Introduction

Approximate inertial manifolds of Burgers equation approached by nonlinear Galerkin’s procedure and its application

We prove that a crossed product algebra arising from a minimal dynamical system on the product of the Cantor set and the circle has real rank zero if and only if that system is rigid. In the case that cocycles take values in the rotation group, it is also shown that rigidity implies tracial rank zero, and in particular, the crossed product algebra is isomorphic to a unital simple AT-algebra of real rank zero. Under the same assumption, we show that two systems are approximately $K$-conjugate if and only if there exists a sequence of isomorphisms between two associated crossed products which approximately maps $C(X\times \T)$ onto $C(X\times \T)$.

In answer to a question of Macpherson and Neumann related to the classification of maximal subgroups of infinite symmetric groups, we characterize set ideals on an infinite set X whose stabilizers in the symmetric group Sym(X) are maximal subgroups. Specifically, for an ideal I containing all subsets of cardinality <∣X∣, the stabilizer S{I} is maximal if and only if the corresponding Boolean dynamical system (X, I, S{I}) is minimal, that in turn can be expressed as an intrinsic structural property of the Boolean space (X, I). The general case is reduced to a minimal dynamical system on a subset. This characterization clarifies many earlier results, and reveals an interesting link between the theory of infinite permutation groups, topological dynamics and ergodic theory. 1991 Mathematics Subject Classification 20B35, 20B27.

In the paper finite-dimensional dynamical control systems described by second order semilinear stationary ordinary differential state equations with delay in control are considered. Using a generalized open mapping theorem, sufficient conditions for constrained local controllability in a given time interval are formulated and proved. These conditions require verification of constrained global controllability of the associated linear first-order dynamical control system. It is generally assumed, that the values of admissible controls are in a convex and closed cone with vertex at zero. Moreover, several remarks and comments on the existing results for controllability of semilinear dynamical control systems are also presented. Finally, simple numerical example which illustrates theoretical considerations is also given. It should be pointed out, that the results given in the paper extend for the case of semilinear second-order dynamical systems constrained controllability conditions, which were previously known only for linear second-order systems.

Dynamical systems of finite-dimensional metric spaces and zero-dimensional covers

In the present paper finite-dimensional dynamical control systems described by linear higherorder ordinary differential state equations with multiple point delays in control are considered. Using algebraic methods, necessary and sufficient condition for relative controllability in a given time interval for linear dynamical system with multiple point delays in control is formulated and proved. This condition is generalization to relative controllability case some previous results concerning controllability of linear dynamical systems without multiple point delays in the control. Simple numerical example, which illustrates theoretical result is also given. Finally, some remarks and comments on the existing results for controllability of dynamical systems with delays in control are also presented.

The usual invariance results for asymptotic stability for continuous autonomous finite-dimensional dynamical systems involve a positive definite Lyapunov function whose time derivative along the system motions is negative semidefinite (along with certain invariance conditions). This is equivalent to requiring that along the system motions, the Lyapunov function is nonincreasing at all times with increasing time. In this paper we establish an invariance result for asymptotic stability for continuous and discontinuous nonautonomous finite-dimensional dynamical systems, which requires a positive definite Lyapunov function which when evaluated along the system motions is nonincreasing only on certain unbounded discrete time sets E with increasing time. This allows that between the time instants determined by E, the Lyapunov function may increase (i.e., over some finite time intervals, the system may exhibit unstable behavior). We also show that the usual invariance theorem for asymptotic stability reduces to the invariance theorem for continuous dynamical systems presented herein. In addition, we establish a variant to the above result involving estimates of the asymptotic behavior of the system's motions. We apply our results to four examples. One of these involves the stabilization of conservative mechanical systems using energy dissipation intermittently.

The main objective of this article is to review the major progress that has been made on controllability of dynamical systems over the past number of years. Controllability is one of the fundamental concepts in the mathematical control theory. This is a qualitative property of dynamical control systems and is of particular importance in control theory. A systematic study of controllability was started at the beginning of sixties in the last century, when the theory of controllability based on the description in the form of state space for both time-invariant and time-varying linear control systems was worked out. Roughly speaking, controllability generally means, that it is possible to steer a dynamical control system from an arbitrary initial state to an arbitrary final state using the set of admissible controls. It should be mentioned, that in the literature there are many different definitions of controllability, which strongly depend on a class of dynamical control systems and on the other hand on the form of admissible controls. Controllability problems for different types of dynamical systems require the application of numerous mathematical concepts and methods taken directly from differential geometry, functional analysis, topology, matrix analysis and theory of ordinary and partial differential equations and theory of difference equations. In the paper we use mainly state-space models of dynamical systems, which provide a robust and universal method for studying controllability of various classes of systems. Controllability plays an essential role in the development of modern mathematical control theory. There are various important relationships between controllability, stability and stabilizability of linear both finite-dimensional and infinite-dimensional control systems. Controllability is also strongly related to the theory of realization and so called minimal realization and canonical forms for linear time-invariant control systems such as the Kalmam canonical form, the Jordan canonical form or the Luenberger canonical form. It should be mentioned, that for many dynamical systems there exists a formal duality between the concepts of controllability and observability. Moreover, controllability is strongly connected with the minimum energy control problem for many classes of linear finite dimensional, infinite dimensional dynamical systems, and delayed systems both deterministic and stochastic. Finally, it is well known, that controllability concept has many important applications not only in control theory and systems theory, but also in such areas as industrial and chemical process control, reactor control, control of electric bulk power systems, aerospce engineering and recently in quantum systems theory.

