics as spectral theory, stability theory, bifurcation theory, the theory of bounded/recurrent motions, etc. Much work in this field concerns the asymptotic properties of the solutions of a nonautonomous differential or discrete system. However, that is by no means always the case, and the reader will find papers in this volume which are concerned only at a distance or not at all with asymptotic matters.
There is a close relation between the field of nonautonomous dynamical systems and that of stochastic dynamical systems. They can be distinguished to a certain extent by the observation that a nonautonomous dynamical system often arises from the study of a differential or discrete system whose coefficients depend on time, but in a non-stochastic way. The limiting case is that of periodic coefficients, but one is also interested in equations whose coefficients exhibit weaker recurrence properties; for example almost periodicity, Birkhoff recurrence, Poisson recurrence, etc. A distinction also occurs on the methodological level in that topological methods tend to find more application in the former field as compared to the latter (while analytical and ergodic tools are heavily used in both). In any case, some people use the term “random dynamics” to refer to both fields in a more or less interchangeable way.
For the full preface, please click on the Full Text "PDF" button above.
In recent years, the area of nonautonomous dynamical systems has matured into a field with recognizable contours together with well-defined themes and methods. Its development has been strongly stimulated by various problems of applied mathematics, and it has in its turn influenced such areas of applied and pure mathematics as spectral theory, stability theory, bifurcation theory, the theory of bounded/recurrent motions, etc. Much work in this field concerns the asymptotic properties of the solutions of a nonautonomous differential or discrete system. However, that is by no means always the case, and the reader will find papers in this volume which are concerned only at a distance or not at all with asymptotic matters.
There is a close relation between the field of nonautonomous dynamical systems and that of stochastic dynamical systems. They can be distinguished to a certain extent by the observation that a nonautonomous dynamical system often arises from the study of a differential or discrete system whose coefficients depend on time, but in a non-stochastic way. The limiting case is that of periodic coefficients, but one is also interested in equations whose coefficients exhibit weaker recurrence properties; for example almost periodicity, Birkhoff recurrence, Poisson recurrence, etc. A distinction also occurs on the methodological level in that topological methods tend to find more application in the former field as compared to the latter (while analytical and ergodic tools are heavily used in both). In any case, some people use the term “random dynamics” to refer to both fields in a more or less interchangeable way.
For the full preface, please click on the Full Text "PDF" button above.
2008, 9(3&4, May): 431-461
doi: 10.3934/dcdsb.2008.9.431
+[Abstract](2345)
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Abstract:
We consider a singularly perturbed system with two normally hyperbolic centre manifolds. We derive one bifurcation function, the zeros of which correspond to heteroclinic connections near such a connection for the unperturbed system, and a second bifurcation function the zeros of which correspond to the vectors in the intersection of the tangent spaces to the centre-unstable and centre-stable manifolds along the heteroclinic connections.
We consider a singularly perturbed system with two normally hyperbolic centre manifolds. We derive one bifurcation function, the zeros of which correspond to heteroclinic connections near such a connection for the unperturbed system, and a second bifurcation function the zeros of which correspond to the vectors in the intersection of the tangent spaces to the centre-unstable and centre-stable manifolds along the heteroclinic connections.
2008, 9(3&4, May): 463-492
doi: 10.3934/dcdsb.2008.9.463
+[Abstract](3151)
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Abstract:
Nonautonomous differential equations on finite-time intervals play an increasingly important role in applications that incorporate time-varying vector fields, e.g. observed or forecasted velocity fields in meteorology or oceanography which are known only for times $t$ from a compact interval. While classical dynamical systems methods often study the behaviour of solutions as $t \to \pm\infty$, the dynamic partition (originally called the EPH partition) aims at describing and classifying the finite-time behaviour. We discuss fundamental properties of the dynamic partition and show that it locally approximates the nonlinear behaviour. We also provide an algorithm for practical computations with dynamic partitions and apply it to a nonlinear 3-dimensional example.
Nonautonomous differential equations on finite-time intervals play an increasingly important role in applications that incorporate time-varying vector fields, e.g. observed or forecasted velocity fields in meteorology or oceanography which are known only for times $t$ from a compact interval. While classical dynamical systems methods often study the behaviour of solutions as $t \to \pm\infty$, the dynamic partition (originally called the EPH partition) aims at describing and classifying the finite-time behaviour. We discuss fundamental properties of the dynamic partition and show that it locally approximates the nonlinear behaviour. We also provide an algorithm for practical computations with dynamic partitions and apply it to a nonlinear 3-dimensional example.
