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.

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.

Mechanical Systems With Impacts: Comparison of Several Numerical Methods

1999 ◽  
Author(s):  
C. H. Lamarque ◽  
O. Janin
Keyword(s):  
Numerical Methods ◽  
Numerical Solution ◽  
Infinite Number ◽  
Mechanical Systems ◽  
Numerical Results ◽  
Degree Of Freedom ◽  
Periodic Behavior ◽  
Numerical Tests ◽  
Runge Kutta ◽  
Theoretical Results

Abstract We study the performances of several numerical methods (Paoli-Schatzman, Newmark, Runge-Kutta) in order to compute periodic behavior of a simple one-degree-of-freedom impacting oscillator. Some theoretical results are given and numerical tests are performed. We compare mathematical and numerical results using our simple example exhibiting either finite or infinite number of impacts per period. Comparison of exact and numerical solution provides a practical order for each scheme. We conclude about the use of the different numerical methods.

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.

The global dynamics of a discrete nonlinear hierarchical population system

2019 ◽  
Vol 12 (02) ◽  
pp. 1950022
Author(s):  
Ze-Rong He ◽  
Huai Chen ◽  
Shu-Ping Wang
Keyword(s):  
Dynamical Systems ◽  
Lyapunov Function ◽  
Population Model ◽  
Numerical Experiments ◽  
Discrete Dynamical Systems ◽  
Reproductive Number ◽  
Global Dynamics ◽  
Population System ◽  
The Stability ◽  
Theoretical Results

This paper is concerned with the global dynamics of a hierarchical population model, in which the fertility of an individual depends on the total number of higher-ranking members. We investigate the stability of equilibria, nonexistence of periodic orbits and the persistence of the population by means of eigenvalues, Lyapunov function, and several results in discrete dynamical systems. Our work demonstrates that the reproductive number governs the evolution of the population. Besides the theoretical results, some numerical experiments are also presented.

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.

NUMERICAL COMPUTATIONS OF CONNECTING ORBITS IN DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS

1996 ◽  
Vol 06 (07) ◽  
pp. 1281-1293 ◽  
Author(s):  
FENGSHAN BAI ◽  
GABRIEL J. LORD ◽  
ALASTAIR SPENCE
Keyword(s):  
Dynamical Systems ◽  
Fixed Points ◽  
Periodic Orbits ◽  
Discrete System ◽  
Numerical Technique ◽  
Autonomous Systems ◽  
Discrete Systems ◽  
Discrete Dynamical Systems ◽  
Continuous Systems ◽  
Connecting Orbits

The aim of this paper is to present a numerical technique for the computation of connections between periodic orbits in nonautonomous and autonomous systems of ordinary differential equations. First, the existence and computation of connecting orbits between fixed points in discrete dynamical systems is discussed; then it is shown that the problem of finding connections between equilibria and periodic solutions in continuous systems may be reduced to finding connections between fixed points in a discrete system. Implementation of the method is considered: the choice of a linear solver is discussed and phase conditions are suggested for the discrete system. The paper concludes with some numerical examples: connections for equilibria and periodic orbits are computed for discrete systems and for nonautonomous and autonomous systems, including systems arising from the discretization of a partial differential equation.

Numerical analysis of two-parameter singularly perturbed boundary value problems via fitted splines

Analysis ◽  
2017 ◽  
Vol 37 (3) ◽  
Author(s):  
A. S. V. Ravi Kanth ◽  
P. Murali Mohan Kumar
Keyword(s):  
Numerical Analysis ◽  
Error Analysis ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Singularly Perturbed ◽  
Spline Functions ◽  
Uniform Mesh ◽  
Theoretical Results ◽  
Two Parameter ◽  
Parametric Cubic Spline

AbstractIn this paper, we consider two-parameter singularly perturbed boundary value problems via fitted splines. We propose an exponentially fitting factor to the highest order derivative of the problem and discretize by parametric cubic spline functions on a uniform mesh. An error analysis of the method is presented. The efficiency of the method is demonstrated by test examples which support the theoretical results.

ON THE DYNAMICS OF SOME NUMERICAL METHODS

2000 ◽  
Vol 11 (07) ◽  
pp. 1481-1487 ◽  
Author(s):  
E. AHMED ◽  
A. S. HEGAZI
Keyword(s):  
Partial Differential Equations ◽  
Dynamical Systems ◽  
Numerical Methods ◽  
Differential Equations ◽  
Reaction Diffusion ◽  
Discrete Dynamical Systems ◽  
Point Of View ◽  
Partial Differential ◽  
Dynamical Behaviors ◽  
Telegraph Equations

From numerical methods point of view of dynamical systems, we have determined dynamical behaviors of the corresponding systems (i.e., chaotic, stable, bifurcations possibility, etc.). New versions of numerical methods are derived and we have compared the dynamical behaviors of the continuous dynamical systems with their corresponding discrete dynamical systems. An application of partial differential equations is given for reaction-diffusion and telegraph equations.

Upon Impact Numerical Modeling of Foam Materials

2017 ◽  
Vol 54 (2) ◽  
pp. 195-202
Author(s):  
Vasile Nastasescu ◽  
Silvia Marzavan
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Modeling ◽  
Numerical Analysis ◽  
Numerical Methods ◽  
Boundary Conditions ◽  
General Character ◽  
Foam Material ◽  
Various Boundary Conditions ◽  
Specific Behaviour

The paper presents some theoretical and practical issues, particularly useful to users of numerical methods, especially finite element method for the behaviour modelling of the foam materials. Given the characteristics of specific behaviour of the foam materials, the requirement which has to be taken into consideration is the compression, inclusive impact with bodies more rigid then a foam material, when this is used alone or in combination with other materials in the form of composite laminated with various boundary conditions. The results and conclusions presented in this paper are the results of our investigations in the field and relates to the use of LS-Dyna program, but many observations, findings and conclusions, have a general character, valid for use of any numerical analysis by FEM programs.

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