Types Of Dynamical Systems Research Articles

κ-Minkowski spacetime through exotic “oscillator”

We have proposed a generally covariant non-relativistic particle model that can represent the κ-Minkowski noncommutative spacetime. The idea is similar in spirit to the noncommutative particle coordinates in the lowest Landau level. Physically our model yields a novel type of dynamical system (termed here as exotic “oscillator”), that obeys a harmonic oscillator like equation of motion with a frequency that is proportional to the square root of energy. On the other hand, the phase diagram does not reveal a closed structure since there is a singularity in the momentum even though energy remains finite. The generally covariant form is related to a generalization of the Snyder algebra in a specific gauge and yields the κ-Minkowski spacetime after a redefinition of the variables. Symmetry considerations are also briefly discussed in the Hamiltonian formulation. Regarding continuous symmetry, the angular momentum acts properly as the generator of rotation. Interestingly, both the discrete symmetries, parity and time reversal, remain intact in the κ-Minkowski spacetime.

PERIOD-DOUBLING SCENARIO WITHOUT FLIP BIFURCATIONS IN A ONE-DIMENSIONAL MAP

In this work a one-dimensional piecewise-smooth dynamical system, representing a Poincaré return map for dynamical systems of the Lorenz type, is investigated. The system shows a bifurcation scenario similar to the classical period-doubling one, but which is influenced by so-called border collision phenomena and denoted as border collision period-doubling bifurcation scenario. This scenario is formed by a sequence of pairs of bifurcations, whereby each pair consists of a border collision bifurcation and a pitchfork bifurcation. The mechanism leading to this scenario and its characteristic properties, like symmetry-breaking and symmetry-recovering as well as emergence of coexisting attractors, are investigated.

Recurrent Points of Fuzzy Dynamical Systems

The main goal of the present paper is to reconsider classical results from the theory of dynamical systems, making them appropriate for treating systems about which we have imprecise, vague, uncertain or incomplete information. Here we study qualitative properties of dynamical systems related to recurrent points. According to the basic Poincaré recurrence theorem, the set of recurrent points has full measure in any subset of the classical dynamical system space. In general, this property is invalid for fuzzy dynamical systems. In this paper, we consider fuzzy dynamical systems of two types. It is demonstrated that for fuzzy dynamical systems of the first type, the Poincaré property of recurrent points remains true if the fuzzy measure in the system space satisfies some, rather weak, additional conditions. It is also proved that infinitely recurrent points have the same property. At the same time, for fuzzy dynamical systems of the second type, the Poincaré property of recurrent points is invalid, and it is possible to find only sufficiently big subsets of recurrent points. In contrast to fuzzy dynamical systems of the first type, infinitely recurrent points of fuzzy dynamical systems of the second type do not have even this property.

THE DYNAMICAL SYSTEMS APPROACH TO LAGRANGIAN TRANSPORT IN OCEANIC FLOWS

▪ Abstract Chaotic advection and, more generally, ideas from dynamical systems, have been fruitfully applied to a diverse, and varied, collection of mixing and transport problems arising in engineering applications over the past 20 years. Indeed, the “dynamical systems approach” was developed, and tested, to the point where it can now be considered a standard tool for understanding mixing and transport issues in many disciplines. This success for engineering-type flows motivated an effort to apply this approach to transport and mixing problems in geophysical flows. However, there are fundamental difficulties arising in this endeavor that must be properly understood and overcome. Central to this approach is that the starting point for analysis is a velocity field (i.e., the “dynamical system”). In many engineering applications this can be obtained sufficiently accurately, either analytically or computationally, so that it describes particle trajectories for the actual flow. However, in geophysical flows (and we concentrate here almost exclusively on oceanographic flows), the wide range of dynamically significant time and length scales makes the justification of any velocity field, in the sense of reproducing particle trajectories for the actual flow, a much more difficult matter. Nevertheless, the case for this approach is compelling due to the advances in observational capabilities in oceanography (e.g., drifter deployments, remote sensing capabilities, satellite imagery, etc.), which reveal space-time structures that are highly suggestive of the structures one visualizes in the global, geometrical study of dynamical systems theory. This has been pursued in recent years through a combination of laboratory studies, kinematic models, and dynamically consistent models that have all been compared with observational data. During the course of these studies it has become apparent that a new type of dynamical system is necessary to consider in these studies (i.e., a finite time, aperiodically time-dependent velocity field defined as a data set), which requires the development of new analytical and computational tools, as well as the necessity to discard some of the standard ideas and results from dynamical systems theory. In this article we review a number of the key developments to date in this young, but rapidly developing, area at the interface between geophysical fluid dynamics and applied and computational mathematics. We also describe the wealth of new directions for research that this approach unlocks.

