scholarly journals Twist orbits for non continuous maps of degree one

1993 ◽  
Vol 47 (3) ◽  
pp. 415-426
Author(s):  
Francisco Esquembre
Keyword(s):  
Dynamical Systems ◽  
Rotation Number ◽  
Discrete Dynamical Systems ◽  
Continuous Maps ◽  
Rotation Set

The existence of twist orbits and twist cycles with a given rotation number is considered for discrete dynamical systems generated by iteration of liftings of maps of the circle into itself. The class of maps for which such orbits exist for every number in the interior of the rotation set is extended to contain an important subclass of non-continuous maps.

ROTATION VECTORS FOR TORAL MAPS AND FLOWS: A TUTORIAL

1995 ◽  
Vol 05 (02) ◽  
pp. 321-348 ◽  
Author(s):  
JAMES A. WALSH
Keyword(s):  
Dynamical Systems ◽  
Rotation Number ◽  
Chaotic Behavior ◽  
Invariant Tori ◽  
Mode Locking ◽  
Rotation Vector ◽  
Higher Genus ◽  
Rotation Vectors ◽  
Dissipative Dynamical Systems ◽  
Rotation Set

This paper is an introduction to the concept of rotation vector defined for maps and flows on the m-torus. The rotation vector plays an important role in understanding mode locking and chaos in dissipative dynamical systems, and in understanding the transition from quasiperiodic motion on attracting invariant tori in phase space to chaotic behavior on strange attractors. Throughout this article the connection between the rotation vector and the dynamics of the map or flow is emphasized. We begin with a brief introduction to the dimension one setting, in which case the rotation vector reduces to the well known rotation number of H. Poincaré. A survey of the main results concerning the rotation number and bifurcations of circle maps is presented. The various definitions of rotation vector in the higher dimensional setting are then introduced with emphasis again placed on how certain properties of the rotation set relate to the dynamics of the map or flow. The dramatic differences between results in dimension two and results in higher dimensions are also presented. The tutorial concludes with a brief introduction to extensions of the concept of rotation vector to the setting of dynamical systems defined on surfaces of higher genus.

CHAOTIFICATION OF DISCRETE DYNAMICAL SYSTEMS GOVERNED BY CONTINUOUS MAPS

2005 ◽  
Vol 15 (02) ◽  
pp. 547-555 ◽  
Author(s):  
YUMING SHI ◽  
GUANRONG CHEN
Keyword(s):  
Dynamical Systems ◽  
Feedback Control ◽  
Relevant Literature ◽  
Discrete Dynamical Systems ◽  
One Dimensional ◽  
Control Techniques ◽  
Controlled Systems ◽  
Finite Dimensional ◽  
Higher Dimensional ◽  
Continuous Maps

This paper is concerned with chaotification of discrete dynamical systems in finite-dimensional real spaces, via feedback control techniques. A chaotification theorem for one-dimensional discrete dynamical systems and a chaotification theorem for general higher-dimensional discrete dynamical systems are established, respectively. The controlled systems are proved to be chaotic in the sense of Devaney. In particular, the maps corresponding to the original systems and designed controllers are only required to satisfy some mild assumptions on two very small disjoint closed subsets in the domains of interest. This condition is weaker than those in the existing relevant literature.

ON THE STRUCTURE OF THE ω-LIMIT SETS FOR CONTINUOUS MAPS OF THE INTERVAL

1999 ◽  
Vol 09 (09) ◽  
pp. 1719-1729 ◽  
Author(s):  
LLUÍS ALSEDÀ ◽  
MOIRA CHAS ◽  
JAROSLAV SMÍTAL
Keyword(s):  
Dynamical Systems ◽  
Topological Entropy ◽  
Discrete Dynamical Systems ◽  
Limit Sets ◽  
Interval Maps ◽  
Positive Topological Entropy ◽  
Countable Ordinal ◽  
Continuous Maps

We introduce the notion of the center of a point for discrete dynamical systems and we study its properties for continuous interval maps. It is known that the Birkhoff center of any such map has depth at most 2. Contrary to this, we show that if a map has positive topological entropy then, for any countable ordinal α, there is a point xα∈I such that its center has depth at least α. This improves a result by [Sharkovskii, 1966].

