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In recent years we have seen an explosion in the literature related to dynamical systems with infinite-dimensional state space, Sell and You [13], Chicone and Latushkin [3], Kloeden and Rasmussen [...

In this article, a new class of generalized f-projected dynamical systems is introduced and studied in Banach spaces. A global existence and uniqueness result of generalized f-projected dynamical system is proved, which generalizes the existence result of Xia and Vincent [Y.S. Xia and T.L. Vincent, On the stability of global projected dynamical systems, J. Optim. Theory Appl. 106 (2009), pp. 129–150]. The global convergence stability of the generalized f-projected dynamical system and the sensitivity result of solutions set with perturbations of the constraint sets are also obtained under some suitable conditions.

In this article, we provide a matrix method in order to compute orbits of parallel and sequential dynamical systems on Boolean functions. In this sense, we develop algorithms for systems defined over directed and undirected graphs when the evolution operator is a general minterm or maxterm and, likewise, when it is constituted by independent local Boolean functions, so providing a new tool for the study of orbits of these dynamical systems.

In this paper, we find additional conditions to be satisfied by a globally periodic discrete dynamical system, so that its good set (the set of initial conditions providing well-defined solutions) is an open set of ℝ k or ℂ k . We will pay especial attention to the rational case and several examples will be given.

The necessity of considering trajectories just of sets rather than of points occurs both in dynamical systems theory by itself and in many evolutionary problems reducible to dynamical systems. In the case of one-dimensional dynamical systems, we present a number of conditions for the trajectory of an interval to be asymptotically periodic. The obtained results find significant applications in the theory of continuous time difference equations and some classes of boundary value problems for partial differential equations.

Linear error models are an integral part of several parameter identification methods for feedforward and feedback control systems and lead in connection with the L 2-norm to a convex distance measure which has to be minimised for identification purposes. The parameters are hereby often subject to specific restrictions whose intersections span a convex solution set with non-differentiability points on its boundary. For solving these well conditioned problems on-line the paper formulates the solution of the bounded convex minimisation problem as a stable equilibrium set of a proper system of differential equations. The vector field of the corresponding system of differential equations is based on a projection of the negative gradient of the distance measure. A general drawback of this approach is the discontinuous right-hand side of the differential equation caused by the projection transformation. The consequence are difficulties for the verification of the existence, uniqueness and stability of a solution...

The non-regular, incomplete and singular differential dynamical systems are discussed from a geometric point of view. The constitutive space of a dynamical system is considered as a subset of the tangent bundle to the manifold (the generalised hypersurface) being the ambient space of the system. The system is the essentially non-regular differential dynamical system if there is observed reduction of the configuration space to the 'proper' configuration space of the system. The dynamical system, which is not observed by the solutions in the entire configuration space, is called the incomplete system. The system such that the dynamic space (the solution space) is not the topologically closed subspace of the ambient space of the system is called the singular dynamical system. The points in the topological closure of the configuration space that are not contained in the solution space of the dynamical system are called the impasse points of the system. The points in the topological closure of the solution space that are not contained in the solution space of the system are called the singularity points of the dynamical system. The systematic classification of the impasse points in dynamical systems is included which extends the classification proposed for the systems defined by the differential algebraic equations. The well posed and the relatively well-posed dynamical systems are defined. These are the systems, which are considered most often in the modelling of real systems. The classification which includes the non-regular, the incomplete, the well posed, and the relatively well-posed dynamical systems is proposed. The trajectory available impasse points have been extended for the general differential dynamical systems. The trajectory available impasse points are important notion in the modelling and in the simulation of real systems which exhibit the discontinuous trajectory behaviour. At the time when the trajectory of the system approached the singularity point, one usually observes the discontinuity in the dynamic behaviour of the system or the 'switching' of the trajectories which is possibly continuous takes place in the system. The examples are enclosed which illustrate the considerations.

The problems concerning dynamical systems modelling are discussed from a geometric point of view. The constitutive space of a dynamical system is considered as a subset of the tangent bundle to the manifold (the generalized hypersurface) being the available space of the system. The constitutive space of the (differential) dynamical system implicitly encloses all the necessary information which is sufficient to select the solution space and the infinitesimal generator of the system which enables the design of the dynamic behavior of the system. The differential inclusions then appear in a natural way in the modelling of physical systems as implicitly written differential systems, where sets of differential algebraic relations describe the constitution of the system. The constructive definition of a differential dynamical system considered here has been obtained through generalization of the mathematical model of a non-linear electrical network extended next to lumped mechanical systems and control systems. The constructive approach to differential dynamical systems presented in this paper corresponds to the procedure commonly used when a mathematical model of a physical system is set up. The subbundles of the tangent bundles to manifolds being mathematical models of the configuration spaces of dynamical systems are the basic mathematical tools. A general iterative procedure for selecting the solution space of the dynamical system is proposed, which extends the formulation given in the references. Examples illustrating the considerations are given.

There is a method for searching first integrals for polynomial ordinary differential equations (usually called Darboux method) based on the knowledge of several of their invariant algebraic hypersurfaces. We extend this method to discrete dynamical systems, providing a way of searching invariants for them and we give several examples of application.

The book Theory and Applications of Difference Equations and Discrete Dynamical Systems, edited by Ziyad AlSharawi, Jim Cushing and Saber Elaydi, features selected contributions from the 19th Inter...

