A YING–YANG THEORY IN NONLINEAR DISCRETE DYNAMICAL SYSTEMS

2010 ◽  
Vol 20 (04) ◽  
pp. 1085-1098 ◽  
Author(s):  
ALBERT C. J. LUO
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Periodic Solutions ◽  
Nonlinear Dynamical Systems ◽  
Discrete Dynamical System ◽  
Discrete Dynamical Systems ◽  
Hénon Map ◽  
Henon Map ◽  
Nonlinear Dynamical ◽  
Iterative Maps

This paper presents a Ying–Yang theory for nonlinear discrete dynamical systems considering both positive and negative iterations of discrete iterative maps. In the existing analysis, the solutions relative to "Yang" in nonlinear dynamical systems are extensively investigated. However, the solutions pertaining to "Ying" in nonlinear dynamical systems are investigated. A set of concepts on "Ying" and "Yang" in discrete dynamical systems are introduced to help one understand the hidden dynamics in nonlinear discrete dynamical systems. Based on the Ying–Yang theory, the periodic and chaotic solutions in nonlinear discrete dynamical system are discussed, and all possible, stable and unstable periodic solutions can be analytically predicted. A discrete dynamical system with the Henon map is investigated, as an example.

On Stable and Unstable Periodic Solutions of N-Dimensional Discrete Dynamical Systems

2009 ◽  
Author(s):  
Albert C. J. Luo ◽  
Yu Guo
Keyword(s):  
Dynamical Systems ◽  
Periodic Solutions ◽  
Discrete System ◽  
Discrete Dynamical System ◽  
Discrete Dynamical Systems ◽  
Hénon Map ◽  
Henon Map ◽  
Discrete Maps ◽  
Mapping Structures ◽  
Bifurcation And Stability

This paper presents a methodology to analytically predict the stable and unstable periodic solutions for n-dimensional discrete dynamical systems. The positive and negative iterative mappings of discrete maps are introduced for the mapping structure of the periodic solutions. The complete bifurcation and stability of the stable and unstable periodic solutions relative to the positive and negative mapping structures are presented. A discrete dynamical system with the Henon map is investigated as an example. The Poincare mapping sections relative to the Neimark bifurcation of periodic solutions are presented, and the chaotic layers for the discrete system with the Henon map are observed.

PARAMETER CHARACTERISTICS FOR STABLE AND UNSTABLE SOLUTIONS IN NONLINEAR DISCRETE DYNAMICAL SYSTEMS

2010 ◽  
Vol 20 (10) ◽  
pp. 3173-3191 ◽  
Author(s):  
ALBERT C. J. LUO ◽  
YU GUO
Keyword(s):  
Dynamical Systems ◽  
Periodic Solutions ◽  
Discrete Systems ◽  
Discrete Dynamical Systems ◽  
Hénon Map ◽  
Henon Map ◽  
Mapping Structures ◽  
Stable Periodic ◽  
Unstable Solutions ◽  
Nonlinear Discrete Systems

This paper studies complete stable and unstable periodic solutions for n-dimensional nonlinear discrete dynamical systems. The positive and negative iterative mappings of discrete systems are used to develop mapping structures of the stable and unstable periodic solutions. The complete bifurcation and stability analysis are presented for the stable and unstable periodic solutions which are based on the positive and negative mapping structures. A comprehensive investigation on the Henon map is carried out for a better understanding of complexity in nonlinear discrete systems. Given is the bifurcation scenario based on positive and negative mappings of the Henon map, and the analytical predictions of the corresponding periodic solutions are achieved. The corresponding eigenvalue analysis of the periodic solutions is presented. The Poincare mapping sections relative to the Neimark bifurcations of periodic solutions are presented. A parameter map for periodic and chaotic solutions is developed. The complete unstable and stable periodic solutions in nonlinear discrete systems are presented for the first time. The results presented in this paper provide a new idea for one to rethink the current existing theories.

Carathe´odory Controllability Criterion for Nonlinear Dynamical Systems

1978 ◽  
Vol 100 (3) ◽  
pp. 209-213 ◽  
Author(s):  
G. Langholz ◽  
M. Sokolov
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Control Theory ◽  
Linear Systems ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical ◽  
Modern Control Theory ◽  
Open Question ◽  
Modern Control

The question of whether a system is controllable or not is of prime importance in modern control theory and has been actively researched in recent years. While it is a solved problem for linear systems, it is still an open question when dealing with bilinear and nonlinear systems. In this paper, a controllability criterion is established based on a theorem by Carathe´odory. By associating a given dynamical system with a certain Pfaffian equation, it is argued that the system is controllable (uncontrollable) if its associated Pfaffian form is nonintegrable (integrable).

scholarly journals Discrete Dynamical Systems: A Brief Survey

2018 ◽  
Vol 14 (1) ◽  
pp. 35-51
Author(s):  
Sara Fernandes ◽  
Carlos Ramos ◽  
Gyan Bahadur Thapa ◽  
Luís Lopes ◽  
Clara Grácio
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Discrete Dynamical System ◽  
Discrete Dynamical Systems ◽  
Complex Numbers ◽  
Survey Paper ◽  
Number Systems ◽  
Mathematical Formalization ◽  
Research Students ◽  
Work Done

