Robustness of Global Attractors for Reversible Gray–Scott Systems

Global asymptotic dynamics of a typical cubic-autocatalytic reaction-diffusion system, the reversible Gray?CScott system of three components, are investigated. The upper semicontinuity (robustness) of the global attractors in the H ¹ product space for the solution semiflows with respect to the reverse reaction rates as they converge to zero is proved. Through an approach of transformative decomposition, the hurdle of the perturbed singularity between the reversible and non-reversible systems is overcome by showing the uniform dissipation, the uniformly bounded evolution of the union of global attractors, and the uniform convergence property of the bundle of reversible and non-reversible semiflows.     

Attractors of non-Newtonian fluids

The existence of global attractors is demonstrated for the dynamical systems generated by motions of nonlinear bipolar and non-Newtonian viscous fluids and upper bounds are obtained for the Hausdorff and fractal dimensions of the attractors for the bipolar case.     

Exponential attractors of optimal Lyapunov dimension for Navier-Stokes equations

In this paper we present a new construction of exponential attractors based on the control of Lyapunov exponents over a compact, invariant set. The fractal dimension estimate of the exponential attractor thus obtained is of the same order as the one for global attractors estimated through Lyapunov exponents. We discuss various applications to Navier-Stokes systems.     

Bifurcations and chaos in parametrically excited single-degree-of-freedom systems

The behavior of single-degree-of-freedom systems possessing quadratic and cubic nonlinearities subject to parametric excitation is investigated. Both fundamental and principal parametric resonances are considered. A global bifurcation diagram in the excitation amplitude and excitation frequency domain is presented showing different possible stable steady-state solutions (attractors). Fractal basin maps for fundamental and principal parametric resonances when three attractors coexist are presented in color. An enlargement of one region of the map for principal parametric resonance reveals a Cantor-like set of fractal boundaries. For some cases, both periodic and chaotic attractors coexist.     

Attractors of Reaction Diffusion Systems on Infinite Lattices

In this paper, we study global attractors for implicit discretizations of a semilinear parabolic system on the line. It is shown that under usual dissipativity conditions there exists a global (Z _u ,Z _ρ )-attractor $A$ in the sense of Babin-Vishik and Mielke-Schneider. Here Z _ρ is a weighted Sobolev space of infinite sequences with a weight that decays at infinity, while the space Z _u carries a locally uniform norm obtained by taking the supremum over all Z _ρ norms of translates. We show that the absorbing set containing $A$ can be taken uniformly bounded (in the norm of Z _u ) for small time and space steps of the discretization. We establish the following upper semicontinuity property of the attractor $A$ for a scalar equation: if $A$ _N is the global attractor for a discretization of the same parabolic equation on the finite segment [?N,N] with Dirichlet boundary conditions, then the attractors $A$ _N (properly embedded into the space Z _u ) tend to $A$ as N→∞ with respect to the Hausdorff semidistance generated by the norm in Z _ρ . We describe a possibility of “embedding” certain invariant sets of some planar dynamical systems into the global attractor $A$ . Finally, we give an example in which the global attractor $A$ is infinite-dimensional.     

The effect of the random parameter on the basins and attractors of the elastic impact system

The global dynamic behavior of the elastic impact system with one random parameter is researched in this paper. First, based on the orthogonal polynomial approximation, the stochastic system is transformed into an equivalent deterministic system. Then the composite cell coordinate system method is used to derive the basins and attractors of the equivalent system. It is found that the random parameter affects the global dynamics of the system tremendously. Under the same conditions, the basins and attractors of the stochastic system are entirely different from that of the deterministic system.     

Reducible Volterra and Levin–Nohel Retarded Equations with Infinite Delay

We consider two cases of reducible Volterra and Levin–Nohel retarded equations with infinite delay. In these cases reducibility arises from the use of a special type of memory functions with an exponential behavior. We address global questions like the existence of Liapunov functions and, consequently, of attractors for the nonlinear systems generated by these equations as well as the attractors for the reduced systems. For the reducible Volterra equations we exhibit cases of nontrivial Hamiltonian behaviour and for the reducible Levin–Nohel equation we identify Hopf and saddle connection bifurcations.     

Dimensions of attractors for discretizations for Navier-Stokes equations

In this paper, we discretize the 2-D incompressible Navier-Stokes equations with the periodic boundary condition by the finite difference method. We prove that with a shift for discretization, the global solutions exist. After proving some discrete Sobolev inequalities in the sense of finite differences, we prove the existence of the global attractors of the discretized system, and we estimate the upper bounds for the Hausdorff and the fractal dimensions of the attractors. These bounds are indepent of the mesh sizes and are considerably close to those of the continuous version.     

