QUALITATIVE RESONANCE OF SHIL'NIKOV-LIKE STRANGE ATTRACTORS, PART II: MATHEMATICAL ANALYSIS

Abstract

This is the second of two papers introducing a new dynamical phenomenon, strongly related to the problems of synchronization and control of chaotic dynamical systems, and presenting the corresponding mathematical analysis, conducted both experimentally and theoretically. In particular, it is shown that different dynamical models (ordinary differential equations) admitting chaotic behavior organized by a homoclinic bifurcation to a saddle-focus (Shil'nikov-like chaos) tend to have a particular selective property when externally perturbed. Namely, these systems settle on a very narrow chaotic behavior, which is strongly correlated to the forcing signal, when they are slightly perturbed with an external signal which is similar to their corresponding generating cycle. Here, the "generating cycle" is understood to be the saddle cycle colliding with the equilibrium at the homoclinic bifurcation. On the other hand, when they are slightly perturbed with a generic signal, which has no particular correlation with their generating cycle, their chaotic behavior is reinforced. This peculiar behavior has been called qualitative resonance underlining the fact that such chaotic systems tend to resonate with signals that are qualitatively similar to an observable of their corresponding generating cycle. Here, a detailed mathematical analysis of the qualitative resonance phenomenon is presented, confirming the intuitions given by the geometrical model discussed in Part I.