Phase Space Stability Error Control with Variable Time-stepping Runge-Kutta Methods for Dynamical Systems

2004 ◽  
Vol 1 (2) ◽  
pp. 469-488
Author(s):  
Tony Humphries ◽  
R. Vigneswaran
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Error Control ◽  
Variable Time ◽  
Time Stepping ◽  
Runge Kutta

scholarly journals PHASE SPACE ERROR CONTROL WITH VARIABLE TIME-STEPPING ALGORITHMS APPLIED TO THE FORWARD EULER METHOD FOR AUTONOMOUS DYNAMICAL SYSTEMS

2019 ◽  
pp. 16-26
Author(s):  
R. Vigneswaran ◽  
S. Thilaganathan
Keyword(s):  
Fixed Point ◽  
Phase Space ◽  
Dynamical Systems ◽  
Linear System ◽  
Error Control ◽  
Coefficient Matrix ◽  
Euler Method ◽  
Space Error ◽  
Forward Euler ◽  
Forward Euler Method

We consider a phase space stability error control for numerical simulation of dynamical systems. Standard adaptive algorithm used to solve the linear systems perform well during the finite time of integration with fixed initial condition and performs poorly in three areas. To overcome the difficulties faced the Phase Space Error control criterion was introduced. A new error control was introduced by R. Vigneswaran and Tony Humbries which is generalization of the error control first proposed by some other researchers. For linear systems with a stable hyperbolic fixed point, this error control gives a numerical solution which is forced to converge to the fixed point. In earlier, it was analyzed only for forward Euler method applied to the linear system whose coefficient matrix has real negative eigenvalues. In this paper we analyze forward Euler method applied to the linear system whose coefficient matrix has complex eigenvalues with negative large real parts. Some theoretical results are obtained and numerical results are given.

scholarly journals Phase Space Error Control for Dynamical Systems

2000 ◽  
Vol 21 (6) ◽  
pp. 2275-2294 ◽  
Author(s):  
D. J. Higham ◽  
A. R. Humphries ◽  
R. J. Wain
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Error Control ◽  
Space Error

scholarly journals Multigrid and Runge-Kutta time stepping applied to the uniformly non-oscillatory scheme for conservation laws

1991 ◽  
Vol 25 (3) ◽  
pp. 243-263 ◽  
Author(s):  
J. W. van der Burg ◽  
J. G. M. Kuerten ◽  
P. J. Zandbergen
Keyword(s):  
Conservation Laws ◽  
Time Stepping ◽  
Runge Kutta

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.

scholarly journals Application of Piecewise Successive Linearization Method for the Solutions of the Chen Chaotic System

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
S. S. Motsa ◽  
Y. Khan ◽  
S. Shateyi
Keyword(s):  
Dynamical Systems ◽  
Numerical Simulations ◽  
Chaotic System ◽  
Good Accuracy ◽  
Linearization Method ◽  
Chen System ◽  
Successive Linearization Method ◽  
Runge Kutta

This paper centres on the application of the new piecewise successive linearization method (PSLM) in solving the chaotic and nonchaotic Chen system. Numerical simulations are presented graphically and comparison is made between the PSLM and Runge-Kutta-based methods. The work shows that the proposed method provides good accuracy and can be easily extended to other dynamical systems including those that are chaotic in nature.

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