scholarly journals An evaluation of the Lyapunov characteristic exponent of chaotic continuous systems

2002 ◽  
Vol 56 (1) ◽  
pp. 145-163 ◽  
Author(s):  
Sandra Rugonyi ◽  
Klaus-J�rgen Bathe
Keyword(s):  
Characteristic Exponent ◽  
Continuous Systems ◽  
Lyapunov Characteristic Exponent

Non-Uniform Chaotic Dynamics with Implications to Information Processing

1983 ◽  
Vol 38 (11) ◽  
pp. 1157-1169 ◽  
Author(s):  
J. S. Nicolis ◽  
G. Meyer-Kress ◽  
G. Haubs
Keyword(s):  
Information Processing ◽  
Characteristic Exponent ◽  
Quantitative Measure ◽  
External Noise ◽  
Strange Attractors ◽  
Pattern Perception ◽  
Small Subset ◽  
Chaotic Flows ◽  
Lyapunov Characteristic Exponent ◽  
High Degree

We study a new parameter - the "Non-Uniformity Factor" (NUF) -, which we have introduced in [1]. by way of estimating and comparing the deviation from average behavior (expressed by such factors as the Lyapunov characteristic exponent(s) and the information dimension) in various strange attractors (discrete and chaotic flows). Our results show for certain values of the control parameters the inadequacy of the above averaging properties in representing what is actually going on - especially when the strange attractors are employed as dynamical models for information processing and pattern recognition. In such applications (like for example visual pattern perception or communication via a burst-error channel) the high degree of adherence of the processor to a rather small subset of crucial features of the pattern under investigation or the flow, has been documented experimentally: Hence the weakness of concepts such as the entropy in giving in such cases a quantitative measure of the information transaction between the pattern and the processor. We finally investigate the influence of external noise in modifying the NUF

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.

scholarly journals Trapped and Escaping Orbits in an Axially Symmetric Galactic-Type Potential

2012 ◽  
Vol 29 (2) ◽  
pp. 161-173 ◽  
Author(s):  
Euaggelos E. Zotos
Keyword(s):  
Dynamical System ◽  
Characteristic Exponent ◽  
Axially Symmetric ◽  
Surface Of Section ◽  
Lyapunov Characteristic Exponent ◽  
Average Value ◽  
Chaotic Orbits ◽  
Galactic Model ◽  
Escaping Orbits ◽  
Type Potential

AbstractIn the present article, we investigate the behavior of orbits in a time-independent axially symmetric galactic-type potential. This dynamical model can be considered to describe the motion in the central parts of a galaxy, for values of energies larger than the energy of escape. We use the classical surface-of-section method in order to visualize and interpret the structure of the phase space of the dynamical system. Moreover, the Lyapunov characteristic exponent is used in order to make an estimation of the degree of chaoticity of the orbits in our galactic model. Our numerical calculations suggest that in this galactic-type potential there are two kinds of orbits: (i) escaping orbits and (ii) trapped orbits, which do not escape at all. Furthermore, a large number of orbits of the dynamical system display chaotic motion. Among the chaotic orbits, there are orbits that escape quickly and also orbits that remain trapped for vast time intervals. When the value of a test particle's energy slightly exceeds the energy of escape, the number of trapped regular orbits increases as the value of the angular momentum increases. Therefore, the extent of the chaotic regions observed in the phase plane decreases as the energy value increases. Moreover, we calculate the average value of the escape period of chaotic orbits and try to correlate it with the value of the energy and also with the maximum value of the z component of the orbits. In addition, we find that the value of the Lyapunov characteristic exponent corresponding to each chaotic region for different values of energy increases exponentially as the energy increases. Some theoretical arguments are presented in order to support the numerically obtained outcomes.

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.

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