scholarly journals On the origin and development of some notions of entropy

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
Francisco Balibrea
Keyword(s):  
Applied Sciences ◽  
Dynamical Systems ◽  
Topological Entropy ◽  
Discrete Dynamical Systems ◽  
Point Of View ◽  
Topological Dynamics ◽  
Theoretical Point ◽  
Compact Metric Space ◽  
Continuous Maps ◽  
And Control

AbstractDiscrete dynamical systems are given by the pair (X, f ) where X is a compact metric space and f : X → X a continuous maps. During years, a long list of results have appeared to precise and understand what is the complexity of the systems. Among them, one of the most popular is that of topological entropy. In modern applications other conditions on X and f have been considered. For example X can be non-compact or f can be discontinuous (only in a finite number of points and with bounded jumps on the values of f or even non-bounded jumps). Such systems are interesting from theoretical point of view in Topological Dynamics and appear frequently in applied sciences such as Electronics and Control Theory. In this paper we are dealing mainly with the original ideas of entropy in Thermodinamics and their evolution until the appearing in the twenty century of the notions of Shannon and Kolmogorov-Sinai entropies and the subsequent topological entropy. In turn such notions have to evolve to other recent situations where it is necessary to give some extended versions of them adapted to the new problems.

scholarly journals Chaos and stability in some random dynamical systems

2012 ◽  
Vol 51 (1) ◽  
pp. 75-82
Author(s):  
Katarína Janková
Keyword(s):  
Dynamical Systems ◽  
Discrete Dynamical Systems ◽  
Random Dynamical Systems ◽  
Random Perturbations ◽  
Continuous Maps ◽  
Small Random Perturbations

ABSTRACT Nonchaotic behavior in the sense of Li and Yorke chaos in discrete dynamical systems generated by a continuous selfmapping of a real compact inter- val means that every trajectory can be approximated by a periodic one. Stability of this behavior was analyzed also for dynamical systems with small random perturbations. In this paper we study similar properties for nonautonomous periodic dynamical systems with random perturbations and for random dynamical systems generated by two continuous maps and their perturbations.

scholarly journals Hidden Attractors in Discrete Dynamical Systems

Entropy ◽  
2021 ◽  
Vol 23 (5) ◽  
pp. 616
Author(s):  
Marek Berezowski ◽  
Marcin Lawnik
Keyword(s):  
Dynamical Systems ◽  
Chaos Theory ◽  
Scientific Literature ◽  
Discrete Systems ◽  
Discrete Dynamical Systems ◽  
Continuous Systems ◽  
Hidden Attractors ◽  
Negative Effects ◽  
Chemical Reactors ◽  
Points Of View

Research using chaos theory allows for a better understanding of many phenomena modeled by means of dynamical systems. The appearance of chaos in a given process can lead to very negative effects, e.g., in the construction of bridges or in systems based on chemical reactors. This problem is important, especially when in a given dynamic process there are so-called hidden attractors. In the scientific literature, we can find many works that deal with this issue from both the theoretical and practical points of view. The vast majority of these works concern multidimensional continuous systems. Our work shows these attractors in discrete systems. They can occur in Newton’s recursion and in numerical integration.

A STUDY OF BIFURCATIONS IN A CIRCULAR REAL CELLULAR AUTOMATON

1993 ◽  
Vol 03 (02) ◽  
pp. 293-321 ◽  
Author(s):  
JÜRGEN WEITKÄMPER
Keyword(s):  
Hopf Bifurcation ◽  
Dynamical Systems ◽  
Cellular Automata ◽  
Cellular Automaton ◽  
Parallel Computers ◽  
Discrete Dynamical Systems ◽  
Two Dimensional ◽  
Bifurcation Structure ◽  
Logistic Mapping ◽  
Henon Mapping

Real cellular automata (RCA) are time-discrete dynamical systems on ℝN. Like cellular automata they can be obtained from discretizing partial differential equations. Due to their structure RCA are ideally suited to implementation on parallel computers with a large number of processors. In a way similar to the Hénon mapping, the system we consider here embeds the logistic mapping in a system on ℝN, N>1. But in contrast to the Hénon system an RCA in general is not invertible. We present some results about the bifurcation structure of such systems, mostly restricting ourselves, due to the complexity of the problem, to the two-dimensional case. Among others we observe cascades of cusp bifurcations forming generalized crossroad areas and crossroad areas with the flip curves replaced by Hopf bifurcation curves.

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