A construction of almost automorphic minimal sets

We describe a general procedure to construct topological extensions of given skew product maps with one-dimensional fibres by blowing up a countable number of single points to vertical segments. This allows to produce various examples of unusual dynamics, including almost automorphic minimal sets of almost periodically forced systems, point-distal but non-distal homeomorphisms of the torus (as first constructed by Rees) or minimal sets of quasiperiodically forced interval maps which are not filled-in.     

Almost automorphic solutions for abstract functional differential equations

For abstract linear functional differential equations with an almost automorphic forcing term, we establish a result on the existence of almost automorphic solutions, which extends the classical theorem due to Massera on the existence of periodic solutions for linear periodic ordinary differential equations.     

Invariant manifolds, global attractors, almost automorphic and almost periodic solutions of non-autonomous differential equations

The paper is devoted to the study of non-autonomous evolution equations: invariant manifolds, compact global attractors, almost periodic and almost automorphic solutions. We study this problem in the framework of general non-autonomous (cocycle) dynamical systems. First, we prove that under some conditions such systems admit an invariant continuous section (an invariant manifold). Then, we obtain the conditions for the existence of a compact global attractor and characterize its structure. Third, we derive a criterion for the existence of almost periodic and almost automorphic solutions of different classes of non-autonomous differential equations (both ODEs (in finite and infinite spaces) and PDEs).     

Two New Recurrent Levels for <Emphasis Type="Italic">C</Emphasis><Superscript>0</Superscript>-Flows

The central problem in dynamical systems is the asymptotic behavior or topological structure of the orbits. Nevertheless only orbits of points with certain recurrence and form a set of full measure are truly of importance. Of course, such a set is desired to be as small (in the sense of set inclusion) as possible. In this paper we discuss such two sets: the set of weakly almost periodic points and the set of quasi-weakly almost periodic points. While the two sets are different from each other by definitions, we prove that their closures both coincide with the measure center (or the minimal center of attraction) of the dynamical systems. Generally, a point may have three levels of orbit-structure: the support of an invariant measure generated by the point, its minimal center of attraction and its ω-limit set. We study the three levels of orbit-structure for weakly almost periodic points and quasi-weakly almost periodic points. We prove that quasi-weakly almost periodic points possess especially rich topological orbit-structures. We also present a necessary and sufficient condition for a point to belong to its own minimal center of attraction.     

Substitution-like minimal sets

Substitution-like minimal sets are a class of symbolic minimal sets on two symbols which includes the discrete substitution minimal sets on two symbols. They are almost automorphic extensions of then-adic integers and they are constructed by using special subsets of then-adics from which the almost automorphic points are determined by following orbits in then-adics. Through their study a complete classification is obtained for substitution minimal sets of constant length on two symbols. Moreover, the classification scheme is such that for specific substitutions the existence or non-existence of an isomorphism can be determined in a finite number of steps.     

Stability and extensibility results for abstract skew-product semiflows

In this paper we present new stability and extensibility results for skew-product semiflows with a minimal base flow. In particular, we describe the structure of uniformly stable and uniformly asymptotically stable sets admitting backwards orbits and the structure of omega-limit sets. As an application, the occurrence of almost periodic and almost automorphic dynamics for monotone non-autonomous infinite delay functional differential equations is analyzed.     

Population models in almost periodic environments

We establish the basic theory of almost periodic sequences on ?₊. Dichotomy techniques are then utilized to find sufficient conditions for the existence of a globally attracting almost periodic solution of a semilinear system of difference equations. These existence results are, subsequently, applied to discretely reproducing populations with and without overlapping generations. Furthermore, we access evidence for attenuance and resonance in almost periodically forced population models.     

Independence of almost periodicity upon the topology of the acting group

Answering a question first explicitly stated by de Vries in 1993, we observe that for an arbitrary topological dynamical system the property of being an almost periodic point does not depend on the topology of the acting group. In other words, the traditional distinction made between the notions of an almost periodic point and of a discretely almost periodic point is unnecessary.     

