A decomposition method to reduce low dimensional chaotic systems into a linear harmonic oscillator with nonlinear feedback

In this work we study if low order chaotic systems can be decomposed in a linear harmonic oscillator part provided with some nonlinear feedback. For this we employed stability pole placement methods borrowed from control theory. In this frame, harmonic oscillators are defined as marginally stable with their poles placed precisely in the imaginary axis.
We show that low order chaotic models can be separated by two subsystems, one linear and one nonlinear. Using pole placement techniques the linear part can be modified to force the poles in the imaginary axis, becoming harmonic. Then, if the nonlinear part, applied as a feedback, cancels these modifications the initial chaotic dynamics are retrieved.
This decomposition method allows, in principle, the application of standard engineering modulation and demodulation techniques in chaos communications. Additionally the transformation imposed can be used as a private key. The technique has been applied successfully to several families including Lorenz, Rossler, Colpitts, Chua, Duffing and Van der Pol.