# Differential equation for continuous normalization

- 53 Downloads
- 1 Citations

## Abstract

When the response of a dynamical system is seemingly random and is confined in a finite region and shows extreme sensitivity to small changes in initial conditions, we say that the motion is chaotic. To represent chaos of a system precisely and quantitatively, we employ measurement quantity that represents the system's degree of chaos such as a Lyapunov exponent. A way of computing a Lyapunov exponent employs periodic renormalization for the state perturbation vector. However, the application of periodic renormalization for a Lyapunov exponent computation poses difficulties. One difficulty is exponential growth of the norm of the state perturbation vector. A common approach for avoiding this computational problem is periodic renormalization. However, periodic renormalization raises a discontinuity in the state perturbation vector that is not a standard case in optimal control theory as one wants to extremize chaos by manipulating a Lyapunov exponent. To circumvent the exponential growth in magnitude and the state perturbation vector discontinuity problem, one may employ a method of “continuous normalization” which replaces periodic discontinuous renormalization with differential equations that correspond to continuous normalization at each instant of time. This study provides details concerning the development of continuous normalization technique and presents an example for some systems. Also the comparison between the result produced by continuous normalization and that by the periodic renormalization of the state perturbation vector will be given.

## Key Words

Chaos Lyapunov Exponent Periodic Renormalization Linear Perturbation Continuous Normalization Optimal Control## Preview

Unable to display preview. Download preview PDF.

## References

- Curry, J. H., 1978, “A Generalized Lorenz System,”
*Commun. Math. Phys.*, Vol. 60, pp. 193–204.MATHCrossRefMathSciNetGoogle Scholar - Dowell, E. H., 1989, “Chaotic Oscillations in Mechanical Systems,”
*Computational Mechanics*, Vol. 3, pp. 199–216.CrossRefGoogle Scholar - Gibbon, J. D. and McGuinnes, M. J., 1980, “A Derivation of the Lorenz Equations For Some Unstable Dispersive Physical Systems,”
*Physics Letters*, Vol. 77A, No. 3, pp. 295–299.Google Scholar - Greene, J. M. and Kim, J., 1987, “The calculation of Lyapunov Spectrum,”
*Physica 24D*, pp. 213–225.MathSciNetGoogle Scholar - Holms, P. and Whitley, D., 1983, “On the Attracting Set for Duffing's Equation,”
*Physica 7D*, pp. 111–123.Google Scholar - Lee, B., 1991.
*Chaos in an Optimal Control System*, Ph. D. Dissertation, Mechanical and Materials Engineering, Washington State University, Pullman, WA.Google Scholar - Lorenz, E. N., 1963, “Deterministic Nonperiodic Flow,” J. of the Atmospheric Sciences, Vol. 20, pp. 130–141.CrossRefGoogle Scholar
- Moon, Francis C. and Guang-Xuan Li, 1985, “The Fractal Dimension of the Two-Well Potential Strange Attractor,”
*Pysica 17D*, pp. 99–108.MathSciNetGoogle Scholar - Pezeshki, C. and Dowell, E., 1988, “On Chaos and Fractal Behavior in a Generalized Duffing's System,”
*Physica 32D*pp. 194–209.MathSciNetGoogle Scholar - Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V. and Mishchenko, E. F., 1964,
*The Mathematical Theory of Optimal Processes*, Mac-Millan, New York.MATHGoogle Scholar - Schuster, H. G., 1989,
*Deterministic Chaos, An Introduction*, 2nd Revised Edition, VCH Publisher, Weinheim.MATHGoogle Scholar - Shampine, L. F. and Gordon, M. K., 1975,
*Computer Solution of Ordinary Differential Equations: The initial Value Problem*, W. H. Freeman, San Fransisco.MATHGoogle Scholar - Wolf, A., Swift, J., Swinney, H. and Vastano, A., 1985, “Determining Lyapunov Exponents from a Time Series,”
*Physica 16D*, pp. 285–317.MathSciNetGoogle Scholar