AUTOMATA ON FRACTAL SETS OBSERVED IN HYBRID DYNAMICAL SYSTEMS

2008 ◽  
Vol 18 (12) ◽  
pp. 3665-3678 ◽  
Author(s):  
JUN NISHIKAWA ◽  
KAZUTOSHI GOHARA
Keyword(s):  
Dynamical System ◽  
Information Processing ◽  
Dynamical Systems ◽  
Feedback System ◽  
The Other ◽  
Hybrid Dynamical Systems ◽  
Discrete Dynamics ◽  
Hybrid Dynamical System ◽  
Fractal Sets ◽  
Fractal Set

We studied a hybrid dynamical system composed of a higher module with discrete dynamics and a lower module with continuous dynamics. Two typical examples of this system were investigated from the viewpoint of dynamical systems. One example is a nonfeedback system whose higher module stochastically switches inputs to the lower module. The dynamics was characterized by attractive and invariant fractal sets with hierarchical clusters addressed by input sequences. The other example is a feedback system whose higher module switches in response to the states of the lower module at regular intervals. This system converged into various switching attractors that correspond to infinite switching manifolds, which define each feedback control rule at the switching point. We showed that the switching attractors in the feedback system are subsets of the fractal sets in the nonfeedback system. The feedback system can be considered an automaton that generates various sequences from the fractal set by choosing the typical switching manifold. We can control this system by adjusting the switching interval to determine the fractal set as a constraint and by adjusting the switching manifold to select the automaton from the fractal set. This mechanism might be the key to developing information processing that is neither too soft nor too rigid.

CLOSURES OF FRACTAL SETS IN NONLINEAR DYNAMICAL SYSTEMS WITH SWITCHED INPUTS

2001 ◽  
Vol 11 (08) ◽  
pp. 2205-2215 ◽  
Author(s):  
RYOICHI WADA ◽  
KAZUTOSHI GOHARA
Keyword(s):  
Dynamical System ◽  
Phase Space ◽  
Dynamical Systems ◽  
Limit Cycle ◽  
Nonlinear Dynamical Systems ◽  
Duffing Oscillator ◽  
Nonlinear Dynamical ◽  
Fractal Sets ◽  
Fractal Set

This paper studies closures of fractal sets observed in nonlinear dynamical systems excited stochastically by switched inputs. The Duffing oscillator and the forced dumped pendulum are analyzed as examples. The dynamics of the system is characterized by a fractal set in the phase space. We can numerically construct a closure that encloses the fractal set. Furthermore, it is shown that the closure is a limit cycle attractor of a dynamical system defined by the switching manifold.

An Extended Continuation Problem for Bifurcation Analysis in the Presence of Constraints

2010 ◽  
Vol 6 (3) ◽  
Author(s):  
Harry Dankowicz ◽  
Frank Schilder
Keyword(s):  
Dynamical System ◽  
Periodic Orbits ◽  
Bifurcation Analysis ◽  
The Other ◽  
Extended Formulation ◽  
Hybrid Dynamical System ◽  
Families Of Periodic Orbits ◽  
Continuation Problem ◽  
The One ◽  
Additional Constraints

This paper presents an extended formulation of the basic continuation problem for implicitly defined, embedded manifolds in Rn. The formulation is chosen so as to allow for the arbitrary imposition of additional constraints during continuation and the restriction to selective parametrizations of the corresponding higher-codimension solution manifolds. In particular, the formalism is demonstrated to clearly separate between the essential functionality required of core routines in application-oriented continuation packages, on the one hand, and the functionality provided by auxiliary toolboxes that encode classes of continuation problems and user definitions that narrowly focus on a particular problem implementation, on the other hand. Several examples are chosen to illustrate the formalism and its implementation in the recently developed continuation core package COCO and auxiliary toolboxes, including the continuation of families of periodic orbits in a hybrid dynamical system with impacts and friction as well as the detection and constrained continuation of selected degeneracies characteristic of such systems, such as grazing and switching-sliding bifurcations.