This paper presents a sequential identification scheme for continuous nonlinear dynamical systems using neural networks. The nonlinearities of the dynamical systems are assumed to be unknown. The identification model is a gaussian radial basis function neural network that grows gradually to span the appropriate state space and of sufficient complexity to provide an approximation to the dynamical system. The sequential identification algorithm for continuous dynamical nonlinear systems is developed in the continuous-time framework instead of in discrete time. The approach, different from the conventional methods of optimizing a cost function, attempts to ensure stability of the overall system while the neural network learns the system dynamics. The stability and convergence of the overall identification scheme are guaranteed by parameter adjustment laws developed using the Lyapunov synthesis approach. To ensure that the modelling error can be reduced arbitrarily, a one-to-one mapping is proposed so that th...

The integral-functional equations with the unknown variable in the upper limit of integration are introduced and studied. They arise in the optimization problems for integral dynamical models and allow to analyze the variable aftereffect duration of dynamical system

This paper presents an analysis of the stability for the continuous time adjustment processes proposed for the spatial oligopolistic market equilibrium problem by Nagurney, Dupuis, and Zhang [13] and for the classical (aspatial) oligopoly problem by Dupuis and Nagurney [6], modeled as projected dynamicalsystems. Here two distinct approaches are presented, a monotonicity approach and a regularity approach. The monotonicity approach utilizes monotonicity conditions on the underlying functions, whereas the regularity approach utilizes the regularity concept of oligopolistic market equilibrium, adapted from Zhang and Nagurney [20], which enables us to explore the stability of the adjustment process—a nonstandard dynamical system, via its minimal face flow—a standard dynamical system. These approaches to stability analysis are shown to generalize some of the results in the existing literature

We consider small random perturbations of dynamical systems on a d-dimensional Euclidean space R d when the origin oϵR d is a hyperbolic equilibrium point of unperturbed dynamical systems. The objects under consideration are empirical measures which are marginal measures of empirical processes at the exit time from a bounded domain and .

The stabilization problem of general second-order single-input dynamical systems is discussed. This class of linear or non-linear dynamical systems contains some uncertainties. A new robust discontinuous controller is proposed to guarantee asymptotic stability of general dynamical systems with bounded uncertainties. Then, to guarantee the continuity of the control, a quasi-saturation function is introduced into the control law and guaranteed eventual stability-in-the-large (ESL) of the dynamical systems is proved. Some examples and simulations are given to illustrate the effectiveness of the robust continuous controller. The control signals are indeed continuous and smooth while maintaining the desired performance of the system.

Two techniques for the approximate analysis of damped linear dynamical systems are compared.

The paper examines several concepts of dynamical systems and chaos in both finite and discrete domains where concepts and results are simpler and more intuitive. Some of the results are then compared to their continuous counterparts to determine the errors inherent in computer simulations of continuous dynamical systems.

The theory of dynamical systems has seen spectacular developments in the past 100 years, beginning with the contributions of Poincare and Lyapunov to a contemporary complete understanding of the at...

(1991). Theory vs. Computation in Some Very Simple Dynamical Systems. The College Mathematics Journal: Vol. 22, No. 1, pp. 42-44.

The problem of determining whether or not a dynamical system ℒ controllable has been an active research topic in recent years. However, the complementary problem of determining just why a system ℒ uncontrollable has received relatively little attention. In this paper the physical acid mathematical mechanism of uncontrollability is studied with particular emphasis on how uncontrollability of a general class of non-linear dynamical systems can originate with the inadvertent selection of a non-minimal dimension ‘ state vector ’. The state-apace geometry corresponding to this particular mode of uncontrollability, which we call ‘state over-description’, is examined in detail. A fundamental necessary condition for the existence of state over-description in non-linear dynamical systems is derived and its application is illustrated by five examples. Application of the present results to the special case of time-invariant linear dynamical systems will be presented in a subsequent paper.

An approximation of the law of the solution of a stochastic differential equation, using the Fokker-Plank equation, has been discussed in the literature by several methods. The Gauss-Galerkin method provides approximations to the well-known Gauss-Christoffel measure of the exact distribution. However, it results in a very ill-conditioned dynamical system. We introduce a simplified dynamical system which leads to a numerically stable procedure for a larger number of nodes. As an outcome of the analysis, we also show that the approximation is a probability measure

A set-theoretic approach to uncertain dynamical systems by using convex polytopes is treated. In some points, this approach is superior to the one using ellipsoids, though it also has some disadvantages. First, some fundamental operations are discussed, such as addition, intersection, and the linear transformation of convex polytopes. The results are applied to the analysis of reachable sets for discrete-time linear dynamical systems and also to the state estimation.

Some abstract ideas on dissipative systems for dynamical systems are extended to weak dynamical systems. Applications are given for a linearly damped nonlinear wave equation and some distributed control problems.

In this paper, the probability of controllability of a class of linear dynamical systems is investigated by applying simple combinatorial reasoning (in the manner of Porter 1972) to the structure of the mode-controllability matrices of systems of this class. The results of this investigation are discussed in relation to the results concerning the probability of stability of linear dynamical systems obtained by Ashby (1952) and Porter (1972).

In this paper, simple formulas for the probability of stability of a class of linear dynamical systems arc presented. The theoretical results thus obtained are compared with certain empirical results obtained by Ashby (1952).

In many complex control processes, the problem of determining optimal control policies can be considerably simplified from the stand-points of computation and implementation, if the mathematical model can be adequately described by a finite-state sequential system, i.e. a discrete-time system whose input (control) and state spaces are finite-state. However, in physical situations, one usually formulates the mathematical models in the form of a set of ordinary differential equations resulting from certain physical laws. Therefore, it is desirable to have systematic methods far approximating such mathematical models by finite-state systems. This paper describes a simple method for approximating a finite set of first-order ordinary differential equations by the finite-state system. The validity and application of the method are discussed and illustrated by examples. Also, the main problem associated with the implementation of control systems using finite-state models will be discussed briefly.