Forward Attractors Research Articles

Minimal random attractors

It is well-known that random attractors of a random dynamical system are generally not unique. We show that for general pullback attractors and weak attractors, there is always a minimal (in the sense of smallest) random attractor which attracts a given family of (possibly random) sets. We provide an example which shows that this property need not hold for forward attractors. We point out that our concept of a random attractor is very general: The family of sets which are attracted is allowed to be completely arbitrary.

Large amplitude free vibration of a viscoelastic beam carrying a lumped mass–spring–damper

Numerous engineering structures can be modeled as flexible beams carrying appendages. Physical time-dependent phenomena in such structures appear by considering nonlinear models. Accordingly, the free vibration of a nonlinear Rayleigh beam carrying a linear mass–spring–damper is investigated in this research. The nonlinearity is due to the axial stress changes, and the viscoelastic characteristic of the beam is described by the Kelvin–Voigt model. The governing nonlinear equations of motion are derived by using Hamilton’s principle and solved by the multiple scales method. The presence of the linear damper in the mass–spring–damper system causes a contradiction in the time response of the mass obtained from the classical procedure of the multiple scales method. Therefore, the problem is solved by some manipulations in the process of the multiple scales method. Then, the complex-valued resonance frequencies and mode-shapes are derived by applying the method of power series with the aid of Green function concept. Considering the solvability condition, the nonlinear time-dependent resonance frequencies and the vibration response are obtained. The stability of the solution is studied by using the forward and pullback attractor ideas. Finally, the vibration of the system involving 3:1 internal resonance is investigated, and the stability boundaries of the trivial and non-trivial steady-state solutions are derived based on Lyapunov’s first method. It is shown that the main parameters of the MSD can provide 3:1 internal resonance condition.

Minimal forward random point attractors need not exist

It is well-known that random attractors of a random dynamical system are generally not unique. It was shown in [ 1 ] that if there exist more than one pullback or weak random attractor which attracts a given family of (possibly random) sets, then there exists a minimal (in the sense of smallest) one. This statement does not hold for forward random attractors. The same paper contains an example of a random dynamical system and a deterministic family of sets which has more than one forward attractor which attracts the given family but no minimal one. The question whether one can find an example which has multiple forward point attractors but no minimal one remained open. Here we provide such an example.

Dynamics for the damped Boussinesq equation with delay

This paper is concerned with the dynamical behavior of the damped Boussinesq equation containing a delay forcing term. It shows that, under suitable dissipative conditions, such a system possesses a pullback attractor and a uniform forward attractor in [Formula: see text].

The structure of global attractors for non-autonomous perturbations of discrete gradient-like dynamical systems

In this paper we give the complete description of the structure of compact global (forward) attractors for non-autonomous perturbations of discrete autonomous gradient-like dynamical systems under the assumption that the original discrete autonomous system has a finite number of hyperbolic stationary solutions. We prove that the perturbed non-autonomous (in particular -periodic, quasi-periodic, Bohr almost periodic, almost automorphic, recurrent in the sense of Birkhoff) system has exactly the same number of invariant sections (in particular the perturbed systems has the same number of -periodic, quasi-periodic, Bohr almost periodic, almost automorphic, recurrent in the sense of Birkhoff solutions). It is shown that the compact global (forward) attractor of non-autonomous perturbed system coincides with the union of unstable manifolds of this finite number of invariant sections.

Forward attraction in nonautonomous difference equations

Two types of attractors consisting of families of sets that are mapped into each other under the dynamics have been defined for nonautonomous difference equations, one using pullback convergence with information about the system in the past and the other using forward convergence with information about the system in the future. In both cases, the component sets are constructed using a pullback argument within a positively invariant family of sets. The forward attractor so constructed also uses information about the past, which is very restrictive and not essential for determining future behaviour. Here an alternative is investigated, essentially the omega-limit set of the system, which Chepyzhov and Vishik called the uniform attractor. It is shown here that this set is asymptotically positively invariant, thus providing it with an hitherto missing form of invariance, if in somewhat weaker than usual, that one expects an attractor to possess. As a consequence this set provides useful information about the behaviour in current time during the approach to the limit.

Nonautonomous and Random Attractors

The theories of nonautonomous and random dynamical systems have undergone extensive, often parallel, developments in the past two decades. In particular, new concepts of nonautonomous and random attractors have been introduced. These consist of families of sets that are mapped onto each other as time evolves and have two forms: a forward attractor based on information about the system in the future and a pullback attractor that uses information about the past of the system. Both reduce to the usual attractor consisting of a single set in the autonomous case.

Construction of nonautonomous forward attractors

Construction of nonautonomous forward attractors

Random interval homeomorphisms

We investigate homeomorphisms of a compact interval, applied randomly. We consider this system as a skew product with the two-sided Bernoulli shift in the base. If on the open interval there is a metric in which almost all maps are contractions, then (with mild additional assumptions) there exists a global pullback attractor, which is a graph of a function from the base to the fiber. It is also a forward attractor. However, the value of this function depends only on the past, so when we take the one-sided shift in the base, it disappears. We illustrate those phenomena on an example, where there are two piecewise linear homeomorphisms, one moving points to the right and the other one to the left.