Border-collision period-doubling scenario.

Using a one-dimensional dynamical system, representing a Poincaré return map for dynamical systems of the Lorenz type, we investigate the border-collision period-doubling bifurcation scenario. In contrast to the classical period-doubling scenario, this scenario is formed by a sequence of pairs of bifurcations, whereby each pair consists of a border-collision bifurcation and a pitchfork bifurcation. The characteristic properties of this scenario, like symmetry-breaking and symmetry-recovering as well as emergence of coexisting attractors and noninvariant attractive sets, are investigated.

Regularized Lotka-Volterra Dynamical System as Continuous Proximal-Like Method in Optimization

We introduce and study a new type of dynamical system which combines the continuous gradient method with a nonlinear Lotka-Volterra (LV) type of differential system within a logarithmic-quadratic proximal scheme. We prove a global existence and viability result for the resulting trajectory which holds for a general smooth function. The asymptotic behavior of the produced trajectory is analyzed and global convergence of the trajectory to a minimizer of the convex minimization problem over the nonnegative orthant is established. The implicit discretization which is at the origin of the proposed continuous dynamical system is an interior proximal scheme for minimizing a closed proper convex function, and convergence results and properties of the resulting discrete scheme are also established. We show finally that the trajectories of the family of regularized Lotka-Volterra systems, parametrized by the positive parameter associated with the quadratic proximal term, are uniformly convergent to the solution of the classical LV-dynamical system, as the parameter associated with the proximal term approaches zero.

Discrete element method

Discrete element method (DEM) has been extensively used in the laboratory of particulate and multiphase processing at the University of New South Wales (UNSW) to study the fundamentals of particulate matter at a particle scale. This paper briefly reviews the work in the laboratory, which covers the development of simulation techniques and their application to the study of particle packing and flow, transport properties and constitutive relationships of typical static or dynamic particulate systems. It is concluded, through representative comparison between simulated and measured results under different conditions, that DEM, as a major technique for discrete particle simulation, is an effective method for particle scale research of particulate matter.

Self-supervised dynamical systems

A new type of dynamical systems which capture the interactions via information flows typical for active multi-agent systems is introduced. The mathematical formalism is based upon coupling the classical dynamical system (with random components caused by uncertainties in initial conditions as well as by Langevin forces) with the corresponding Liouville or the Fokker–Planck equations describing evolution of these uncertainties in terms of probability density. The coupling is implemented by information-based supervising forces which fundamentally change the patterns of probability evolution. It is demonstrated that the probability density can approach prescribed attractors while exhibiting such patterns as shock waves, solitons and chaos in probability space. Applications of these phenomena to information-based neural nets, expectation-based cooperation, self-programmed systems, control chaos using terminal attractors as well as to games with incomplete information, are addressed. A formal similarity between the mathematical structure of the introduced dynamical systems and quantum mechanics is discussed.

Stability of a Class of Real-Time Switched Piezoelectric Shunts

Experimental and anecdotal evidences have shown that state-switched control strategies for piezoelectric actuators can be advantageous. However, most discussions of the stability of these systems has relied on heuristic, or physically motivated, arguments. In this paper, we show that recent open-circuit-short-circuit (OC-SC) state-switching control laws can be viewed as hybrid dynamical systems of Witsenhausen type. Within this framework, the closed-loop stability of OC/SC switching is rigorously established using the method of multiple Lyapunov functions.

