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 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.

scholarly journals Information Gain in Event Space Reflects Chance and Necessity Components of an Event

Information ◽  
2019 ◽  
Vol 10 (11) ◽  
pp. 358 ◽  
Author(s):  
Georg F. Weber
Keyword(s):  
Phase Space ◽  
Ergodic Theorem ◽  
Information Gain ◽  
Characteristic Exponent ◽  
Vector Spaces ◽  
Event Space ◽  
Lyapunov Characteristic Exponent ◽  
Multiplicative Ergodic Theorem ◽  
Non Linear Systems ◽  
Eigenvectors And Eigenvalues

Information flow for occurrences in phase space can be assessed through the application of the Lyapunov characteristic exponent (multiplicative ergodic theorem), which is positive for non-linear systems that act as information sources and is negative for events that constitute information sinks. Attempts to unify the reversible descriptions of dynamics with the irreversible descriptions of thermodynamics have replaced phase space models with event space models. The introduction of operators for time and entropy in lieu of traditional trajectories has consequently limited—to eigenvectors and eigenvalues—the extent of knowable details about systems governed by such depictions. In this setting, a modified Lyapunov characteristic exponent for vector spaces can be used as a descriptor for the evolution of information, which is reflective of the associated extent of undetermined features. This novel application of the multiplicative ergodic theorem leads directly to the formulation of a dimension that is a measure for the information gain attributable to the occurrence. Thus, it provides a readout for the magnitudes of chance and necessity that contribute to an event. Related algorithms express a unification of information content, degree of randomness, and complexity (fractal dimension) in event space.

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.

scholarly journals ANALYSIS OF ROUND OFF ERRORS WITH REVERSIBILITY TEST AS A DYNAMICAL INDICATOR

2012 ◽  
Vol 22 (09) ◽  
pp. 1250215 ◽  
Author(s):  
DAVIDE FARANDA ◽  
MARTÍN FEDERICO MESTRE ◽  
GIORGIO TURCHETTI
Keyword(s):  
Phase Space ◽  
Initial Conditions ◽  
Characteristic Exponent ◽  
Random Perturbations ◽  
Symplectic Maps ◽  
Lyapunov Characteristic Exponent ◽  
The Mean ◽  
Dynamical Properties ◽  
Discrete Time Dynamical Systems ◽  
Number Of Iterations

We compare the divergence of orbits and the reversibility error for discrete time dynamical systems. These two quantities are used to explore the behavior of the global error induced by round off in the computation of orbits. The similarity of results found for any system we have analyzed suggests the use of the reversibility error, whose computation is straightforward since it does not require the knowledge of the exact orbit, as a dynamical indicator. The statistics of fluctuations induced by round off for an ensemble of initial conditions has been compared with the results obtained in the case of random perturbations. Significant differences are observed in the case of regular orbits due to the correlations of round off error, whereas the results obtained for the chaotic case are nearly the same. Both the reversibility error and the orbit divergence computed for the same number of iterations on the whole phase space provide an insight on the local dynamical properties with a detail comparable with other dynamical indicators based on variational methods such as the finite time maximum Lyapunov characteristic exponent, the mean exponential growth factor of nearby orbits and the smaller alignment index. For 2D symplectic maps, the differentiation between regular and chaotic regions is well full-filled. For 4D symplectic maps, the structure of the resonance web as well as the nearby weakly chaotic regions are accurately described.

Generalized canonical quantization of dynamical systems with constraints and curved phase space

1990 ◽  
Vol 332 (3) ◽  
pp. 723-736 ◽  
Author(s):  
I.A. Batalin ◽  
E.S. Fradkin ◽  
T.E. Fradkina
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Canonical Quantization

A Dynamical Systems Approach to Failure Prognosis

2004 ◽  
Vol 126 (1) ◽  
pp. 2-8 ◽  
Author(s):  
David Chelidze ◽  
Joseph P. Cusumano
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Damage Evolution ◽  
Systems Approach ◽  
Remaining Useful Life ◽  
Electromechanical System ◽  
Time To Failure ◽  
Damage Tracking ◽  
Failure Prognosis ◽  
Short Time

In this paper, a previously published damage tracking method is extended to provide failure prognosis, and applied experimentally to an electromechanical system with a failing supply battery. The method is based on a dynamical systems approach to the problem of damage evolution. In this approach, damage processes are viewed as occurring in a hierarchical dynamical system consisting of a “fast,” directly observable subsystem coupled to a “slow,” hidden subsystem describing damage evolution. Damage tracking is achieved using a two-time-scale modeling strategy based on phase space reconstruction. Using the reconstructed phase space of the reference (undamaged) system, short-time predictive models are constructed. Fast-time data from later stages of damage evolution of a given system are collected and used to estimate a tracking function by calculating the short time reference model prediction error. In this paper, the tracking metric based on these estimates is used as an input to a nonlinear recursive filter, the output of which provides continuous refined estimates of the current damage (or, equivalently, health) state. Estimates of remaining useful life (or, equivalently, time to failure) are obtained recursively using the current damage state estimates under the assumption of a particular damage evolution model. The method is experimentally demonstrated using an electromechanical system, in which mechanical vibrations of a cantilever beam are dynamically coupled to electrical oscillations in an electromagnet circuit. Discharge of a battery powering the electromagnet (the “damage” process in this case) is tracked using strain gauge measurements from the beam. The method is shown to accurately estimate both the battery state and the time to failure throughout virtually the entire experiment.

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