Specific Dynamical System Research Articles

Perturbations and Hopf bifurcation of the planar discontinuous dynamical system

The objective of the paper is to obtain results on the behavior of a specific plane discontinuous dynamical system in the neighbourhood of the singular point. A new technique of investigation is presented. Conditions for existence of the foci and centres are proposed. The focus-centre problem and Hopf bifurcation are considered. Appropriate examples are given to ilustrate the bifurcation theorem.

Particle filters for system identification with application to chaos prediction

The theory of the particle filter, or sequential Monte Carlo methods, has made substantial progress the last decade. The number of applications has increased substantially the last three years, in particular in navigation and telecommunication areas. In this contribution, we will first point out how the particle filter can be used for system identification, using a quite general problem formulation, and it is pointed out in which kind of application the particle filter can be an attractive alternative to classical system identification methods.This is then demonstrated on prediction of time series arising from chaotic dynamical systems. The specific dynamical system considered is the so called logistical map with an unknown parameter, which belongs to the chaotic regime.

Dynamic systems methods for models of developmental psychopathology.

A survey of dynamic systems (DS) methods appropriate for testing systems-based models in developmental psychopathology is provided. The rationale for developing new methods for the field is reviewed first. In line with other scholars, we highlight the fundamental incompatibility between developmentalists' organismic, open systems models and the mechanistic research methods with which these models are tested. Key DS principles are explained and their commensurability with developmental psychopathologists' core theoretical concerns are discussed. Next, a survey of research designs and methodological techniques currently being used and refined by developmental DS researchers is provided. The strengths and limitations of each approach are discussed throughout this review. Finally, we elaborate on one specific dynamic systems method, state space grids, which addresses many of the limitations of previous DS techniques and may prove most useful for the discipline. This approach was developed as a middle road between DS methods that are mathematically heavy on the one hand and purely descriptive on the other. Examples of developmental and clinical studies that have applied state space grids are reviewed and suggestions for future analyses are made. We conclude with some implications for the application of this new methodology for studying change processes in clinical research.

Finding zeros by multilevel subdivision techniques

We present a new set oriented numerical method for the detection of all the zeros of a given (smooth) function g : Rn → Rn within a compact region. The underlying idea is to view classical iteration schemes as specific dynamical systems and to apply adaptive multilevel subdivision techniques for the computation of their fixed points. These techniques have previously been developed for the identification of arbitrary invariant sets of dynamical systems. Based on this idea we construct a subdivision algorithm which allows us to create close coverings of the set of zeros of g. We prove convergence of this algorithm in a general abstract context and propose three different realizations which are motivated by the theoretical results. Finally, we discuss the numerical efficiency of these algorithms by a comparison with a standard zero finding procedure using a routine from the NAG library.

Design rules for application specific dynamical systems

This paper explores different forms of custom dynamical systems for decision making. Curiously, covering and association constraints can be rather easily met, and it can sometimes be guaranteed that all feasible solutions (hence including the optimal solution) exist among the attractors. As yet, no general theory exists for designing such systems, but some titillating special cases can be described. Theorems for the pure differential equation layer can lead to simple and direct electronic implementation of new digital devices. The key strength of the application specific dynamical systems (ASDS) approach lies in the fact that all variables are updated in parallel. Thus ASDS method offers the potential of high speed solutions using modern high density integrated circuits hardware. Scope and purpose Custom dynamical systems can be devised to solve certain operations research (OR) problems. Such systems might be called ASDS after the application specific integrated circuits (ASIC) of modern digital electronics. ASDS can be regarded in three ways, from the pure to the practical: differential equations as a mathematical abstraction, difference equations running on general purpose computers, and integrated digital circuits with finite bit accuracy. In OR, solutions are represented by constant attractors (stable equilibrium states) of ASDS in all three layers. In general, the structure of the problem determines a dynamical system format and problem parameters such as instances of costs are used to define an initial state in state space. A trajectory then evolves and asympotically approaches one of several attractors, that is, a decision. Different initial states derived from different benefits and costs characteristic of the problem lead to different decisions. Applications include nearest neighbor finders (as in error-correction of binary codes), resource to task assignments, and optimization of functions of constrained binary domains.

Single-trigger stochastic resonance

We discuss a recently discovered mechanism for stochastic resonance which involves only a single reference state with excitable dynamics and deterministic reinjection. While calculations based on an analogy with the shot effect capture the broadest features, detailed comparisons with specific dynamical systems suggest the need for certain refinements of the theory.

