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.

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 Wavelet Based Procedure to Study Vibrations and Stability

1999 ◽  
Author(s):  
S. Pernot ◽  
C. H. Lamarque
Keyword(s):  
Stability Analysis ◽  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Periodic Solutions ◽  
Time Varying ◽  
Differential Systems ◽  
Varying Coefficients ◽  
Linear Differential Systems ◽  
Time Varying Coefficients ◽  
Linear Differential

Abstract A Wavelet-Galerkin procedure is introduced in order to obtain periodic solutions of multidegrees-of-freedom dynamical systems with periodic time-varying coefficients. The procedure is then used to study the vibrations of parametrically excited mechanical systems. As problems of stability analysis of nonlinear systems are often reduced after linearization to problems involving linear differential systems with time-varying coefficients, we demonstrate the method provides efficient practical computations of Floquet exponents and consequently allows to give estimators for stability/instability levels. A few academic examples illustrate the relevance of the method.

scholarly journals Periodic Solutions for a System of Difference Equations

2009 ◽  
Vol 2009 ◽  
pp. 1-9 ◽  
Author(s):  
Shugui Kang ◽  
Bao Shi
Keyword(s):  
Nonlinear Systems ◽  
Critical Point ◽  
Periodic Solutions ◽  
Difference Equations ◽  
Critical Point Theory ◽  
Existence Theorems ◽  
Point Theory ◽  
Second Order ◽  
System Of Difference Equations

This paper deals with the second-order nonlinear systems of difference equations, we obtain the existence theorems of periodic solutions. The theorems are proved by using critical point theory.

ON DYNAMICAL SYSTEMS GENERATED BY TWO ALTERNATING VECTOR FIELDS

1996 ◽  
Vol 06 (11) ◽  
pp. 2015-2030 ◽  
Author(s):  
A. KLÍČ ◽  
P. POKORNÝ
Keyword(s):  
Dynamical Systems ◽  
Fixed Points ◽  
Periodic Solutions ◽  
Time Evolution ◽  
Autonomous System ◽  
Vector Fields ◽  
Numerical Study ◽  
Method Of Averaging ◽  
Controlling Of Chaos

Dynamical systems with time evolution determined by two alternating vector fields are investigated both analytically and numerically. When the two vector fields are related by an involutory diffeomorphism G then the fixed points of G (either isolated or non-isolated) are shown to give rise to branches of periodic solutions of the resulting non-autonomous system. The method of averaging is used for small switching periods. Detailed numerical study of both conservative (“blinking vortex”) and dissipative (“blinking nodes”, “blinking cycles” and “blinking Lorenz”) systems shows that the technique of blinking can be used to initiating and controlling of chaos.

Hybrid Dynamical Systems

2012 ◽  
Author(s):  
Rafal Goebel ◽  
Ricardo G. Sanfelice ◽  
Andrew R. Teel
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Asymptotic Stability ◽  
Discrete Time ◽  
Mathematical Analysis ◽  
Continuous Time ◽  
Hybrid Control ◽  
Control Algorithms ◽  
Hybrid Dynamical Systems ◽  
Approximation Techniques

Hybrid dynamical systems exhibit continuous and instantaneous changes, having features of continuous-time and discrete-time dynamical systems. Filled with a wealth of examples to illustrate concepts, this book presents a complete theory of robust asymptotic stability for hybrid dynamical systems that is applicable to the design of hybrid control algorithms—algorithms that feature logic, timers, or combinations of digital and analog components. With the tools of modern mathematical analysis, this book unifies and generalizes earlier developments in continuous-time and discrete-time nonlinear systems. It presents hybrid system versions of the necessary and sufficient Lyapunov conditions for asymptotic stability, invariance principles, and approximation techniques, and examines the robustness of asymptotic stability, motivated by the goal of designing robust hybrid control algorithms. This self-contained and classroom-tested book requires standard background in mathematical analysis and differential equations or nonlinear systems. It will interest graduate students in engineering as well as students and researchers in control, computer science, and mathematics.

Complex Networks Approach for Dynamical Characterization of Nonlinear Systems

2019 ◽  
Vol 29 (13) ◽  
pp. 1950188 ◽  
Author(s):  
Vander L. S. Freitas ◽  
Juliana C. Lacerda ◽  
Elbert E. N. Macau
Keyword(s):  
Time Series ◽  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Complex Networks ◽  
Discrete Time ◽  
Betweenness Centrality ◽  
Bifurcation Diagrams ◽  
Computationally Expensive ◽  
Nonlinear Maps

Bifurcation diagrams and Lyapunov exponents are the main tools for dynamical systems characterization. However, they are often computationally expensive and complex to calculate. We present two approaches for dynamical characterization of nonlinear systems via the generation of an undirected complex network that is built from their time series. Periodic windows and chaos can be detected by analyzing network statistics like average degree, density and betweenness centrality. Results are assessed in two discrete time nonlinear maps.

Analysis of Nonlinear Systems by Truncated Point Mappings

1989 ◽  
Vol 42 (11S) ◽  
pp. S83-S92 ◽  
Author(s):  
Ramesh S. Guttalu ◽  
Henryk Flashner
Keyword(s):  
Nonlinear Systems ◽  
Periodic Solutions ◽  
Parametric Analysis ◽  
Analytical Techniques ◽  
Mapping Method ◽  
Order Of Approximation ◽  
Van Der Pol ◽  
Point Mapping ◽  
Mathieu’S Equation ◽  
Van Der Pol Oscillators

This paper summarizes results obtained by the authors regarding the utility of truncated point mappings which have been recently published in a series of papers. The method described here is applicable to the analysis of multidimensional, multiparameter, periodic nonlinear systems by means of truncated point mappings. Based on multinomial truncation, an explicit analytical expression is determined for the point mapping in terms of the states and parameters of the system to any order of approximation. By combining this approach with analytical techniques, such as the perturbation method employed here, we obtain a powerful tool for finding periodic solutions and for analyzing their stability. The versatility of truncated point mapping method is demonstrated by applying it to study the limit cycles of van der Pol and coupled van der Pol oscillators, the periodic solutions of the forced Duffing’s equation and for a parametric analysis of periodic solutions of Mathieu’s equation.

Sign in / Sign up

Export Citation Format

Share Document