NONLINEAR DYNAMICS AND CHAOS PART I: A GEOMETRICAL APPROACH

1998 ◽  
Vol 2 (4) ◽  
pp. 505-532 ◽  
Author(s):  
Alfredo Medio
Keyword(s):  
Nonlinear Dynamics ◽  
Dynamical Systems ◽  
Characteristic Exponent ◽  
Chaotic Attractors ◽  
Modern Theory ◽  
Geometrical Approach ◽  
Lyapunov Characteristic Exponent ◽  
Geometrical Properties ◽  
Nonlinear Dynamics And Chaos ◽  
Different Types

This paper is the first part of a two-part survey reviewing some basic concepts and methods of the modern theory of dynamical systems. The survey is introduced by a preliminary discussion of the relevance of nonlinear dynamics and chaos for economics. We then discuss the dynamic behavior of nonlinear systems of difference and differential equations such as those commonly employed in the analysis of economically motivated models. Part I of the survey focuses on the geometrical properties of orbits. In particular, we discuss the notion of attractor and the different types of attractors generated by discrete- and continuous-time dynamical systems, such as fixed and periodic points, limit cycles, quasiperiodic and chaotic attractors. The notions of (noninteger) fractal dimension and Lyapunov characteristic exponent also are explained, as well as the main routes to chaos.

Time Domain Approaches to the Stability Analysis of Flexible Dynamical Systems

2015 ◽  
Vol 11 (4) ◽  
Author(s):  
Jielong Wang ◽  
Xiaowen Shan ◽  
Bin Wu ◽  
Olivier A. Bauchau
Keyword(s):  
Stability Analysis ◽  
Dynamical Systems ◽  
Time Domain ◽  
Floquet Theory ◽  
Characteristic Exponent ◽  
Periodic Coefficient ◽  
Theoretical Background ◽  
System Stability ◽  
Lyapunov Characteristic Exponent ◽  
The Stability

This paper presents two approaches to the stability analysis of flexible dynamical systems in the time domain. The first is based on the partial Floquet theory and proceeds in three steps. A preprocessing step evaluates optimized signals based on the proper orthogonal decomposition (POD) method. Next, the system stability characteristics are obtained from partial Floquet theory through singular value decomposition (SVD). Finally, a postprocessing step assesses the accuracy of the identified stability characteristics. The Lyapunov characteristic exponent (LCE) theory provides the theoretical background for the second approach. It is shown that the system stability characteristics are related to the LCE closely, for both constant and periodic coefficient systems. For the latter systems, an exponential approximation is proposed to evaluate the transition matrix. Numerical simulations show that the proposed approaches are robust enough to deal with the stability analysis of flexible dynamical systems and the predictions of the two approaches are found to be in close agreement.

Three Types of Chaotic Attractors in 3 D Maps

1993 ◽  
Vol 48 (5-6) ◽  
pp. 666-668 ◽  
Author(s):  
Michael Klein ◽  
Achim Kittel ◽  
Gerold Baier
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Three Dimensions ◽  
Chaotic Attractors ◽  
Stable Fixed Point ◽  
Fractal Structures ◽  
One Dimensional ◽  
Different Types

Abstract Coupling a one-dimensional chaotic forcing to a stable fixed point in the plane may generate different fractal attractors embedded in three dimensions. The system with real eigenvalues of the fixed point gives rise to simple chaotic attractors with three different types of fractal structures. We show that the competition of local exponents provides a generic criterion for the classification of the fractal structures in dynamical systems.

Restricting Chaos to a Small Region in the Phase Space

1998 ◽  
Vol 08 (02) ◽  
pp. 401-407
Author(s):  
Zhihua Wu ◽  
Zhaoxuan Zhu ◽  
Chengfu Zhang
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Computer Simulations ◽  
Characteristic Exponent ◽  
Small Region ◽  
Dissipative Systems ◽  
Simple Method ◽  
Controlled System ◽  
Lyapunov Characteristic Exponent ◽  
Uncontrolled System

The idea of restricting chaos in dissipative systems to a small region in the phase space is proposed. The possibility of realization of this idea is demonstrated by applying a simple method summed up from computer simulations successfully to three different dynamical systems. It is found that not only does the trajectory of the controlled system occupy a region smaller than that of the uncontrolled chaotic system, the corresponding attractor of the Poincaré map is also smaller than that of the uncontrolled system. In addition, but also the maximum Lyapunov characteristic exponent of the system is greatly lowered.

NONLINEAR DYNAMICS AND CHAOS PART II: ERGODIC APPROACH

1999 ◽  
Vol 3 (1) ◽  
pp. 84-114 ◽  
Author(s):  
Alfredo Medio
Keyword(s):  
Dynamical Systems ◽  
Chaotic Systems ◽  
Nonlinear Dynamical Systems ◽  
Measure Theory ◽  
Modern Theory ◽  
Physical Measure ◽  
Nonlinear Dynamical ◽  
Nonlinear Dynamics And Chaos ◽  
Ergodic Measures ◽  
Theoretical Apparatus

This is the second part of a two-part survey of the modern theory of nonlinear dynamical systems. We focus on the study of statistical properties of orbits generated by maps, a field of research known as ergodic theory. After introducing some basic concepts of measure theory, we discuss the notions of invariant and ergodic measures and provide examples of economic applications. The question of attractiveness and observability, already considered in Part I, is revisited and the concept of natural, or physical, measure is explained. This theoretical apparatus then is applied to the question of predictability of dynamical systems, and the notion of metric entropy is discussed. Finally, we consider the class of Bernoulli dynamical systems and discuss the possibility of distinguishing orbits of deterministic chaotic systems and realizations of stochastic processes.

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