2008, 9(3&4, May): 493-516
doi: 10.3934/dcdsb.2008.9.493
+[Abstract](2505)
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Abstract:
We study the minimal subsets of the projective flow defined by a two-dimensional linear differential system with almost periodic coefficients. We show that such a minimal set may exhibit Li-Yorke chaos and discuss specific examples in which this phenomenon is present. We then give a classification of these minimal sets, and use it to discuss the bounded mean motion property relative to the projective flow.
We study the minimal subsets of the projective flow defined by a two-dimensional linear differential system with almost periodic coefficients. We show that such a minimal set may exhibit Li-Yorke chaos and discuss specific examples in which this phenomenon is present. We then give a classification of these minimal sets, and use it to discuss the bounded mean motion property relative to the projective flow.
2008, 9(3&4, May): 517-523
doi: 10.3934/dcdsb.2008.9.517
+[Abstract](2305)
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Abstract:
An orientation-preserving homeomorphism of the plane having a two-cycle has also a fixed point. This result goes back to Brouwer. Gagliardo and Kottman and later M. Brown have developed topological strategies to locate the fixed point from the position of the cycle. We employ these ideas to study certain classes of homeomorphisms which are useful in the theory of periodic differential equations.
An orientation-preserving homeomorphism of the plane having a two-cycle has also a fixed point. This result goes back to Brouwer. Gagliardo and Kottman and later M. Brown have developed topological strategies to locate the fixed point from the position of the cycle. We employ these ideas to study certain classes of homeomorphisms which are useful in the theory of periodic differential equations.
2008, 9(3&4, May): 525-539
doi: 10.3934/dcdsb.2008.9.525
+[Abstract](2699)
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Abstract:
We study the asymptotic behaviour of a non-autonomous stochastic reaction-diffusion equation with memory. In fact, we prove the existence of a random pullback attractor for our stochastic parabolic PDE with memory. The randomness enters in our model as an additive Hilbert valued noise. We first prove that the equation generates a random dynamical system (RDS) in an appropriate phase space. Due to the fact that the memory term takes into account the whole past history of the phenomenon, we are not able to prove compactness of the generated RDS, but its asymptotic compactness, ensuring thus the existence of the random pullback attractor.
We study the asymptotic behaviour of a non-autonomous stochastic reaction-diffusion equation with memory. In fact, we prove the existence of a random pullback attractor for our stochastic parabolic PDE with memory. The randomness enters in our model as an additive Hilbert valued noise. We first prove that the equation generates a random dynamical system (RDS) in an appropriate phase space. Due to the fact that the memory term takes into account the whole past history of the phenomenon, we are not able to prove compactness of the generated RDS, but its asymptotic compactness, ensuring thus the existence of the random pullback attractor.
2008, 9(3&4, May): 541-554
doi: 10.3934/dcdsb.2008.9.541
+[Abstract](3163)
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Abstract:
The spectral theory of the one-dimensional Schrödinger operator with a quasi-periodic potential can be fruitfully studied considering the corresponding differential system. In fact the presence of an exponential dichotomy for the system is equivalent to the statement that the energy $E$ belongs to the resolvent of the operator. Starting from results already obtained for the spectrum in the continuous case, we show that in the discrete case a generic bounded measurable Schrödinger cocycle has Cantor spectrum.
The spectral theory of the one-dimensional Schrödinger operator with a quasi-periodic potential can be fruitfully studied considering the corresponding differential system. In fact the presence of an exponential dichotomy for the system is equivalent to the statement that the energy $E$ belongs to the resolvent of the operator. Starting from results already obtained for the spectrum in the continuous case, we show that in the discrete case a generic bounded measurable Schrödinger cocycle has Cantor spectrum.
2008, 9(3&4, May): 555-580
doi: 10.3934/dcdsb.2008.9.555
+[Abstract](2327)
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Abstract:
We provide an analysis of the error in approximating Lyapunov exponents of dissipative PDEs on inertial manifolds using QR techniques. The reduction in the number of modes needed for an inertial form facilitates the error analysis. Numerical computations on the Kuramoto-Sivashinsky equation illustrate the results.
We provide an analysis of the error in approximating Lyapunov exponents of dissipative PDEs on inertial manifolds using QR techniques. The reduction in the number of modes needed for an inertial form facilitates the error analysis. Numerical computations on the Kuramoto-Sivashinsky equation illustrate the results.