COMPUTATIONAL ENVIRONMENTS FOR EXPLORING DYNAMICAL SYSTEMS

Nonlinear dynamical-system models are ubiquitous throughout the sciences and engineering. Efficient computational analysis of specific dynamical systems is often a critical component for the successful completion of a research or design project. The enormous increase in the computational capability of desktop computers presents an opportunity for corresponding increases in our ability to explore dynamical models efficiently. Doing so requires that a suitable computational environment be established that gives the user interactive access to algorithms, graphical displays and data management tools. This paper is an overview of what such an environment might look like. The program kaos is a prototype that is oriented towards the computation of multiparameter bifurcation diagrams.

Task‐dynamic modeling of interarticulator coordination

Speech production involves the formation and release of constrictions at different points in the vocal tract. Progress is reported on a task‐dynamic computational model of autonomous coordination among the components of articulatory synergies (e.g., lips and jaw) that participate in motions along functionally defined tract variables (e.g., lip aperture and protrusion). For each speech gesture, a time‐invariant dynamical system (damped, second‐order) is specified at the tract‐variable level, and is transformed into a gesturally and posturally specific dynamical system for the synergy components. Articulatory movement patterns emerge as implicit consequences of the gesture‐specific tract‐variable control structures and the ongoing postural state of the articulators. Explicit trajectory planning is not required. Significantly, the modeled coordinative processes are exactly the same during simulations of unperturbed, mechanically perturbed, and coproduced speech gestures. All simulations are implemented using...

Monocyclicity and acyclicity of second order dynamic systems: PMM vol. 43, no. 5, 1979, pp. 940–945

The problem of monocyclicity and acyclicity of second order dynamic systems is considered. Results of investigations [1, 2] are extended to a wider class of specific dynamic systems.

Qualitative Analysis of Interconnected Systems Described on Banach Spaces: Well Posedness, Instability and Lagrange Stability

AbstractNew qualitative results for interconnected dynamical systems described on Banach spaces are established. Items of interest include well posedness, instability, complete instability, and Lagrange stability (i.e., boundedness and ultimate boundedness of solutions). In all cases the objective is to analyze complex systems in terms of lower order subsystems and in terms of the system interconnecting structure. The results are applied to a specific hybrid dynamical system.

Suboptimal Control Laws for Differential Games

This paper is concerned with optimal parameter selection in differential games. Necessary conditions and sufficient conditions are derived for the existence of a saddle point for a general two-person zero-sum form. The specified forms for the controls consist of weighted sums of the state variables the weighting factors being products of known timevarying functions and of constants to be determined in an optimal manner. The general results are applied to linear-quadratic games (games with linear dynamics and quadratic performance indices). For this class of differential games, an additional development is given to obtain control law parameters that are independent of initial conditions so that a saddle point with respect to the expected value of the performance index is obtained.The results are applied to specific scalar and vector dynamic systems, and numerical solutions are presented. The approach and the results of this paper are especially useful when only partial information about possible initial states is available to the players and when on-line computational facilities are limited. The results should broaden the class of problems to which differential game theory can be realistically applied. By restricting the form of feedback gains, it is possible in some cases to obtain control laws that are only nominally suboptimal, but are both computationally feasible and implementable in practice.

An Intelligent System For Automated MathematicalModelling And Simulation Of DynamicalEngineering Systems

We describe in this paper a computer program for Automated Mathematical Modelling (AMM) and Automated Simulation (AS) of dynamical engineering systems using Artificial Intelligence (AI) techniques. This computer program is an implementation of a new method for AMM using Fuzzy Logic techniques and Fractal Theory, and a new method for AS using Expert Systems technology. Our new method for AMM consists of three main parts: Time Series Analysis, Developing a set of Admissible Models and Selecting the Best model. Our method for Time Series Analysis consists in the use of the fractal dimension of a set of points as a measure of the geometrical complexity of the time series. Our method for developing a set of admissible models is based on the use of Fuzzy Logic techniques to simulate the reasoning process of the human experts in mathematical modelling of engineering systems. The selection of the best model for the engineering systems is done using heuristics from the experts and statistical calculations. The simulation of the best model can be done by using a new method developed by the authors, that enables automated computer exploration of all the dynamical behaviors of the engineering system. Given a mathematical model for a specific Dynamical Engineering System (DES), this method will automatically select (using a rule base) the best set of parameter values to perform numerical simulations of the system. This simulations will in turn enable the identification of all the dynamical behaviors of the engineering system.