scholarly journals Heteroclinic Cycles Imply Chaos and Are Structurally Stable

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Xiaoying Wu
Keyword(s):  
Dynamical Systems ◽  
Structural Stability ◽  
Periodic Points ◽  
Discrete Dynamical Systems ◽  
Symbolic System ◽  
Heteroclinic Cycles ◽  
Novel Method ◽  
The One ◽  
Theoretical Results ◽  
Structurally Stable

This paper is concerned with the chaos of discrete dynamical systems. A new concept of heteroclinic cycles connecting expanding periodic points is raised, and by a novel method, we prove an invariant subsystem is topologically conjugate to the one-side symbolic system. Thus, heteroclinic cycles imply chaos in the sense of Devaney. In addition, if a continuous differential map h has heteroclinic cycles in ℝ n , then g has heteroclinic cycles with h − g C 1 being sufficiently small. The results demonstrate C 1 structural stability of heteroclinic cycles. In the end, two examples are given to illustrate our theoretical results and applications.

Chaotic Dynamics of Partial Difference Equations with Polynomial Maps

2021 ◽  
Vol 31 (09) ◽  
pp. 2150133
Author(s):  
Haihong Guo ◽  
Wei Liang
Keyword(s):  
Dynamical Systems ◽  
Computer Simulations ◽  
Difference Equations ◽  
Chaotic Dynamics ◽  
Discrete Dynamical Systems ◽  
Partial Difference Equations ◽  
Partial Difference ◽  
Polynomial Maps ◽  
Expansion Theory ◽  
Theoretical Results

In this paper, chaotic dynamics of a class of partial difference equations are investigated. With the help of the coupled-expansion theory of general discrete dynamical systems, two chaotification schemes for partial difference equations with polynomial maps are established. These controlled equations are proved to be chaotic either in the sense of Li–Yorke or in the sense of both Li–Yorke and Devaney. One example is provided to illustrate the theoretical results with computer simulations for demonstration.

scholarly journals Scalar field cosmology — geometry of dynamics

2014 ◽  
Vol 11 (02) ◽  
pp. 1460012 ◽  
Author(s):  
Marek Szydłowski ◽  
Orest Hrycyna ◽  
Aleksander Stachowski
Keyword(s):  
Phase Space ◽  
Scalar Field ◽  
Dynamical Systems ◽  
Potential Function ◽  
Structural Stability ◽  
Saddle Points ◽  
Scalar Fields ◽  
Fine Tuning ◽  
Scalar Field Cosmology ◽  
Structurally Stable

We study the Scalar Field Cosmology (SFC) using the geometric language of the phase space. We define and study an ensemble of dynamical systems as a Banach space with a Sobolev metric. The metric in the ensemble is used to measure a distance between different models. We point out the advantages of visualization of dynamics in the phase space. It is investigated the genericity of some class of models in the context of fine tuning of the form of the potential function in the ensemble of SFC. We also study the symmetries of dynamical systems of SFC by searching for their exact solutions. In this context, we stressed the importance of scaling solutions. It is demonstrated that scaling solutions in the phase space are represented by unstable separatrices of the saddle points. Only critical point itself located on two-dimensional stable submanifold can be identified as scaling solution. We have also found a class of potentials of the scalar fields forced by the symmetry of differential equation describing the evolution of the Universe. A class of potentials forced by scaling (homology) symmetries was given. We point out the role of the notion of a structural stability in the context of the problem of indetermination of the potential form of the SFC. We characterize also the class of potentials which reproduces the ΛCDM model, which is known to be structurally stable. We show that the structural stability issue can be effectively used is selection of the scalar field potential function. This enables us to characterize a structurally stable and therefore a generic class of SFC models. We have found a nonempty and dense subset of structurally stable models. We show that these models possess symmetry of homology.

Shadowing and structural stability for operators

2020 ◽  
pp. 1-20
Author(s):  
NILSON C. BERNARDES ◽  
ALI MESSAOUDI
Keyword(s):  
Banach Space ◽  
Dynamical Systems ◽  
Structural Stability ◽  
Invertible Operator ◽  
Hyperbolic Operator ◽  
Weighted Shifts ◽  
Shadowing Property ◽  
Finite Dimensional ◽  
Structurally Stable

A well-known result in the area of dynamical systems asserts that any invertible hyperbolic operator on any Banach space is structurally stable. This result was originally obtained by Hartman in 1960 for operators on finite-dimensional spaces. The general case was independently obtained by Palis and Pugh around 1968. We will exhibit a class of examples of structurally stable operators that are not hyperbolic, thereby showing that the converse of the above-mentioned result is false in general. We will also prove that an invertible operator on a Banach space is hyperbolic if and only if it is expansive and has the shadowing property. Moreover, we will show that if a structurally stable operator is expansive, then it must be uniformly expansive. Finally, we will characterize the weighted shifts on the spaces $c_{0}(\mathbb{Z})$ and $\ell _{p}(\mathbb{Z})$ ( $1\leq p<\infty$ ) that satisfy the shadowing property.

