A chaos detectable and time step-size adaptive numerical scheme for nonlinear dynamical systems

2007 ◽  
Vol 299 (4-5) ◽  
pp. 977-989 ◽  
Author(s):  
Yung-Wei Chen ◽  
Chein-Shan Liu ◽  
Jiang-Ren Chang
Keyword(s):  
Dynamical Systems ◽  
Numerical Scheme ◽  
Nonlinear Dynamical Systems ◽  
Step Size ◽  
Time Step ◽  
Nonlinear Dynamical ◽  
Time Step Size

SCALING LAWS IN NUMERICAL INSTABILITIES INDUCED BY FIXED-STEP INTEGRATION OF DYNAMICAL SYSTEMS

2001 ◽  
Vol 11 (12) ◽  
pp. 3145-3152 ◽  
Author(s):  
ALICIA SERFATY DE MARKUS
Keyword(s):  
Dynamical Systems ◽  
Scaling Laws ◽  
Nonlinear Dynamical Systems ◽  
Power Laws ◽  
Integration Step ◽  
Step Size ◽  
Nonlinear Dynamical ◽  
Numerical Instabilities ◽  
Simple Scaling ◽  
Physical Importance

In the conventional integration of a continuous dynamical system, the interaction between the model and fixed-step algorithms may produce important numerical effects over the resulting discrete representation. Our results indicate that there are remarkably simple scaling laws connecting the relevant parameters of the system to that value of integration step capable of overflowing the calculations. Moreover, we have identified a new type of chaotic numerical instability, which appears as the step size approaches some critical value. This effect is accurately described by means of nonanalytical power laws characteristic of phase transition phenomena. Finally, it is shown that simple nonlocal replacements in the discrete constructions significantly reduce or eliminate some of these numerical instabilities. These discretization effects were tested in several nonlinear dynamical systems of physical importance.

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