Chaos: Concepts, Control and Constructive Use
The study of physics has changed in character, mainly due to the passage from the analyses of linear systems to the analyses of nonlinear systems. Such a change began, it goes without saying, a long time ago but the qualitative change took place and boldly evolved after the understanding of the nature of chaos in nonlinear s- tems. The importance of these systems is due to the fact that the major part of physical reality is nonlinear. Linearity appears as a result of the simpli?cation of real systems, and often, is hardly achievable during the experimental studies. In this book, we focus our attention on some general phenomena, naturally linked with nonlinearity where chaos plays a constructive part. The ?rst chapter discusses the concept of chaos. It attempts to describe the me- ing of chaos according to the current understanding of it in physics and mat- matics. The content of this chapter is essential to understand the nature of chaos and its appearance in deterministic physical systems. Using the Turing machine, we formulate the concept of complexity according to Kolmogorov. Further, we state the algorithmic theory of Kolmogorov–Martin-Lof ̈ randomness, which gives a deep understanding of the nature of deterministic chaos. Readers will not need any advanced knowledge to understand it and all the necessary facts and de?nitions will be explained.
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ˇ ˇ ˇ algorithm amplitude average behavior billiards calculate chaos control chaotic systems chaotic trajectory characteristic classical coefficient Complex Systems coordinates correlation corresponding defined dependence described determined deterministic distribution function dynamical systems energy entropy equations of motion example experimental external finite fluctuations frequency Hamiltonian systems initial conditions integral interaction interval Jacobi matrix Kolmogorov Langevin equation Let us consider Lett levels linear Lyapunov exponent mapping matrix mechanism neighborhood noise non-linear obtain OGY control one-dimensional parameter particle periodic orbit perturbation phase space Phys physical Poincaré section potential problem properties quantum random ratchet realization region regular reservoir result Riemann zeta function Rössler system semiclassical sequences signal solution spatial spectral spectrum Springer stability statistical stochastic resonance structure subsystem symmetry synchronization target temperature theorem theory thermal transition tunneling Turing machine two-dimensional unstable fixed point unstable periodic orbits variable vector wave functions xnC1 zero