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Discrete & Continuous Dynamical Systems - A

2007 , Volume 18 , Issue 2&3

Special Issue on Nonautonomous,
Stochastic and Hamiltonian Dynamical Systems

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Preface
Tomás Caraballo, Amadeu Delshams, Àngel Jorba, Peter E. Kloeden and Rafael Obaya
2007, 18(2&3): i-i doi: 10.3934/dcds.2007.18.2i +[Abstract](114) +[PDF](29.8KB)
Abstract:
The theory of dynamical systems has undergone some spectacular and fascinating developments in the past century, as the readers of this journal are well aware, with the focus predominately on autonomous systems. There are many ways in which one could classify the work that has been done, but one that stands clearly in the forefront is the distinction between dissipative systems with their attractors and conservative systems, in particular Hamiltonian systems.
    Another classification is between autonomous and nonautonomous systems. Of course, the latter subsumes the former as special case, but with the former having special structural features, i.e., the semigroup evolution property, which has allowed an extensive and seemingly complete theory to be developed. Although not as extensive, there have also been significant developments in the past half century on nonautonomous dynamical systems, in particular the skew-product formalism involving a cocycle evolution property which generalizes the semigroup property of autonomous systems. This has been enriched in recent years by advances on random dynamical systems, which are roughly said a measure theoretic version of a skew-product flow. In particular, new concepts of random and nonautonomous attractors have been introduced and investigated.