Petri Nets and Discrete Events Systems

2013 ◽  
pp. 231-240 ◽  
Author(s):  
Juan L. G. Guirao ◽  
Fernando L. Pelayo
Keyword(s):  
Dynamical Systems ◽  
Petri Nets ◽  
Discrete Event Systems ◽  
Discrete Event ◽  
Discrete Dynamical Systems ◽  
Discrete Events ◽  
Key Factors ◽  
Discrete Events Systems ◽  
The One ◽  
The Relationship

This paper provides an overview over the relationship between Petri Nets and Discrete Event Systems as they have been proved as key factors in the cognitive processes of perception and memorization. In this sense, different aspects of encoding Petri Nets as Discrete Dynamical Systems that try to advance not only in the problem of reachability but also in the one of describing the periodicity of markings and their similarity, are revised. It is also provided a metric for the case of Non-bounded Petri Nets.

NUMERICAL ANALYSIS OF DEGENERATE CONNECTING ORBITS FOR MAPS

2004 ◽  
Vol 14 (10) ◽  
pp. 3385-3407 ◽  
Author(s):  
WOLF-JÜRGEN BEYN ◽  
THORSTEN HÜLS ◽  
YONGKUI ZOU
Keyword(s):  
Numerical Analysis ◽  
Dynamical Systems ◽  
Numerical Methods ◽  
Error Analysis ◽  
Error Estimates ◽  
Discrete Dynamical Systems ◽  
Numerical Results ◽  
Connecting Orbits ◽  
Theoretical Results ◽  
Degenerate Cases

This paper contains a survey of numerical methods for connecting orbits in discrete dynamical systems. Special emphasis is put on degenerate cases where either the orbit loses transversality or one of its endpoints loses hyperbolicity. Numerical methods that approximate the connecting orbits by finite orbit sequences are described in detail and theoretical results on the error analysis are provided. For most of the degenerate cases we present examples and numerical results that illustrate the applicability of the methods and the validity of the error estimates.

Structurally stable heteroclinic cycles

1988 ◽  
Vol 103 (1) ◽  
pp. 189-192 ◽  
Author(s):  
John Guckenheimer ◽  
Philip Holmes
Keyword(s):  
Dynamical Systems ◽  
Symmetry Group ◽  
Vector Fields ◽  
Equilibrium Points ◽  
Saddle Points ◽  
Finite Subgroup ◽  
Heteroclinic Cycle ◽  
Heteroclinic Cycles ◽  
Open Set ◽  
Structurally Stable

This paper describes a previously undocumented phenomenon in dynamical systems theory; namely, the occurrence of heteroclinic cycles that are structurally stable within the space of Cr vector fields equivariant with respect to a symmetry group. In the space X(M) of Cr vector fields on a manifold M, there is a residual set of vector fields having no trajectories joining saddle points with stable manifolds of the same dimension. Such heteroclinic connections are a structurally unstable phenomenon [4]. However, in the space XG(M) ⊂ X(M) of vector fields equivariant with respect to a symmetry group G, the situation can be quite different. We give an example of an open set U of topologically equivalent vector fields in the space of vector fields on ℝ3 equivariant with respect to a particular finite subgroup G ⊂ O(3) such that each X ∈ U has a heteroclinic cycle that is an attractor. The heteroclinic cycles consist of three equilibrium points and three trajectories joining them.

scholarly journals Chaotification for Partial Difference Equations via Controllers

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Wei Liang ◽  
Yuming Shi ◽  
Zongcheng Li
Keyword(s):  
Dynamical Systems ◽  
Computer Simulations ◽  
Difference Equations ◽  
Discrete Dynamical Systems ◽  
Partial Difference Equations ◽  
Partial Difference ◽  
Theoretical Results

Chaotification problems of partial difference equations are studied. Two chaotification schemes are established by utilizing the snap-back repeller theory of general discrete dynamical systems, and all the systems are proved to be chaotic in the sense of both Li-Yorke and Devaney. An example is provided to illustrate the theoretical results with computer simulations.

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