Topological Structural Stability of Partial Differential Equations on Projected Spaces

In this paper we study topological structural stability for a family of nonlinear semigroups \(T_h(\cdot )\) on Banach space \(X_h\) depending on the parameter h. Our results shows the robustness of the internal dynamics and characterization of global attractors for projected Banach spaces, generalizing previous results for small perturbations of partial differential equations. We apply the results to an abstract semilinear equation with Dumbbell type domains and to an abstract evolution problem discretized by the finite element method.     

Almost Periodic Solutions and Global Attractors of Non-autonomous Navier–Stokes Equations

The article is devoted to the study of non-autonomous Navier–Stokes equations. First, the authors have proved that such systems admit compact global attractors. This problem is formulated and solved in the terms of general non-autonomous dynamical systems. Second, they have obtained conditions of convergence of non-autonomous Navier–Stokes equations. Third, a criterion for the existence of almost periodic (quasi periodic, almost automorphic, recurrent, pseudo recurrent) solutions of non-autonomous Navier–Stokes equations is given. Finally, the authors have derived a global averaging principle for non-autonomous Navier–Stokes equations.     

LONG TIME BEHAVIOR FOR SOLUTION OF INITIAL BOUNDARY VALUE PROBLEM OF ONE CLASS OF SYSTEMS WITH MULTIDIMENSIONAL INHOMOGENEOUS GBBM EQUATIONS

The following initial-boundary value problem for the systems with multidimensional inhomogeneous generalized Benjamin-Bona-Mahony ( GBBM ) equations is reviewed. The existence of global attractors of this problem was proved by means of a uniform priori estimate for time.     

Complete synchronization of chaotic complex nonlinear systems with uncertain parameters

Our main objective in this work is to investigate complete synchronization (CS) of n-dimensional chaotic complex systems with uncertain parameters. An adaptive control scheme is designed to study the synchronization of chaotic attractors of these systems. We applied this scheme, as an example, to study complete synchronization of chaotic attractors of two identical complex Lorenz systems. The adaptive control functions and the parameters estimation laws are calculated analytically based on the complex Lyapunov function. We show that the error dynamical systems are globally stable. Numerical simulations are computed to check the analytical expressions of adaptive controllers.     

Attractors and Chain Recurrence for Semigroup Actions

Let ${\mathcal{S}}$ be a semigroup acting on a topological space M. We define attractors for the action of ${\mathcal{S}}$ on M. This concept depends on a family ${\mathcal{F}}$ of subsets of ${\mathcal{S}}$ . For certain semigroups and families it recovers the concept of attractors for flows or semiflows. We define and study the complementary repeller of an attractor. We also characterize the set of chain recurrent points in terms of attractors.     

Trajectory and Global Attractors of the Boundary Value Problem for Autonomous Motion Equations of Viscoelastic Medium

We introduce the concept of minimal trajectory attractor generalizing the known concept of trajectory attractor of an abstract evolution equation. We obtain several results on existence and properties of minimal trajectory and global attractors without assumptions of any invariance of the trajectory space of an equation. With the help of these results we prove existence of minimal trajectory and global attractors for weak solutions of the boundary value problem for autonomous motion equations of an incompressible viscoelastic medium with the Jeffreys constitutive law. The work was partially supported by grants 04-01-00081 of Russian Foundation of Basic Research, VZ-010-0 of the Ministry of Education and Science of Russia and CRDF and MK- 3650.2005.1 of President of Russian Federation.     

Entropy and the Hausdorff dimension for infinite-dimensional dynamical systems

We define a sequence of uniform Lyapunov exponents in the setting of Banach spaces and prove that the Hausdorff dimension of global attractors is bounded from above by the Lyapunov dimension of the tangent map. This result generalizes the papers by Douady and Oesterlé (1980) and Ledrappier (1981) in finite dimension and Constantinet al. (1985) for Hilbert spaces.     

Global attractor for Klein-Gordon-Schroedinger lattice system

We considered the longtime behavior of solutions of a coupled lattice dynamical system of Klein-Gordon-Schroedinger equation （KGS lattice system）. We first proved the existence of a global attractor for the system considered here by introducing an equivalent norm and using ＂End Tails＂ of solutions. Then we estimated the upper bound of the Kolmogorov delta-entropy of the global attractor by applying element decomposition and the covering property of a polyhedron by balls of radii delta in the finite dimensional space. Finally, we presented an approximation to the global attractor by the global attractors of finite-dimensional ordinary differential systems.     

Duffing振子奇怪吸引子的简单胞映射计算

本文运用简单胞映射理论,计算出了有阻尼Ｄｕｆｆｉｎｇ振子的强迫振动在一定参数条件下存在的奇怪吸引子,算法简单,结果理想,通过实例证明了简单胞映射理论在计算混沌问题方面的有效性．