Topological dynamical systems of type I

The equivalence of existence of a Borel section to nonexistence of recurrent aperiodic points for homeomorphisms of locally compact topological spaces is proved using the theory of C*-algebras.

DYNAMICS OF CREMONA MAPS FROM PHYSICAL MODELS

A Cremona transformation X=f(x, y), Y=g(x, y) is a rational mapping (meaning that f and g are ratios of polynomials) with rational inverse x=F(X, Y), y=G(X, Y). Discrete dynamical systems defined by such transformations are well studied. They include symmetries of the Yang-Baxter equations and their generalizations. In this paper we comment on two types of dynamical systems based on Cremona transformations. The first is the P1 case of Bellon et al. which pertains to the inversion relation for the matrix of Boltzmann weights of the 4-state chiral Potts model. The resulting dynamical system decouples completely to one in a single variable. The sub case z=x corresponds to the symmetric Ashkin-Teller model. We solve this case explicitly giving orbits as closed formulas in the number n of iterations. The second type of system treated is an extension from the famous example due to McMillan of invariant curves of area preserving maps in two dimensions to the case of invariant curves and surfaces of three dimensional Cremona maps that preserve volume. The trace map of the renormalization of transmission through a Fibonacci chain, first introduced by Kohmoto, Kadanoff and Tang, is considered as an example of such a system.

Analytical approach for simplifying dynamical systems of polynomial type

We present a method to calculate formal symmetries of dynamical systems which are represented by polynomials in n dimensions. The technique is based on Lie transformations for ordinary differential equations. The procedure is illustrated with two applications.

CHAOS AND COMPLEXITY

In this paper we show how to relate a form of high-dimensional complexity to chaotic and other types of dynamical systems. The derivation shows how "near-chaotic" complexity can arise without the presence of homoclinic tangles or positive Lyapunov exponents. The relationship we derive follows from the observation that the elements of invariant finite integer lattices of high-dimensional dynamical systems can, themselves, be viewed as single integers rather than coordinates of a point in n-space. From this observation it is possible to construct high-dimensional dynamical systems which have properties of shifts but for which there is no conventional topological conjugacy to a shift. The particular manner in which the shift appears in high-dimensional dynamical systems suggests that some forms of complexity arise from the presence of chaotic dynamics which are obscured by the large dimensionality of the system domain.

Artificial Reasoners for Human Process Supervision

A methodology for analysing human reasoning when performing a supervision task has been proposed in WAHRPS project (Worlds for assessing human reasoning in process supervision), conducted by INSERM 455 [1]. The main issue is the modelling of different human reasoning styles implemented as Reference Artificial Reasoners (RAR), and the study of decision sequences in front of several dynamic processes. In this paper, an adaptive tool able to generate different types of dynamical systems with increasing difficulty degrees is presented. The design and implementation of a reference artificial reasoner (RAR) based on sequential control, and the implementation of a RAR based on qualitative reasoning is also reported. Finally, a discussion about the comparison between human and artificial decision sequences, in front of the same dynamical process, is included.

Optimal synthesis and stabilization of time-delay dynamical systems

A new approach to the solution of the classical optimal synthesis problem for linear time delay dynamical systems is under consideration. Positional solutions of optimal control problems are constructed by solving special algebraic equations over the optimal Lagrange vector and switching points of the optimal open-loop control at each current moment of a control process. A stabilization problem for two types of dynamical systems with delay (delay in feedback loop, delays in control) is investigated. Algorithms of constructing bounded stabilizing feedbacks based on positional solutions of special auxiliary optimal control problems are justified. Two schemes of realization of stabilizing feedbacks are proposed.