Phase Space Reconstruction

2018 ◽  
Author(s):  
Ray Huffaker ◽  
Marco Bittelli ◽  
Rodolfo Rosa
Keyword(s):  
Dynamical System ◽  
Phase Space ◽  
Dynamical Systems ◽  
Statistical Mechanics ◽  
State Space ◽  
Phase Plane ◽  
Material Point ◽  
The Other ◽  
Straight Line ◽  
Mathematical Construction

In this chapter we introduce an important concept concerning the study of both discrete and continuous dynamical systems, the concept of phase space or “state space”. It is an abstract mathematical construction with important applications in statistical mechanics, to represent the time evolution of a dynamical system in geometric shape. This space has as many dimensions as the number of variables needed to define the instantaneous state of the system. For instance, the state of a material point moving on a straight line is defined by its position and velocity at each instant, so that the phase space for this system is a plane in which one axis is the position and the other one the velocity. In this case, the phase space is also called “phase plane”. It is later applied in many chapters of the book.

Bifurcation Analysis of a Microactuator Using a New Toolbox for Continuation of Hybrid System Trajectories

2008 ◽  
Vol 4 (1) ◽  
Author(s):  
Wonmo Kang ◽  
Phanikrishna Thota ◽  
Bryan Wilcox ◽  
Harry Dankowicz
Keyword(s):  
Dynamical System ◽  
Bifurcation Analysis ◽  
Period Doubling ◽  
Hybrid Dynamical Systems ◽  
Hybrid Dynamical System ◽  
Multiple Impacts ◽  
Grazing Bifurcation ◽  
Periodic Trajectories ◽  
Time Dynamics ◽  
Saddle Node

This paper presents the application of a newly developed computational toolbox, TC-HAT(TCˆ), for bifurcation analysis of systems in which continuous-in-time dynamics are interrupted by discrete-in-time events, here referred to as hybrid dynamical systems. In particular, new results pertaining to the dynamic behavior of a sample hybrid dynamical system, an impact microactuator, are obtained using this software program. Here, periodic trajectories of the actuator with single or multiple impacts per period and associated saddle-node, period-doubling, and grazing bifurcation curves are documented. The analysis confirms previous analytical results regarding the presence of co-dimension-two grazing bifurcation points from which saddle-node and period-doubling bifurcation curves emanate.

scholarly journals Using Hybrid Automata for Diagnosis of Hybrid Dynamical Systems

2015 ◽  
Vol 5 (6) ◽  
pp. 1396
Author(s):  
Lotfi Mhamdi ◽  
Lobna Belkacem ◽  
Hedi Dhouibi ◽  
Zineb Simeu Abazi
Keyword(s):  
Dynamical Systems ◽  
Discrete Model ◽  
Continuous Model ◽  
Dynamic Evolution ◽  
Hybrid Dynamical Systems ◽  
Discrete Dynamics ◽  
Hybrid Automata ◽  
Hybrid Automaton ◽  
Physical Systems

Physical systems can fail. For this reason the problem of identifying and reacting to faults has received a large attention in the control and computer science communities. In this paper we study the fault diagnosis problem and modeling of Hybrid Dynamical Systems (HDS). Generally speaking, HDS is a system mixing continuous and discrete behaviors that cannot be faithfully modeled neither by using formalism with continuous dynamics only nor by a formalism including only discrete dynamics. We use the well known framework of hybrid automata for modeling hybrid systems, because they combine the continous and discretes parts on the same structure. Hybrid automaton is a states-transitions graph, whose dynamic evolution is represented by discretes and continous steps alternations, also, continous evolution happens in the automaton apexes, while discrete evolution is realized by transitions crossing (arcs) of the graph. Their simulation presents many problems mainly the synchronisation between the two models. Stateflow, used to describe the discrete model, is co-ordinated with Matlab, used to describe the continuous model. This article is a description of a case study, which is a two tanks system.

scholarly journals Non-smooth Modeling and Variable Structure Control of a Class of Hybrid Dynamical Systems