 Dynamical system is a mathematical formalization for any fixed rule that is described in time dependent fashion. The time can be measured by either of the number systems - integers, real numbers, complex numbers. A discrete dynamical system is a dynamical system whose state evolves over a state space in discrete time steps according to a fixed rule. This brief survey paper is concerned with the part of the work done by José Sousa Ramos [2] and some of his research students. We present the general theory of discrete dynamical systems and present results from applications to geometry, graph theory and synchronization. Journal of the Institute of Engineering, 2018, 14(1): 35-51

THE ROLE OF CODIMENSION IN DYNAMICAL SYSTEMS

1992 ◽  
Vol 03 (06) ◽  
pp. 1295-1321 ◽  
Author(s):  
JASON A.C. GALLAS
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Period Doubling ◽  
Main Body ◽  
Hénon Map ◽  
Henon Map ◽  
Dynamical Parameters

Isoperiodic diagrams are used to investigate the topology of the codimension space of a representative dynamical system: the Hénon map. The codimension space is reported to be organized in a simple and regular way: instead of “structures-within-structures” it consists of a “structures-parallel-to-structures” sequence of shrimp-shaped isoperiodic islands immersed on a via caotica. The isoperiodic islands consist of a main body of principal periodicity k=1, 2, 3, 4, …, which bifurcates according to a period-doubling route. The Pk=k×2n, n=0, 1, 2, … shrimps are very densely concentrated along a main α-direction, a neighborhood parallel to the line b=−0.583a+1.025, where a and b are the dynamical parameters in Eq. (1). Isoperiodic diagrams allow to interpret and unify some apparently uncorrelated phenomena, such as ‘period-bubbling’, classes of reverse bifurcations and antimonotonicity and to recognize that they are in fact signatures of the complicated way in which period-doubling occurs in higher codimensional systems.

Approximating Lyapunov Exponents of Discrete Dynamical Systems

2019 ◽  
Vol 11 (11) ◽  
pp. 1612-1615
Author(s):  
Wadia Faid Hassan Al-Shameri
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lyapunov Exponents ◽  
Science And Technology ◽  
Discrete Dynamical System ◽  
Discrete Dynamical Systems ◽  
Significant Part ◽  
Matlab Code

Lyapunov exponents play a significant part in revealing and quantifying chaos, which occurs in many areas of science and technology. The purpose of this study was to approximate the Lyapunov exponents for discrete dynamical systems and to present it as a quantifier for inferring and detecting the existence of chaos in those discrete dynamical systems. Finally, the approximation of the Lyapunov exponents for the discrete dynamical system was implemented using the Matlab code listed in the Appendix.

scholarly journals Unfoldings of discrete dynamical systems

1984 ◽  
Vol 4 (3) ◽  
pp. 421-486 ◽  
Author(s):  
Joel W. Robbin
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Conjugacy Class ◽  
Discrete Dynamical System ◽  
Discrete Dynamical Systems ◽  
Topological Conjugacy ◽  
Universal Unfolding ◽  
Moser Iteration

AbstractA universal unfolding of a discrete dynamical system f0 is a manifold F of dynamical systems such that each system g sufficiently near f0 is topologically conjugate to an element f of F with the conjugacy φ and the element f depending continuously on f0. An infinitesimally universal unfolding of f0 is (roughly speaking) a manifold F transversal to the topological conjugacy class of f0. Using Nash-Moser iteration we show infinitesimally universal unfoldings are universal and (in part II) give a class of examples relating to moduli of stability introduced by Palis and De Melo.

Prediction for the Rub-Impact Phenomena in Rotor Systems

2001 ◽  
Author(s):  
Zhiqiang Wu ◽  
Yushu Chen
Keyword(s):  
Dynamical Systems ◽  
Periodic Solutions ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical ◽  
Rotor Systems ◽  
Different Types ◽  
Impact Phenomena ◽  
Direct Guidance

Abstract By the method (Wu, 2001) developed by authors for singularity analyzing of the bifurcation of the periodic solutions in nonlinear dynamical systems with clearance, the bifurcation patterns of non-impact-rub response and a method for predicting rub-impact are given. It is shown that there are much more types of bifurcation patterns when the clearance constraint is take into account. Given their physical meanings of the parameters in practical rotor systems, the resonant periodic solutions of rotor systems consist of 11 different types of bifurcation patterns among of which the following four types are more likely to appear, (1) patterns without impact and jump, (2) jump patterns without impact, (3) impact pattern without jump and (4) patterns with impact and jump. Based on these results, parameter conditions for rub-impact phenomena are derived. These conditions can give more direct guidance to the design of rotor systems. The method proposed here can be used to predict rub-impact phenomena in more complicated rotor systems.

Bekenstein–Hawking entropy for discrete dynamical systems on sites

2019 ◽  
Vol 34 (32) ◽  
pp. 1950265
Author(s):  
Sh. Najmizadeh ◽  
M. Toomanian ◽  
M. R. Molaei ◽  
T. Nasirzade
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Upper Bound ◽  
Discrete Dynamical System ◽  
Discrete Dynamical Systems ◽  
New Class ◽  
A Site

In this paper, we extend the notion of Bekenstein–Hawking entropy for a cover of a site. We deduce a new class of discrete dynamical system on a site and we introduce the Bekenstein–Hawking entropy for each member of it. We present an upper bound for the Bekenstein–Hawking entropy of the iterations of a dynamical system. We define a conjugate relation on the set of dynamical systems on a site and we prove that the Bekenstein–Hawking entropy preserves under this relation. We also prove that the twistor correspondence preserves the Bekenstein–Hawking entropy.

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