Sporadically Fractal Basin Boundaries of Chaotic Systems

We demonstrate a new type of basin boundary for typical chaotic dynamical systems. For the case of a two dimensional map, this boundary has the character of the graph of a function that is smooth and differentiable except on a set of fractal dimensions less than one. In spite of the basin boundary being smooth ``almost everywhere,'' its fractal dimension exceeds one (implying degradation of one's ability to predict the attractor an orbit approaches in the presence of small initial condition uncertainty). We call such a boundary sporadically fractal.

Response cumulants of nonlinear systems subject to external and multiplicative excitations

A general framework is presented for deriving the differential equations governing the evolution of the response cumulants of linear and nonlinear dynamical systems subjected to external and multiplicative non-Gaussian delta-correlated processes. Significant simplifications of these equations are given based on using appropriate recursive relationships for joint cumulants involving products of one or more variables. A compact form of the equations for the response cumulants is presented which provides insight into the structure of the cumulant equations for specific types of dynamical systems. The procedure developed can easily be implemented in computer software to derive symbolic cumulant equations and to estimate numerically the response cumulants of systems with power-law nonlinearities using approximate cumulant-neglect closure schemes. Comparison between the equations for cumulants and the equations for moments are also presented, with particular emphasis on the advantages and disadvantages of each formulation. Suggestions are given regarding the choice to use cumulant or moment equations for analysing the stochastic response of dynamical systems. The preferred formulation is shown to depend on the type of system analysed (linear or nonlinear), the type of system nonlinearity (polynomial or non-polynomial), and the type of excitation (external or multiplicative, delta-correlated or filtered).

Dynamics of intergestural timing: a perturbation study of lip-larynx coordination.

In this study, downward-directed mechanical perturbations were applied to the lower lip during both repetitive (/...paepaepae.../) and discrete (/pe'saepaepl/) utterances in order to examine the perturbation-induced changes of intergestural timing between syllables (i.e., between the bilabial and laryngeal gestures for successive /p/'s) and within phonemes (i.e., between the bilabial and laryngeal gestures within single /p/'s). Our findings led us to several conclusions. First, steady-state (phase-resetting) analyses of the repetitive utterances indicated both that "permanent" phase shifts existed for both the lips and the larynx after the system returned to its pre-perturbation rhythm and that smaller steady-state shifts occurred in the relative phasing of these gestures. These results support the hypothesis that central intergestural dynamics can be reset by peripheral articulatory events. Such resetting was strongest when the perturbation was delivered within a "sensitive phase" of the cycle, during which the downwardly directed lower-lip perturbation opposed the just-initiated, actively controlled bilabial closing gesture for /p/. Although changes in syllable duration were found for other perturbed phases, these changes were simply transient effects and did not indicate a resetting of the central "clock." Second, analyses of the transient portions of the perturbed cycles of the repetitive utterances indicated that the perturbation-induced steady-state phase shifts are almost totally attributable to changes occurring during the first two perturbed cycles. Finally, the transient changes in speech timing induced by perturbations in the discrete sequences appeared to share a common dynamical basis with the changes to the repetitive sequences. We conclude by speculating on the type of dynamical system that could generate these temporal patterns.

Outer Tangency Bifurcations of Chaotic Sets

We present and explain numerical results illustrating the mechanism of a type of discontinuous bifurcation of a chaotic set that occurs in typical dynamical systems. After the bifurcation, the chaotic set acquires new pieces located at a finite distance from its location just before the bifurcation, and these new pieces were not part of a previously existing chaotic set. A scaling law is given describing the creation of unstable periodic orbits following such a bifurcation. We also provide numerical evidence of such a bifurcation for a nonattracting chaotic set of the H\'enon map.

The transition from deterministic chaos to a stochastic process

Deterministic dynamical systems of Langevin type recently introduced by Beck and Roepsdorf (Physica A 145 (1987) 1) generate Langevin dynamics in a suitable limit. We discuss the transition from chaos to randomness in these systems under the point of view of the attractor dimension and dynamical entropy and show that the limit of a stochastic process is reached via a shift of the deterministic properties towards the infinitesimal length scales.