2021 ◽  
Vol 2090 (1) ◽  
pp. 012108
Author(s):  
Yasser A. Bin Salamah
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Feedback Linearization ◽  
Controller Design ◽  
Sliding Mode ◽  
Variable Structure ◽  
Hybrid Dynamical Systems ◽  
Structure Control ◽  
Output Tracking ◽  
Output Tracking Control

Abstract In this work, we propose a modeling formulation and controller design for a class of hybrid dynamical systems. In this formulation, a switching dynamical system is modeled as a dynamical system with discontinuous right hand side. More specifically, the system is transformed to a nonlinear system with discontinuous nonlinearities. Then, a synthesis of feedback linearization and sliding mode control is employed for output tracking control problem. Application and implementation of this approach is illustrated via a chemical process example.

scholarly journals Slow Invariant Manifolds of Slow–Fast Dynamical Systems

2021 ◽  
Vol 31 (07) ◽  
pp. 2150112
Author(s):  
Jean-Marc Ginoux
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Singular Perturbation ◽  
Invariant Manifolds ◽  
The Other ◽  
Singularly Perturbed ◽  
Van Der Pol ◽  
Flow Curvature ◽  
The One

Slow–fast dynamical systems, i.e. singularly or nonsingularly perturbed dynamical systems possess slow invariant manifolds on which trajectories evolve slowly. Since the last century various methods have been developed for approximating their equations. This paper aims, on the one hand, to propose a classification of the most important of them into two great categories: singular perturbation-based methods and curvature-based methods, and on the other hand, to prove the equivalence between any methods belonging to the same category and between the two categories. Then, a deep analysis and comparison between each of these methods enable to state the efficiency of the Flow Curvature Method which is exemplified with paradigmatic Van der Pol singularly perturbed dynamical system and Lorenz slow–fast dynamical system.

Bifurcation Analysis of a Microactuator Using a New Toolbox for Continuation of Hybrid System Trajectories

2007 ◽  
Author(s):  
Wonmo Kang ◽  
Bryan Wilcox ◽  
Harry Dankowicz ◽  
Phanikrishna Thota
Keyword(s):  
Dynamical System ◽  
Bifurcation Analysis ◽  
Period Doubling ◽  
Hybrid Dynamical Systems ◽  
Hybrid Dynamical System ◽  
Multiple Impacts ◽  
Grazing Bifurcation ◽  
Periodic Trajectories ◽  
Time Dynamics ◽  
Saddle Node

This paper presents the application of a newly developed computational toolbox, TC-HAT (TCˆ), for bifurcation analysis of systems in which continuous-in-time dynamics are interrupted by discrete-in-time events, here referred to as hybrid dynamical systems. In particular, new results pertaining to the dynamic behavior of an example hybrid dynamical system, an impact microactuator, are obtained using this software program. Here, periodic trajectories of the actuator with single or multiple impacts per period and associated saddle-node, perioddoubling, and grazing bifurcation curves are documented. The analysis confirms previous analytical results regarding the presence of co-dimension-two grazing bifurcation points from which saddle-node and period-doubling bifurcation curves emanate.

Pure Discrete Spectrum for One-dimensional Substitution Systems of Pisot Type

2002 ◽  
Vol 45 (4) ◽  
pp. 697-710 ◽  
Author(s):  
V. F. Sirvent ◽  
B. Solomyak
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Discrete Spectrum ◽  
Sequence Space ◽  
Arbitrary Number ◽  
The Other ◽  
Suspension Flow ◽  
Partial Result ◽  
One Dimensional ◽  
Symbol Substitution

AbstractWe consider two dynamical systems associated with a substitution of Pisot type: the usual -action on a sequence space, and the -action, which can be defined as a tiling dynamical system or as a suspension flow. We describe procedures for checking when these systems have pure discrete spectrum (the “balanced pairs algorithm” and the “overlap algorithm”) and study the relation between them. In particular, we show that pure discrete spectrum for the -action implies pure discrete spectrum for the -action, and obtain a partial result in the other direction. As a corollary, we prove pure discrete spectrum for every -action associated with a two-symbol substitution of Pisot type (this is conjectured for an arbitrary number of symbols).

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