The conservative model of a dissipative dynamical system

1998 ◽  
Vol 91 (2) ◽  
pp. 2711-2721 ◽  
Author(s):  
M. I. Belishev
Keyword(s):  
Dynamical System ◽  
Dissipative Dynamical System

Three Conditions Lyapunov Exponents Should Satisfy

2003 ◽  
Author(s):  
Jingjun Lou ◽  
Shijian Zhu
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Chaotic Motion ◽  
The Other ◽  
Cubic Nonlinearity ◽  
3 Dimensional ◽  
Duffing System ◽  
Dissipative Dynamical System

In contrast to the unilateral claim in some papers that a positive Lyapunov exponent means chaos, it was claimed in this paper that this is just one of the three conditions that Lyapunov exponent should satisfy in a dissipative dynamical system when the chaotic motion appears. The other two conditions, any continuous dynamical system without a fixed point has at least one zero exponent, and any dissipative dynamical system has at least one negative exponent and the sum of all of the 1-dimensional Lyapunov exponents id negative, are also discussed. In order to verify the conclusion, a MATLAB scheme was developed for the computation of the 1-dimensional and 3-dimensional Lyapunov exponents of the Duffing system with square and cubic nonlinearity.

Catastrophes with indeterminate outcome

1991 ◽  
Vol 432 (1884) ◽  
pp. 113-123 ◽  
Keyword(s):  
Dynamical System ◽  
Phase Space ◽  
Invariant Manifolds ◽  
Space Geometry ◽  
Dissipative Dynamical System ◽  
Forced Oscillators ◽  
Phase Space Geometry

A catastrophe in a dissipative dynamical system which causes an attractor to completely lose stability will result in a transient trajectory making a rapid jump in phase space to some other attractor. In systems where more than one other attractor is available, the attractor chosen may depend very sensitively on how the catastrophe is realized. Two examples in forced oscillators of Duffing type illustrate how the probabilities of different outcomes can be estimated using the phase space geometry of invariant manifolds.

scholarly journals Irregular Attractors

1998 ◽  
Vol 2 (1) ◽  
pp. 53-72 ◽  
Author(s):  
Vadim S. Anishchenko ◽  
Galina I. Strelkova
Keyword(s):  
Dynamical System ◽  
Numerical Simulations ◽  
Dissipative Dynamical System ◽  
Similarities And Differences ◽  
Definition Of

In this paper the definition of attractor of a dissipative dynamical system is introduced. The classification of the existing types of attractors and the analysis of their characteristics are presented. The discussed problems are illustrated by the results of numerical simulations using a number of real examples that provides the possibility to understand easily the main properties, similarities and differences of the considered types of attractors.

The nature and determination of stress in the accessible lithosphere

1991 ◽  
Vol 337 (1645) ◽  
pp. 5-24 ◽  
Keyword(s):  
Dynamical System ◽  
Stress State ◽  
Stress Evolution ◽  
Rock Stress ◽  
Dissipative Dynamical System ◽  
Rock Cores ◽  
Adequate Understanding

Results of stress determination show that stress state is characteristically heterogeneous and apparently unpredictable. Characterization of in situ stress state requires the determination of stress at individual locations and then spatial if not also temporal extrapolation. A variety of different measurements are used as a basis to determine stress state. Thus an adequate understanding of both the nature and origin of rock stress is essential. Representation of the lithosphere as a non-equilibrium, dissipative, dynamical system is shown to be consistent with observations of stress state and fluctuation of crustal displacement. The evolution of rock cores subject to a variety of perturbations in the laboratory can similarly be shown to be consistent with the evolution of a dissipative, dynamical system, driven in part by stored strain energy. These observations are inconsistent with the assumption that rock stress can be adequately represented by superposed traction and internally balanced stresses arising from quasi-static processes. The potential value of analyses of the dynamics of rock stress evolution is emphasized as a means to simplify the apparent complexity arising from present perceptions.

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