2008, 9(3&4, May): 581-593
doi: 10.3934/dcdsb.2008.9.581
+[Abstract](2503)
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Abstract:
Under appropriate regularity conditions it is shown that the continuous dependence of the global attractors $\mathcal{A}_\tau$ of semi dynamical systems $S^{(\tau)}(t)$ in $C([-\tau,0];Z)$ with $Z$ a Banach space and time delay $\tau \in [T_*,T^$*$]$, where $T_* > 0$, is equivalent to the equi-attraction of the attractors. Examples and counter examples posed in this right framework are provided.
Under appropriate regularity conditions it is shown that the continuous dependence of the global attractors $\mathcal{A}_\tau$ of semi dynamical systems $S^{(\tau)}(t)$ in $C([-\tau,0];Z)$ with $Z$ a Banach space and time delay $\tau \in [T_*,T^$*$]$, where $T_* > 0$, is equivalent to the equi-attraction of the attractors. Examples and counter examples posed in this right framework are provided.
2008, 9(3&4, May): 595-633
doi: 10.3934/dcdsb.2008.9.595
+[Abstract](2417)
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Abstract:
We study quasilinear systems of parabolic partial differential equations with fully nonlinear boundary conditions on bounded or exterior domains. Our main results concern the asymptotic behavior of the solutions in the vicinity of an equilibrium. The local center, center–stable, and center–unstable manifolds are constructed and their dynamical properties are established using nonautonomous cutoff functions. Under natural conditions, we show that each solution starting close to the center manifold converges to a solution on the center manifold.
We study quasilinear systems of parabolic partial differential equations with fully nonlinear boundary conditions on bounded or exterior domains. Our main results concern the asymptotic behavior of the solutions in the vicinity of an equilibrium. The local center, center–stable, and center–unstable manifolds are constructed and their dynamical properties are established using nonautonomous cutoff functions. Under natural conditions, we show that each solution starting close to the center manifold converges to a solution on the center manifold.
2008, 9(3&4, May): 635-642
doi: 10.3934/dcdsb.2008.9.635
+[Abstract](2818)
+[PDF](137.0KB)
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In this paper, we generalize the classical Siegel’s theorem for deterministic dynamical systems to that under random perturbations.
In this paper, we generalize the classical Siegel’s theorem for deterministic dynamical systems to that under random perturbations.
2008, 9(3&4, May): 643-659
doi: 10.3934/dcdsb.2008.9.643
+[Abstract](2554)
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Abstract:
In this paper we investigate some relations among the notions of pullback attractor, time-average measure and statistical solution.
Using time-averages and Banach generalized limits we construct a family of probability measures $\{\mu_t\}_{t\in \IR}$ on the pullback attractor $\{A(t)\}_{t\in \R}$ of the dynamical system associated with a two-dimensional nonautonomous Navier-Stokes flow in a bounded domain. The measures satisfy supp$\mu_t \subset A(t)$ for all $t\in \R$ and also the corresponding Liouville equation and energy equation. In the autonomous case, they reduce to some time-average measure $\mu$ with support included in the global attractor and being a stationary statistical solution of the Navier-Stokes flow.
In this paper we investigate some relations among the notions of pullback attractor, time-average measure and statistical solution.
Using time-averages and Banach generalized limits we construct a family of probability measures $\{\mu_t\}_{t\in \IR}$ on the pullback attractor $\{A(t)\}_{t\in \R}$ of the dynamical system associated with a two-dimensional nonautonomous Navier-Stokes flow in a bounded domain. The measures satisfy supp$\mu_t \subset A(t)$ for all $t\in \R$ and also the corresponding Liouville equation and energy equation. In the autonomous case, they reduce to some time-average measure $\mu$ with support included in the global attractor and being a stationary statistical solution of the Navier-Stokes flow.
2008, 9(3&4, May): 661-699
doi: 10.3934/dcdsb.2008.9.661
+[Abstract](2522)
+[PDF](415.3KB)
Abstract:
Linear nonautonomous/random parabolic partial differential equations are considered under the Dirichlet, Neumann or Robin boundary conditions, where both the zero order coefficients in the equation and the coefficients in the boundary conditions are allowed to depend on time. The theory of the principal spectrum/principal Lyapunov exponents is shown to apply to those equations. In the nonautonomous case, the main result states that the principal eigenvalue of any time-averaged equation is not larger than the supremum of the principal spectrum and that there is a time-averaged equation whose principal eigenvalue is not larger than the infimum of the principal spectrum. In the random case, the main result states that the principal eigenvalue of the time-averaged equation is not larger than the principal Lyapunov exponent.