Composition of pseudo almost automorphic and asymptotically almost automorphic functions

This paper is concerned with pseudo almost automorphic functions, which are more general and complicated than pseudo almost periodic functions and asymptotically almost automorphic functions. New results, concerning the composition of pseudo almost automorphic functions, are established.     

On quasi-weakly almost periodic points

The core problem of dynamical systems is to study the asymptotic behaviors of orbits and their topological structures. It is well known that the orbits with certain recurrence and generating ergodic (or invariant) measures are important, such orbits form a full measure set for all invariant measures of the system, its closure is called the measure center of the system. To investigate this set, Zhou introduced the notions of weakly almost periodic point and quasi-weakly almost periodic point in 1990s, and presented some open problems on complexity of discrete dynamical systems in 2004. One of the open problems is as follows: for a quasi-weakly almost periodic point but not weakly almost periodic, is there an invariant measure generated by its orbit such that the support of this measure is equal to its minimal center of attraction (a closed invariant set which attracts its orbit statistically for every point and has no proper subset with this property)? Up to now, the problem remains open. In this paper, we construct two points in the one-sided shift system of two symbols, each of them generates a sub-shift system. One gives a positive answer to the question above, the other answers in the negative. Thus we solve the open problem completely. More important, the two examples show that a proper quasi-weakly almost periodic orbit behaves very differently with weakly almost periodic orbit.     

Eine Bemerkung zu meiner Arbeit „Rhythmische Abbildungen abelscher Gruppen II“

Summary In the paper quoted in the title it was proved that a function on a discrete group is almost automorphic if and only if it is bounded and continuous in the Bohr topology. Here this result is extended to continuous functions on arbitrary topological groups. Taken together with a theorem of Marenko, this implies a theorem first stated, but not proved, by Veech: a function of a real variable is continuous and almost automorphic if and only if it is bounded and Levitan almost periodic.     

Embedding discrete flows on R in a continuous flow

The problem of determining when a given discrete flow on a topological space is embeddable in some continuous flow was mentioned by G. R. Sell (“Topological Dynamics and Ordinary Differential Equations,” Van Nostrand, New York, 1971) in his book on topological dynamics. In this book, the theory of generalized dynamical systems is exploited in the qualitative study of differential equations. Even more complicated is the problem of simultaneously embedding two or more discrete flows in a single continuous flow. We examine both of these problems when the underlying topological space is the space R of the real numbers.     

Dynamics on Ahlfors quasi-circles

The celebrated theory of Denjoy introduced a topological invariant distinguishingC ¹ andC ² diffeomorphisms of the circle. AC ² diffeomorphism of the circle cannot have an infinite minimal set other than the circle itself. However, this is possible forC ¹ diffeomorphisms. In dimension two there is a related invariant distinguishingC ² andC ³ diffeomorphisms. Partially supported by NSF grant No. MCS-83202062.     

Effect of the phase on the dynamics of a perturbed bouncing ball system

The phase control method is a non-feedback control technique which has been used for different purposes in continuous periodically driven dynamical systems. One of the main goals of this paper is to apply this control technique to the bouncing ball system, which can be seen as a paradigmatic periodically driven discrete dynamical system, and has a rather simple physical interpretation. The main idea is to apply a periodic control signal including a phase difference with respect to the periodic forcing of the initial system and to analyze its effect on the dynamics of the bouncing ball system. The numerical simulations we have carried out clearly show the strong effect of the phase of the control signal in suppressing or generating chaotic behavior and in changing the period of a periodic orbit. We have also analyzed the effect of the phase in the phenomenon of the crisis-induced intermittency, showing how the phase enhances the size of the attractor near a crisis and can induce the intermittent behavior. Finally we have analyzed the scaling behavior of the crisis by varying the phase difference between the perturbation and the external forcing.     