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Dynamics of a predator-prey model with non-monotonic response function
H. W. Broer, K. Saleh, V. Naudot and R. Roussarie
2007, 18(2&3): 221-251 doi: 10.3934/dcds.2007.18.221 +[Abstract](218) +[PDF](10926.0KB)
Abstract:
A five-parameter family of planar vector fields, which models the dynamics of certain populations of predators and their prey, is discussed. The family is a variation of the classical Volterra-Lotka system by taking into account group defense strategy, competition between prey and competition between predators. Also we initiate computer-assisted research on time-periodic perturbations, which model seasonal dependence. We are interested in persistent features. For the planar autonomous model this amounts to structurally stable phase portraits. We focus on the attractors, where it turns out that multi-stability occurs. Further, the bifurcations between the various domains of structural stability are investigated. It is possible to fix the values of two of the parameters and study the bifurcations in terms of the remaining three. Here we find several codimension 3 bifurcations that form organizing centres for the global bifurcation set. Studying the time-periodic system, our main interest is the chaotic dynamics. We plot several numerical examples of strange attractors.
Existence and asymptotic behaviour for stochastic heat equations with multiplicative noise in materials with memory
Tomás Caraballo, I. D. Chueshov, Pedro Marín-Rubio and José Real
2007, 18(2&3): 253-270 doi: 10.3934/dcds.2007.18.253 +[Abstract](204) +[PDF](245.4KB)
Abstract:
The existence and uniqueness of solutions for a stochastic reaction-diffusion equation with infinite delay is proved. Sufficient conditions ensuring stability of the zero solution are provided and a possibility of stabilization by noise of the deterministic counterpart of the model is studied.
Existence of exponentially attracting stationary solutions for delay evolution equations
Tomás Caraballo, M. J. Garrido-Atienza and B. Schmalfuss
2007, 18(2&3): 271-293 doi: 10.3934/dcds.2007.18.271 +[Abstract](157) +[PDF](266.3KB)
Abstract:
We consider the exponential stability of semilinear stochastic evolution equations with delays when zero is not a solution for these equations. We prove the existence of a non-trivial stationary solution exponentially stable, for which we use a general random fixed point theorem for general cocycles. We also construct stationary solutions with the stronger property of attracting bounded sets uniformly, by means of the theory of random dynamical systems and their conjugation properties.
The exponential stability of neutral stochastic delay partial differential equations
Tomás Caraballo, José Real and T. Taniguchi
2007, 18(2&3): 295-313 doi: 10.3934/dcds.2007.18.295 +[Abstract](310) +[PDF](221.9KB)
Abstract:
In this paper we analyze the almost sure exponential stability and ultimate boundedness of the solutions to a class of neutral stochastic semilinear partial delay differential equations. These kind of equations arise in problems related to coupled oscillators in a noisy environment, or in viscoeslastic materials under random or stochastic influences.
Qualitative behavior of a class of stochastic parabolic PDEs with dynamical boundary conditions
Igor Chueshov and Björn Schmalfuss
2007, 18(2&3): 315-338 doi: 10.3934/dcds.2007.18.315 +[Abstract](249) +[PDF](315.5KB)
Abstract:
We consider non-linear parabolic stochastic partial differential equations with dynamical boundary conditions and with a noise which acts in the domain but also on the boundary and is presented by the temporal generalized derivative of an infinite dimensional Wiener process. We prove that solutions to this stochastic partial differential equation generate a random dynamical system. Under additional conditions we show that this system is monotone. Our main result states the existence of a compact global (pullback) attractor.
Nonlinear Iwasawa decomposition of control flows
Fritz Colonius and Paulo Régis C. Ruffino
2007, 18(2&3): 339-354 doi: 10.3934/dcds.2007.18.339 +[Abstract](153) +[PDF](218.7KB)
Abstract:
Let $\varphi(t,\cdot,u)$ be the flow of a control system on a Riemannian manifold $M$ of constant curvature. For a given initial orthonormal frame $k$ in the tangent space $T_{x_{0}}M$ for some $x_{0}\in M$, there exists a unique decomposition $\varphi_{t}=\Theta_{t}\circ\rho_{t}$ where $\Theta_{t}$ is a control flow in the group of isometries of $M$ and the remainder component $\rho_{t}$ fixes $x_{0}$ with derivative $D\rho_{t}(k)=k\cdot s_{t}$ where $s_{t}$ are upper triangular matrices. Moreover, if $M$ is flat, an affine component can be extracted from the remainder.
Necessary condition for the basin of attraction of a periodic orbit in non-smooth periodic systems
Peter Giesl
2007, 18(2&3): 355-373 doi: 10.3934/dcds.2007.18.355 +[Abstract](191) +[PDF](230.8KB)
Abstract:
We study a time-periodic non-smooth differential equation $\dot{x}=f(t,x)$, $x\in \mathbb R$. In [4] we have presented a sufficient condition for existence, uniqueness, stability and the basin of attraction of a periodic orbit in such a system, which is a generalized Borg's condition. In this paper we prove that this condition is necessary. The proof involves a generalization of Floquet exponents for periodic orbits of non-smooth differential equations.
Lyapunov's second method for nonautonomous differential equations
Lars Grüne, Peter E. Kloeden, Stefan Siegmund and Fabian R. Wirth
2007, 18(2&3): 375-403 doi: 10.3934/dcds.2007.18.375 +[Abstract](267) +[PDF](550.0KB)
Abstract:
Converse Lyapunov theorems are presented for nonautonomous systems modelled as skew product flows. These characterize various types of stability of invariant sets and pullback, forward and uniform attractors in such nonautonomous systems.
On the inverse Sturm-Liouville problem
Russell Johnson and Luca Zampogni
2007, 18(2&3): 405-428 doi: 10.3934/dcds.2007.18.405 +[Abstract](137) +[PDF](262.1KB)
Abstract:
We pose and solve an inverse problem of an algebro-geometric type for the classical Sturm-Liouville operator. We use techniques of nonautonomous dynamical systems together with methods of classical algebraic geometry.
Numerical approximation of atmospheric-ocean models with subdivision algorithm
David Julitz
2007, 18(2&3): 429-447 doi: 10.3934/dcds.2007.18.429 +[Abstract](218) +[PDF](1279.8KB)
Abstract:
The Lorenz Maas-System is a coupled atmospheric-ocean model. It contains the Maas System which models the ocean (slow variables) and the Lorenz84 System which models the atmosphere (fast variables). Both systems are coupled to each other. Recently this System was used to investigate the long term behavior of climate models with simple numerical methods (see Arnold, Imkeller and Wu [5]). In this paper we will use the established subdivision algorithm to visualize the attractor of this system. Furthermore we will investigate (also with the subdivision algorithm) some reduced versions (statistical and stochastic models) of the Lorenz Maas-System and compare the results to the original 6-dimensional system.
A weak attractor and properties of solutions for the three-dimensional Bénard problem
O. V. Kapustyan, V. S. Melnik and José Valero
2007, 18(2&3): 449-481 doi: 10.3934/dcds.2007.18.449 +[Abstract](180) +[PDF](398.1KB)
Abstract:
In this paper we study the asymptotic behaviour of weak solutions for the three-dimensional Boussinesq equations (also known as the Bénard problem). First, we prove some regularity properties of the weak solutions of the system. Then we construct a one parameter familiy of multivalued semiflows and for them obtain the existence of a global attractor with respect to the weak topology of the phase space. Finally, we obtain a conditional result (valid only under an unproved hypothesis) stating the existence of a global attractor with respect to the strong topology.
Existence and nonexistence of unbounded forwards attractor for a class of non-autonomous reaction diffusion equations
José A. Langa, James C. Robinson, Aníbal Rodríguez-Bernal, A. Suárez and A. Vidal-López
2007, 18(2&3): 483-497 doi: 10.3934/dcds.2007.18.483 +[Abstract](159) +[PDF](206.1KB)
Abstract:
The goal of this work is to study the forward dynamics of positive solutions for the non-autonomous logistic equation $u_{t}-\Delta u=\lambda u-b(t)u^{p}$, with $p>1$, $b(t)>0$, for all $t\in \mathbb{R}$, $\lim_{t\to \infty }b(t)=0$. While the pullback asymptotic behaviour for this equation is now well understood, several different possibilities are realized in the forward asymptotic regime.
Continuous dependence of attractors on parameters of non-autonomous dynamical systems and infinite iterated function systems
David Cheban and Cristiana Mammana
2007, 18(2&3): 499-515 doi: 10.3934/dcds.2007.18.499 +[Abstract](307) +[PDF](242.8KB)
Abstract:
The paper is dedicated to the study of the problem of continuous dependence of compact global attractors on parameters of non-autonomous dynamical systems and infinite iterated function systems (IIFS). We prove that if a family of non-autonomous dynamical systems $(X,\mathbb T_1,\pi_{\lambda}),(Y,\mathbb T_{2},\sigma),h $depending on parameter $\lambda\in\Lambda$ is uniformly contracting (in the generalized sense), then each system of this family admits a compact global attractor $J^{\lambda}$ and the mapping $\lambda \to J^{\lambda}$ is continuous with respect to the Hausdorff metric. As an application we give a generalization of well known Theorem of Bransley concerning the continuous dependence of fractals on parameters.
Exponential stability in non-autonomous delayed equations with applications to neural networks
Sylvia Novo, Rafael Obaya and Ana M. Sanz
2007, 18(2&3): 517-536 doi: 10.3934/dcds.2007.18.517 +[Abstract](186) +[PDF](266.0KB)
Abstract:
We consider the skew-product semiflow induced by a family of finite-delay functional differential equations and we characterize the exponential stability of its minimal subsets. In the case of non-autonomous systems modelling delayed cellular neural networks, the existence of a global exponentially attracting solution is deduced from the uniform asymptotical stability of the null solution of an associated non-autonomous linear system.
Existence, uniqueness and attractivity properties of positive complete trajectories for non-autonomous reaction-diffusion problems
Aníbal Rodríguez-Bernal and Alejandro Vidal–López
2007, 18(2&3): 537-567 doi: 10.3934/dcds.2007.18.537 +[Abstract](163) +[PDF](345.2KB)
Abstract:
We give conditions for the existence of a unique positive complete trajectories for non-autonomous reaction-diffusion equations. Also, attraction properties of the unique complete trajectory is obtained in a pullback sense and also forward in time. As an example, a non-autonomous logistic equation is considered.
Invariant tori in quasi-periodic non-autonomous dynamical systems via Herman's method
Mikhail B. Sevryuk
2007, 18(2&3): 569-595 doi: 10.3934/dcds.2007.18.569 +[Abstract](245) +[PDF](313.8KB)
Abstract:
We consider quasi-periodic (with $N$ basic frequencies) non-autonomous perturbations of Hamiltonian, reversible, volume preserving, and dissipative systems. The unperturbed systems possess analytic families of invariant $n$-tori carrying conditionally periodic motions, are allowed to depend on external parameters, and are assumed to satisfy just very weak nondegeneracy conditions. We construct invariant $(n+N)$-tori in perturbed systems following M.R. Herman's approach: additional external parameters are introduced to remove degeneracies and then are eliminated via an appropriate number-theoretical lemma concerning Diophantine approximations of dependent quantities.
Global attractor and rotation number of a class of nonlinear noisy oscillators
Wenxian Shen
2007, 18(2&3): 597-611 doi: 10.3934/dcds.2007.18.597 +[Abstract](133) +[PDF](207.7KB)
Abstract:
The current paper is concerned with the global dynamics of a class of nonlinear oscillators driven by real or white noises, of which a typical example is a shunted Josephson junction exposed to some random medium. Applying random dynamical systems theory, it is shown that a driven oscillator in the class under consideration with a tempered real noise has a one-dimensional global random attractor provided that the damping is not too small. Moreover, restricted to the global attractor, the oscillator induces a random dynamical system on $S^1$. It is then shown that there is a rotation number associated to the oscillator which characterizes the speed at which the solutions of the oscillator move around the global attractor. The results extend the existing ones for time periodic and quasi-periodic Josephson junctions and can be applied to Josephson junctions driven by white noises.

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