Existence and stability of nontrivial periodic solutions of periodically forced discrete dynamical Systems

1996 ◽  
Vol 2 (3) ◽  
pp. 315-331 ◽  
Author(s):  
Shandelle M. Henson
Keyword(s):  
Dynamical Systems ◽  
Periodic Solutions ◽  
Discrete Dynamical Systems ◽  
Existence And Stability ◽  
Nontrivial Periodic Solutions

scholarly journals On the affine-periodic solutions of discrete dynamical systems

2018 ◽  
Vol 42 (5) ◽  
pp. 2260-2269
Author(s):  
Halis Can KOYUNCUOĞLU ◽  
Murat ADIVAR
Keyword(s):  
Dynamical Systems ◽  
Periodic Solutions ◽  
Discrete Dynamical Systems

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.

scholarly journals AFFINE-PERIODIC SOLUTIONS FOR DISCRETE DYNAMICAL SYSTEMS

2015 ◽  
Vol 5 (4) ◽  
pp. 781-792
Author(s):  
Xin Meng ◽  
◽  
Yong Li ◽  
◽  
◽  
...  
Keyword(s):  
Dynamical Systems ◽  
Periodic Solutions ◽  
Discrete Dynamical Systems

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.

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.

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.

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