Index Evaluation for Dynamical Systems and Its Application to Locating All the Zeros of a Vector Function

1983 ◽  
Vol 50 (4a) ◽  
pp. 858-862 ◽  
Author(s):  
C. S. Hsu ◽  
R. S. Guttalu
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Periodic Solutions ◽  
Vector Function ◽  
Evaluation Method ◽  
Point Mapping ◽  
Strongly Nonlinear ◽  
Index Evaluation

An index evaluation method is discussed in this paper. It can also serve as the basis of a procedure to locate all the zeros of a vector function. An application of the procedure is made to a strongly nonlinear point-mapping dynamical system in order to locate all the periodic solutions of period one and period two, 41 in total number.

A Theory of Index for Point Mapping Dynamical Systems

1980 ◽  
Vol 47 (1) ◽  
pp. 185-190 ◽  
Author(s):  
C. S. Hsu
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Periodic Solutions ◽  
Discrete Time ◽  
Difference Equations ◽  
Vector Fields ◽  
Time Difference ◽  
Point Mapping ◽  
Strongly Nonlinear

Dynamical systems governed by discrete time-difference equations are referred to as point mapping dynamical systems in this paper. Based upon the Poincare´ theory of index for vector fields, a theory of index is established for point mapping dynamical systems. Besides its intrinsic theoretic value, the theory can be used to help search and locate periodic solutions of strongly nonlinear systems.

scholarly journals Periodic Peakon and Smooth Periodic Solutions for KP-MEW(3,2) Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Junning Cai ◽  
Minzhi Wei ◽  
Liping He
Keyword(s):  
Dynamical System ◽  
Periodic Solution ◽  
Dynamical Systems ◽  
Periodic Solutions ◽  
Phase Portrait ◽  
Traveling Wave ◽  
Bifurcation Theory ◽  
Wave System ◽  
Integral Constant ◽  
Straight Line

In this paper, we consider the KP-MEW(3,2) equation by the bifurcation theory of dynamical systems when integral constant is considered. The corresponding traveling wave system is a singular planar dynamical system with one singular straight line. The phase portrait for c < 0 , 0 < c < 1 , and c > 1 is drawn. Exact parametric representations of periodic peakon solutions and smooth periodic solution are presented.

scholarly journals Existence of random invariant periodic curves via random semiuniform ergodic theorem

2016 ◽  
Vol 17 (01) ◽  
pp. 1750007 ◽  
Author(s):  
Kenneth Uda
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Periodic Solutions ◽  
Ergodic Theorem ◽  
Ergodic Theory ◽  
Dissipative Structure ◽  
Random Dynamical Systems ◽  
Random Dynamical System ◽  
Compact Set

We employ an extension of ergodic theory to the random setting to investigate the existence of random periodic solutions of random dynamical systems. Given that a random dynamical system on a cylinder [Formula: see text] has a dissipative structure, we proved that a random invariant compact set can be expressed as a union of finite of number of random periodic curves. The idea in this paper is closely related to the work recently considered by Zhao and Zheng [46].

Bifurcations and Exact Solutions of the Equation of Barotropic FRW Cosmologies

2017 ◽  
Vol 27 (05) ◽  
pp. 1750080 ◽  
Author(s):  
Jibin Li ◽  
Tonghua Zhang
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Exact Solutions ◽  
Periodic Solutions ◽  
Phase Portraits ◽  
Planar Dynamical System ◽  
Heteroclinic Solutions ◽  
Unbounded Solutions ◽  
Level Curves ◽  
Friedmann Robertson Walker

In this paper, we study the equation of barotropic Friedmann–Robertson–Walker cosmologies. By using the method of dynamical systems, we obtain bifurcations of the phase portraits of the corresponding planar dynamical system. Corresponding to different level curves, we derive exact explicit parametric representations of bounded and unbounded solutions, such as periodic solutions, periodic peakon solutions, homoclinic and heteroclinic solutions and compacton solutions.

scholarly journals VII.—On the Adelphic Integral of the Differential Equations of Dynamics

1918 ◽  
Vol 37 ◽  
pp. 95-116 ◽  
Author(s):  
E. T. Whittaker
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Two Degrees Of Freedom ◽  
Image Position ◽  
Kinetic And Potential Energies

§ 1. Ordinary and singular periodic solutions of a dynamical system. — The present paper is concerned with the motion of dynamical systems which possess an integral of energy. To fix ideas, we shall suppose that the system has two degrees of freedom, so that the equations of motion in generalised co-ordinates may be written in Hamilton's formwhere (q1q2) are the generalised co-ordinates, (p1, p2) are the generalised momenta, and where H is a function of (q1, q2, p1, p2) which represents the sum of the kinetic and potential energies.

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.

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

scholarly journals Potential Well in Poincaré Recurrence

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 379
Author(s):  
Miguel Abadi ◽  
Vitor Amorim ◽  
Sandro Gallo
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Stochastic Processes ◽  
Recurrence Time ◽  
Potential Well ◽  
Exponential Approximation ◽  
Mixing Processes ◽  
System Perspective ◽  
Return Times ◽  
Error Terms

From a physical/dynamical system perspective, the potential well represents the proportional mass of points that escape the neighbourhood of a given point. In the last 20 years, several works have shown the importance of this quantity to obtain precise approximations for several recurrence time distributions in mixing stochastic processes and dynamical systems. Besides providing a review of the different scaling factors used in the literature in recurrence times, the present work contributes two new results: (1) For ϕ-mixing and ψ-mixing processes, we give a new exponential approximation for hitting and return times using the potential well as the scaling parameter. The error terms are explicit and sharp. (2) We analyse the uniform positivity of the potential well. Our results apply to processes on countable alphabets and do not assume a complete grammar.

A DYNAMICAL SYSTEM WITH CHAOTIC BEHAVIOR

1989 ◽  
Vol 03 (15) ◽  
pp. 1185-1188 ◽  
Author(s):  
J. SEIMENIS
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Infinite Number ◽  
Chaotic Behavior ◽  
Equations Of Motion ◽  
Hamiltonian Dynamical Systems

We develop a method to find solutions of the equations of motion in Hamiltonian Dynamical Systems. We apply this method to the system [Formula: see text] We study the case a → 0 and we find that in this case the system has an infinite number of period dubling bifurcations.

scholarly journals The Dynamics of Musical Participation

2021 ◽  
pp. 102986492098831
Author(s):  
Andrea Schiavio ◽  
Pieter-Jan Maes ◽  
Dylan van der Schyff
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Systems Theory ◽  
Dynamical Systems Theory ◽  
Coordination Dynamics ◽  
Musical Experience ◽  
The Core ◽  
Interactive Dynamics ◽  
Interdisciplinary Scholarship ◽  
Core Features

In this paper we argue that our comprehension of musical participation—the complex network of interactive dynamics involved in collaborative musical experience—can benefit from an analysis inspired by the existing frameworks of dynamical systems theory and coordination dynamics. These approaches can offer novel theoretical tools to help music researchers describe a number of central aspects of joint musical experience in greater detail, such as prediction, adaptivity, social cohesion, reciprocity, and reward. While most musicians involved in collective forms of musicking already have some familiarity with these terms and their associated experiences, we currently lack an analytical vocabulary to approach them in a more targeted way. To fill this gap, we adopt insights from these frameworks to suggest that musical participation may be advantageously characterized as an open, non-equilibrium, dynamical system. In particular, we suggest that research informed by dynamical systems theory might stimulate new interdisciplinary scholarship at the crossroads of musicology, psychology, philosophy, and cognitive (neuro)science, pointing toward new understandings of the core features of musical participation.

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