Linear nonautonomous/random parabolic partial differential equations are considered under the Dirichlet, Neumann or Robin boundary conditions, where both the zero order coefficients in the equation and the coefficients in the boundary conditions are allowed to depend on time. The theory of the principal spectrum/principal Lyapunov exponents is shown to apply to those equations. In the nonautonomous case, the main result states that the principal eigenvalue of any time-averaged equation is not larger than the supremum of the principal spectrum and that there is a time-averaged equation whose principal eigenvalue is not larger than the infimum of the principal spectrum. In the random case, the main result states that the principal eigenvalue of the time-averaged equation is not larger than the principal Lyapunov exponent.
2008, 9(3&4, May): 701-730
doi: 10.3934/dcdsb.2008.9.701
+[Abstract](2647)
+[PDF](351.9KB)
Abstract:
In the extension of the concepts of saddle-node, transcritical and pitchfork bifurcations to the non-autonomous case, one considers the change in the number and attraction properties of the minimal sets for the skew-product flow determined by the initial one-parametric equation. In this work conditions on the coefficients of the equation ensuring the existence of a global bifurcation phenomenon of each one of the types mentioned are established. Special attention is paid to show the importance of the non-trivial almost automorphic extensions and pinched sets in describing the dynamics at the bifurcation point.
In the extension of the concepts of saddle-node, transcritical and pitchfork bifurcations to the non-autonomous case, one considers the change in the number and attraction properties of the minimal sets for the skew-product flow determined by the initial one-parametric equation. In this work conditions on the coefficients of the equation ensuring the existence of a global bifurcation phenomenon of each one of the types mentioned are established. Special attention is paid to show the importance of the non-trivial almost automorphic extensions and pinched sets in describing the dynamics at the bifurcation point.
2008, 9(3&4, May): 731-741
doi: 10.3934/dcdsb.2008.9.731
+[Abstract](2724)
+[PDF](181.1KB)
Abstract:
This paper gives a version of the Takens time delay embedding theorem that is valid for non-autonomous and stochastic infinite-dimensional dynamical systems that have a finite-dimensional attractor. A key ingredient is a result showing that a single linear map from the phase space into a sufficiently high dimensional Euclidean space is one-to-one between most realizations of the attractor and its image.
This paper gives a version of the Takens time delay embedding theorem that is valid for non-autonomous and stochastic infinite-dimensional dynamical systems that have a finite-dimensional attractor. A key ingredient is a result showing that a single linear map from the phase space into a sufficiently high dimensional Euclidean space is one-to-one between most realizations of the attractor and its image.
2008, 9(3&4, May): 743-761
doi: 10.3934/dcdsb.2008.9.743
+[Abstract](3207)
+[PDF](295.6KB)
Abstract:
The dynamical behavior of non-autonomous strongly damped wave-type evolutionary equations with linear memory, critical nonlinearity, and time-dependent external forcing is investigated. The time-dependent external forcing is assumed to be only translation-bounded, instead of translation-compact. First, the asymptotic regularity of solutions is proved, and then the existence of the compact uniform attractor together with its structure and regularity is obtained.
The dynamical behavior of non-autonomous strongly damped wave-type evolutionary equations with linear memory, critical nonlinearity, and time-dependent external forcing is investigated. The time-dependent external forcing is assumed to be only translation-bounded, instead of translation-compact. First, the asymptotic regularity of solutions is proved, and then the existence of the compact uniform attractor together with its structure and regularity is obtained.
Kernel sections for processes and nonautonomous lattice systems
2008, 9(3&4, May): 763-785
doi: 10.3934/dcdsb.2008.9.763
+[Abstract](2621)
+[PDF](297.6KB)
Abstract:
In this paper, we first establish a set of sufficient and necessaryconditions for the existence of globally attractive kernel sectionsfor processes defined on a general Banach space and a weighted spaceℓ$_\rho ^p$ of infinite sequences ($p\geq 1)$, respectively.Then we obtain an upper bound of the Kolmogorov$\varepsilon$-entropy of kernel sections for processes on theHilbert space ℓ$_\rho ^2 $. As applications, we investigatecompact kernel sections for first order, partly dissipative, andsecond order nonautonomous lattice systems on weighted spacescontaining bounded sequences.
In this paper, we first establish a set of sufficient and necessaryconditions for the existence of globally attractive kernel sectionsfor processes defined on a general Banach space and a weighted spaceℓ$_\rho ^p$ of infinite sequences ($p\geq 1)$, respectively.Then we obtain an upper bound of the Kolmogorov$\varepsilon$-entropy of kernel sections for processes on theHilbert space ℓ$_\rho ^2 $. As applications, we investigatecompact kernel sections for first order, partly dissipative, andsecond order nonautonomous lattice systems on weighted spacescontaining bounded sequences.
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