Clustering of periodic orbits of a hyperbolic automorphism on the torus T2

2021 ◽  
Vol 15 (1) ◽  
pp. 51-60
Author(s):  
Minh Hien Huynh ◽  
◽  
Van Nam Vo ◽  
Tinh Le ◽  
Thi Dai Trang Nguyen
Keyword(s):  
Dynamical System ◽  
Periodic Orbits ◽  
Toral Automorphism ◽  
Number Of Clusters ◽  
Periodic Sequences ◽  
Axiom A ◽  
Hyperbolic Automorphism ◽  
Hyperbolic Toral Automorphism ◽  
Symbolic Dynamical System

This paper deals with clustering of periodic orbits of the hyperbolic toral automorphism induced by matrix A. We prove that Ta satisfies the Axiom A. The clustering of periodic orbits of Ta is ivestigated via the notion of 'p-closeness' of periodic sequences of the respective symbolic dynamical system. We also provide the number of clusters of periodic sequences with given periods in the case of 2-closeness.

On the Potential for Entangled States Between Chaotic Systems

2014 ◽  
Vol 24 (06) ◽  
pp. 1450077 ◽  
Author(s):  
Matthew A. Morena ◽  
Kevin M. Short
Keyword(s):  
Dynamical System ◽  
Periodic Orbits ◽  
Chaotic Systems ◽  
Entangled States ◽  
Future Research ◽  
Unstable Periodic Orbits ◽  
Qualitative Information ◽  
Chaotic Dynamical System ◽  
Naturally Occurring ◽  
External Intervention

We report on the tendency of chaotic systems to be controlled onto their unstable periodic orbits in such a way that these orbits are stabilized. The resulting orbits are known as cupolets and collectively provide a rich source of qualitative information on the associated chaotic dynamical system. We show that pairs of interacting cupolets may be induced into a state of mutually sustained stabilization that requires no external intervention in order to be maintained and is thus considered bound or entangled. A number of properties of this sort of entanglement are discussed. For instance, should the interaction be disturbed, then the chaotic entanglement would be broken. Based on certain properties of chaotic systems and on examples which we present, there is further potential for chaotic entanglement to be naturally occurring. A discussion of this and of the implications of chaotic entanglement in future research investigations is also presented.

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.

Geodesic laminations as geometric realizations of Pisot substitutions

2000 ◽  
Vol 20 (4) ◽  
pp. 1253-1266 ◽  
Author(s):  
VÍCTOR F. SIRVENT
Keyword(s):  
Dynamical System ◽  
Geodesic Lamination ◽  
Geodesic Laminations ◽  
Symbolic Dynamical System ◽  
Geometrical Realization

We construct a geodesic lamination on the hyperbolic disk and a dynamical system defined on this lamination. We prove that this dynamical system is a geometrical realization of the symbolic dynamical system that arises from the following Pisot substitution: $1\rightarrow 12, \dotsc, (n-1) \rightarrow 1n, n\rightarrow 1$.

On a One-Parameter Discrete-Time Z4-Equivariant Cubic Dynamical System

2019 ◽  
Vol 29 (04) ◽  
pp. 1950052
Author(s):  
Armands Gritsans
Keyword(s):  
Dynamical System ◽  
Periodic Orbits ◽  
Discrete Time ◽  
Chaotic Behavior ◽  
Invariant Curves ◽  
Discrete Time System ◽  
Time System ◽  
Multiple Periodic Orbits ◽  
The One ◽  
Planar Hamiltonian System

The lemniscate sine and cosine are solutions of a [Formula: see text]-equivariant planar Hamiltonian system for all of which nontrivial solutions are nonhyperbolic periodic orbits. The forward Euler scheme is applied to this system and the one-parameter discrete-time [Formula: see text]-equivariant cubic dynamical system is obtained. The discrete-time system depending upon a parameter exhibits rich dynamics: numerical simulation shows that the system has attracting closed invariant curves, multiple periodic orbits and attracting sets exhibiting chaotic behavior. The approximating system of ordinary differential equations is constructed. We discuss the existence of closed invariant curves for the discrete-time system.

scholarly journals The orbit of a Hölder continuous path under a hyperbolic toral automorphism

1983 ◽  
Vol 3 (3) ◽  
pp. 345-349 ◽  
Author(s):  
M. C. Irwin
Keyword(s):  
Orbit Closure ◽  
Continuous Path ◽  
Toral Automorphism ◽  
Hölder Continuous ◽  
Linear Automorphism ◽  
Hyperbolic Toral Automorphism

AbstractLet f:T3→T3 be a hyperbolic toral automorphism lifting to a linear automorphism with real eigenvalues. We prove that there is a Hölder continuous path in T3 whose orbit-closure is 1-dimensional. This strengthens results of Hancock and Przytycki concerning continuous paths, and contrasts with results of Franks and Mañé concerning rectifiable paths.

scholarly journals PERIODIC ORBITS OF A DYNAMICAL SYSTEM RELATED TO A KNOT

2011 ◽  
Vol 20 (03) ◽  
pp. 411-426 ◽  
Author(s):  
LILYA LYUBICH
Keyword(s):  
Dynamical System ◽  
Finite Group ◽  
Finite Type ◽  
Periodic Orbits ◽  
Weak Topology ◽  
Commutator Subgroup ◽  
Alexander Polynomial ◽  
Shift Of Finite Type ◽  
A Finite Group

Following [6] we consider a knot group G, its commutator subgroup K = [G, G], a finite group Σ and the space Hom (K, Σ) of all representations ρ : K → Σ, endowed with the weak topology. We choose a meridian x ∈ G of the knot and consider the homeomorphism σx of Hom (K, Σ) onto itself: σxρ(a) = ρ(xax-1) ∀ a ∈ K, ρ ∈ Hom (K, Σ). As proven in [5], the dynamical system ( Hom (K, Σ), σx) is a shift of finite type. In the case when Σ is abelian, Hom (K, Σ) is finite. In this paper we calculate the periods of orbits of ( Hom (K, ℤ/p), σx), where p is prime, in terms of the roots of the Alexander polynomial of the knot. In the case of two-bridge knots we give a complete description of the set of periods.

Exact periodic orbits in an asymmetric dynamical system

1984 ◽  
Vol 41 (7) ◽  
pp. 234-236
Author(s):  
N. Caranicolas
Keyword(s):  
Dynamical System ◽  
Periodic Orbits
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