Almost automorphic solutions for some partial functional differential equations

In this work, we study the existence of almost automorphic solutions for some partial functional differential equations. We prove that the existence of a bounded solution on R⁺ implies the existence of an almost automorphic solution. Our results extend the classical known theorem by Bohr and Neugebauer on the existence of almost periodic solutions for inhomegeneous linear almost periodic differential equations. We give some applications to hyperbolic equations and Lotka-Volterra type equations used to describe the evolution of a single diffusive animal species.     

Almost periodic dynamical behaviors for generalized Cohen–Grossberg neural networks with discontinuous activations via differential inclusions

In this paper, we investigate the almost periodic dynamical behaviors for a class of general Cohen–Grossberg neural networks with discontinuous right-hand sides, time-varying and distributed delays. By means of retarded differential inclusions theory and nonsmooth analysis theory with generalized Lyapunov approach, we obtain the existence, uniqueness and global stability of almost periodic solution to the neural networks system. It is worthy to pointed out that, without assuming the boundedness or monotonicity of the discontinuous neuron activation functions, our results will also be valid. Finally, we give some numerical examples to show the applicability and effectiveness of our main results.     

Almost automorphic symbolic minimal sets without unique ergodicity

Given a metrizable monothetic groupG with generatorg and a suitable closed nowhere dense subsetC of positive Haar measure, we associate a natural compact metric space whose points are almost automorphic symbolic minimal sets. It is then shown that those minimal sets which have positive topological entropy and fail to be uniquely ergodic form a esidual set. The example due to P. Julius [2] of a Toeplitz sequence of positive entropy which, is uniquely ergodic shows that the “residual” conclusion is sharp.     

A global solution curve for a class of periodic problems, including the pendulum equation

Using continuation methods and bifurcation theory, we study the exact multiplicity of periodic solutions, and the global solution structure, for a class of periodically forced pendulum-like equations. Our results apply also to the first order equations. We also show that by choosing a forcing term, one can produce periodic solutions with any number of Fourier coefficients arbitrarily prescribed.     

Dynamics of a class of ODEs more general than almost periodic

A temporally global solution, if it exists, of a nonautonomous ordinary differential equation need not be periodic, almost periodic or almost automorphic when the forcing term is periodic, almost periodic or almost automorphic, respectively. An alternative class of functions extending periodic and almost periodic functions which has the property that a bounded temporally global solution solution of a nonautonomous ordinary differential equation belongs to this class when the forcing term does is introduced here. Specifically, the class of functions consists of uniformly continuous functions, defined on the real line and taking values in a Banach space, which have pre-compact ranges. Besides periodic and almost periodic functions, this class also includes many nonrecurrent functions. Assuming a hyperbolic structure for the unperturbed linear equation and certain properties for the linear and nonlinear parts, the existence of a special bounded entire solution, as well the existence of stable and unstable manifolds of this solution are established. Moreover, it is shown that this solution and these manifolds inherit the temporal behaviour of the vector field equation. In the stable case it is shown that this special solution is the pullback attractor of the system. A class of infinite dimensional examples involving a linear operator consisting of a time independent part which generates a C₀-semigroup plus a small time dependent part is presented and applied to systems of coupled heat and beam equations.     

自全国各地100多位理事Road Machinery Branch of China Highway and Transportation Society He

In this paper, a three dimensional ratio-dependent chemostat model with periodically pulsed input is considered. By using the discrete dynamical system determined by the stroboscopic map and Floquet theorem, an exact periodic solution with positive concentrations of substrate and predator in the absence of prey is obtained. When β is less than some critical value the boundary periodic solution (xs(t), 0, zs(t)) is locally stable, and when β is larger than the critical value there are periodic oscillations in substrate, prey and predator. Increasing the impulsive period τ, the system undergoes a series of period-doubling bifurcation leading to chaos, which implies that the dynamical behaviors of the periodically pulsed ratio-dependent predator-prey ecosystem are very complex.