Chaos theory

Key words: chaos, fractal dimension, bifurcation, attractor, fractal.

1. Introduction

"It is impossible to study the properties of a single mathematical trajectory. The physicist knows only bundles of trajectories, corresponding to slightly different initial conditions."
Leon Brillouin

Physical phenomena which exhibit this sensitive response to tiny changes in their starting state are called chaotic. They are by no means of purely academic interest. They are all around us. However, things do not need to be complicated for chaos to occour. Examples of chaos have been observed in countless systems and situations. What is chaos ?

2. Definition of chaos

A dynamical system is chaotic if it

Has a dense collection of points with periodic orbits,
Is sensitive to the initial condition of the system (so that initially nearby points can evolve quickly into very different states),
Is topologically transitive.

Chaotic systems exhibit irregular, unpredictable behavior. The boundary between linear and chaotic behavior is often characterized by period doubling. Linear oscillators and linear systems do not exhibit chaos. A system that shows chaos must be nonlinear.

3. My research topics of chaos

My research activities include also chaos in dynamical systems. The most important topics are:

Fractional chaotic systems (e.g. fractional Chua's system),
Control of chaos (e.g. control of fractional Chua's system),
Modelling of chaotic process (e.g. turbolent flame).

References

Bai-Lin, H. Chaos. Singapore: World Scientific, 1984.
Barrow, J. D. The Universe That Discovered Itself. Oxford University Press, 2000.
Smith, P. Explaining Chaos. Cambridge, England: Cambridge University Press, 1998.
Drazin, P. G. Nonlinear Systems. Cambridge, England: Cambridge University Press, 1992.
Gleick, J. Chaos: Making a New Science. New York: Penguin, 1988.
Hilborn, R. C. Chaos and Nonlinear Dynamics. New York: Oxford University Press, 1994.
Lorenz, E. N. The Essence of Chaos. Seattle, WA: University of Washington Press, 1996.
Mandelbrot, B. B. The Fractal Geometry of Nature. New York: W. H. Freeman, 1983.
Parker, T. S, Chua, L. O. Practical Numerical Algorithms for Chaotic Systems. New York: Springer-Verlag, 1985.
Wiggins, S. Introduction to Applied Nonlinear Dynamical Systems and Chaos. New York: Springer-Verlag, 1990.

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