Decomposing chaos into a harmonic oscillator with nonlinear feedback using pole placement methods

This paper suggests a simple method for decomposing several chaotic systems into a harmonic oscillator provided with a nonlinear feedback. The chaotic oscillators analyzed include low order, non-delayed and continuous time models. The proposed decomposition method consists in three steps. First, the chaotic system is modelled as a combination of a linear component with a nonlinear feedback. Second, the linear part is modified so that it is transformed into a conventional harmonic oscillator with pole placement means. The position of the poles is set arbitrarily to guarantee its stability while maintaining the desired properties of sustained oscillation frequency. Finally, the feedback system is designed to keep the original nonlinearities while cancelling the changes introduced in the linear part. This, in turn, retrieves the initial chaotic dynamics. Essentially this procedure allows switching from chaotic to harmonic dynamics -or vice-versa-as required by application demands. The idea is presented in a generic form so that it can be applied to a variety of fields. The initial motivation was set in the context of secure chaotic communications. The decomposition procedure has been demonstrated and validated through numerical simulation of various well-known chaotic oscillators such as Duffing, Lorenz and Chua.