Morse Decompositions for Periodic General Dynamical Systems and Differential Inclusions

We first establish the Morse decomposition theory of periodic invariant sets for non-autonomous periodic general dynamical systems (set-valued dynamical systems). Then we discuss the stability of Morse decompositions of periodic uniform forward attractors. We also apply the abstract results to non-autonomous periodic differential inclusions with only upper semi-continuous right-hand side. We show that Morse decompositions are robust with respect to both internal and external perturbations (upper semi-continuity of Morse sets). Finally as an application we study the effect of small time delays to asymptotic behavior of control systems from the point of view of Morse decompositions.

Localization and Perron–Frobenius Theory for Directed Polymers

We consider directed polymers in a random potential given by a deterministic profile with a strong maximum at the origin taken with random sign at each integer time. We study two main objects based on paths in this random potential. First, we use the random potential and averaging over paths to define a parabolic model via a random Feynman--Kac evolution operator. We show that for the resulting cocycle, there is a unique positive cocycle eigenfunction serving as a forward and pullback attractor. Secondly, we use the potential to define a Gibbs specification on paths for any bounded time interval in the usual way and study the thermodynamic limit and existence and uniqueness of an infinite volume Gibbs measure. Both main results claim that the local structure of interaction leads to a unique macroscopic object for almost every realization of the random potential.

Pullback and forward attractors for dissipative lattice dynamical systems with additive noises

This work investigates the dissipative dynamical system in the infinite lattice ℤ. The dynamics of each node depends on itself and nearby nodes by a nonlinear function. When each node is perturbed with weighted Gaussian white noise, a unique pullback attractor and forward attractor exists whose domain of attraction are random tempered sets. Furthermore, we prove that the pullback and forward attractors are equivalent to a random equilibrium which is also tempered. Both convergence to the pullback and forward attractors are exponentially fast.

Lyapunov's second method for nonautonomous differential equations

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.

Nonautonomous attractors of switching systems

Switching systems formed by switching between a finite number of autonomous ordinary differential equations are formulated as abstract nonautonomous dynamical systems driven by the shift operator on the space of switching controls. A metric is proposed for such switching controls and the space of switching controls with a given positive dwell time is shown to be compact. This allows known results on pullback attractors for nonautonomous dynamical systems to be applied to switching systems including conditions which ensure that a pullback attractor exists and is also a forward attractor. In particular, this provides a conceptual framework for investigating the asymptotic behaviour of switching systems beyond the usual case in which the asymptotic stability of the zero solution under switching is considered.

Pullback and forward attractors for a 3D LANS$-\alpha$ model with delay

We analyze the asymptotic behaviour of a 3D Lagrangian averaged Navier-Stokes $\alpha$-model (3D LANS$-\alpha$) with delays. In fact, we apply the theory of pullback attractors to ensure the existence of a pullback attractor, and at the same time, we also prove the existence of a uniform (forward) attractor in the sense of Chepyzhov and Vishik. Instead of working directly with the 3D LANS$-\alpha$ model, we establish a general theory for an abstract delay model and then we apply the general results to our particular situation.

On the dynamics of nonautonomous periodic general dynamical systems and differential inclusions

We establish the existence and stability results for periodic nonautonomous uniform forward attractors of periodic general dynamical systems (set-valued dynamical systems). We also investigate the dynamical behavior of nonautonomous periodic differential inclusion x ′ ( t ) ∈ f ( t , x ( t ) ) on R m with only upper semi-continuous right-hand side by applying the abstract results. Firstly, we show that if the system has a compact uniformly attracting set, then it has a periodic nonautonomous uniform forward attractor A . Secondly, we prove that A is robust with respect to both internal and external perturbations. Finally, we apply the robustness result to discuss the effects of small time delays to asymptotic stability of the system.

Relation between Different Types of Global Attractors of Set-Valued Nonautonomous Dynamical Systems

The article is devoted to the study of the relation between forward and pullback attractors of set-valued nonautonomous dynamical systems (cocycles). Here it is proved that every compact global forward attractor is also a pullback attractor of the set-valued nonautonomous dynamical system. The inverse statement, generally speaking, is not true, but we prove that every global pullback attractor of an α-condensing set-valued cocycle is always a local forward attractor. The obtained general results are applied while studying periodic and homogeneous systems. We give also a new criterion of the absolute asymptotic stability of nonstationary discrete linear inclusions.

Uniform attractors of periodic and asymptotically periodic dynamical systems

The uniform forward global attractors of periodic systems and the comparison of the dynamics between asymptotically periodic non-autonomous dynamical systems and their corresponding limiting periodic processes are investigated. The existence and Lyapunov stability of such attractors are established and a Lyapunov functional charecterization of uniform forward global attractors of non-autonomous periodic processes is presented. The results are illustrated with examples of ordinary and delay differential equations.

PULLBACK AND FORWARD ATTRACTORS FOR A DAMPED WAVE EQUATION WITH DELAYS

The existence of a pullback (and also a uniform forward) attractor is proved for a damped wave equation containing a delay forcing term which, in particular, covers the models of sine–Gordon type. The result follows from the existence of a compact set which is uniformly attracting for the two-parameter semigroup associated to the model.

Uniform attractors of nonautonomous index 2 differential algebraic equations under discretization

The long‐time behaviour of Runge–Kunge discretizations is investigated when applied to a smooth nonautonomous index 2 differential algebraic equation (DAE) with a cocycle structure, i.e. a DAE driven by an autonomous dynamical system, which is assumed to have a uniform attractor. It is shown that the cocycle structure of the continuous dynamics is preserved under discretization and that a uniform forward or pullback attractor of the DAE persists under discretization by a Runge–Kutta scheme with the component subsets of the numerical attractor converging upper semicontinuously to their continuous time counterparts.