The aim of this chapter is to show how the information extracted from a physical experiment or numerical computation may be analysed using modern signal processing techniques and indicate how the results can be related to ideas from finite-dimensional dynamical systems. The ultimate objective is to see whether a direct connection can be made between the information contained in the time series obtained from a continuous infinite-degree-of-freedom system and the well-developed field of chaos in ordinary differential equations and maps. Before we proceed with the description of the modern nonlinear techniques, we will first of all briefly discuss the traditional linear methods of signal processing. Next we will introduce the concepts of phase portrait reconstruction using the straight-forward ‘method of delays’. The principles are extended by a discussion of the singular value decomposition technique which provides a systematic way of selecting a coordinate system in which the attractor can be constructed and which also deals with the universal problem of noise in experimental observations. A brief discussion of the construction of Poincare maps is then presented as a useful technique for reducing the dimension of the reconstructed attractor. Finally, we discuss the merits and practical limitations of methods of obtaining quantitative estimates from these nonlinear signal processing techniques.

Finite dimensional dynamical systems with one constraint that is quadratic both in canonical momenta and coordinates are considered. The four simplest cases with non-trivial topology are chosen and the Dirac method of quantisation is applied to them. The Wheeler-DeWitt equation is solved by separation of variables. The resulting wavefunctions are interpreted in the regions where they become semiclassical as well as with the help of the DeWitt probability current. The interpretation of initial data for the Wheeler-DeWitt equation is shown to be incompatible with the interpretation of the semiclassical wavepackets in the cases when there is no time function on the configuration spacetime. Thus, the standard way of constructing a unitary theory, with standard time evolution, within the Dirac method, breaks down in a similar way to the reduction one.

Controllability of dynamical systems with constraints

We consider a finite-dimensional deterministic dynamical system with the global attractor 𝒜 which supports a unique ergodic probability measure P. The measure P can be considered as the uniform long-term mean of the trajectories staying in a bounded domain D containing 𝒜. We perturb the dynamical system by a multiplicative heavy tailed Lévy noise of small intensity ϵ > 0 and solve the asymptotic first exit time and location problem from D in the limit of ϵ↘0. In contrast to the case of Gaussian perturbations, the exit time has an algebraic exit rate as a function of ϵ, just as in the case when 𝒜 is a stable fixed point studied earlier in [9, 14, 19, 26]. As an example, we study the first exit problem from a neighborhood of the stable limit cycle for the Van der Pol oscillator perturbed by multiplicative α-stable Lévy noise.

In this article, we present a brief overview of some of the recent progress made in identifying and generating finite dimensional integrable nonlinear dynamical systems, exhibiting interesting oscillatory and other solution properties, including quantum aspects. Particularly we concentrate on Lienard type nonlinear oscillators and their generalizations and coupled versions. Specific systems include Mathews-Lakshmanan oscillators, modified Emden equations, isochronous oscillators and generalizations. Nonstandard Lagrangian and Hamiltonian formulations of some of these systems are also briefly touched upon. Nonlocal transformations and linearization aspects are also discussed.

In a chain of nonlinear oscillators, linearly coupled to their nearest neighbors, all travelling waves of small amplitude are found as solutions of finite dimensional reversible dynamical systems. The coupling constant and the inverse wave speed form the parameter space. The groundstate consists of a one-parameter family of periodic waves. It is realized in a certain parameter region containing all cases of light coupling. Beyond the border of this region the complexity of wave-forms increases via a succession of bifurcations. In this paper we give an appropriate formulation of this problem, prove the basic facts about the reduction to finite dimensions, show the existence of the ground states and discuss the first bifurcation by determining a normal form for the reduced system. Finally we show the existence of nanopterons, which are localized waves with a noncancelling periodic tail at infinity whose amplitude is exponentially small in the bifurcation parameter.

Concerning the learning problems of recurrent neural networks (RNNs), this paper deals with the problem of approximating a dynamical system (DS) by an RNN as one extension of the problem of approximating trajectories by an RNN. In particular, we systematically investigate how an RNN can produce a DS on the visible state space to approximate a target DS. First, it is proved that RNNs without hidden units uniquely produce a certain class of DSs. Next, a neural dynamical system (NDS) is proposed as such a DS that an RNN with hidden units can produce on the visible state space, and affine neural dynamical systems (A-NDSs) are constructed as concrete examples of NDSs. Moreover, we prove that any DS on a Euclidean space can be finitely approximated by some A-NDS with any precision, and propose adopting an A-NDS as such a DS that an RNN with hidden units produces to approximate a target DS. © 2000 Scripta Technica, Syst Comp Jpn, 31(4): 77–86, 2000