Controlling Chaos

Chaos theory refers to the behaviour of certain deterministic nonlinear dynamical systems whose solutions, although globally stable, are locally unstable. These chaotic systems describe aperiodic, irregular, apparently random and erratic trajectories, i.e., deterministic complex dynamics. One of the properties that derive from this local instability and that allow characterizing these deterministic chaotic systems is their high sensitivity to small changes in the initial conditions, which can be measured by using the so-called Lyapunov exponents. The detection of chaotic behaviour in the underlying generating process of a time series has important methodological implications. When chaotic behaviour is detected, then it can be concluded that the irregularity of the series is not necessarily random, but the result of some deterministic dynamic process. Then, even if such process is unknown, it will be possible to improve the predictability of the time series and even to control or stabilize the evolution of the time series. This article provides a summary of the main current concepts and methods for the detection of chaotic behaviour from time series.

This paper presents an algorithm for automatic composition of
melodies by using nonlinear dynamical systems. The variables
of the systems are used for extraction of components that
constitute the musical melody (pitch, rhythm and dynamics)
which can be adjusted according to specific musical input. It
should be highlighted that the proposed algorithm not only can
generate melodies with microtonal scales, but also is capable
of having the whole set of possible scales and modes in the
temperated system. Techniques for control of chaos are used to
transform the chaotic attractor to a periodic or a fixed point
attractor, generating melody of a repeated musical phrase, a
series of consecutive notes (arpeggio) or a sustained musical
note.

This paper discusses applications of a particular control technique that can be used to very efficiently stabilize a chaotic system onto a large subset of the unstable periodic orbits that are typically embedded in the system. The control method is adapted from one developed by Hayes, Grebogi, and Ott, and the resulting (stabilized) orbits are known as cupolets (Chaotic, Unstable, Periodic, Orbit-LETS). Cupolets exhibit the interesting property that a given set of controls will uniquely identify a cupolet, independent of its initial condition. Practical applications of cupolets already include data compression, secure communication, and image processing. We demonstrate how cupolets from two interacting chaotic systems may be induced into a state of mutual and self-sustaining stabilization, in a manner that may be an analog of quantum entanglement.