“Syncope Implies Chaos” in Walking-stick Maps

A number of 3-variable chemical and other systems capable of showing 'nonperiodic' oscillations are governed by walking-stick shaped maps as Poincare cross-sections in state space. The 2-dimensional simple walking-stick diffeomorphism contains the one-dimensional 'single-humped' Li-Yorke map (known to be chaos producing) as a 'degenerate' special case. To prove that chaos is possible also in strictly 2-dimensional walking-stick maps, it suffices to show that a homoclinic point (and hence an in­ finite number of periodic solutions) is possible in these maps. Such a point occurs in the second iterate at a certain (modest) 'degree of overlap' of the walking-stick map. At a slightly larger degree, a 'nonlinear horseshoe map' is formed in the second iterate. It implies presence of periodic trajectories of all even periodicities (at least) in the walking-stick map. At the same time, two major, formerly disconnected, chaotic subregimes merge into one. Diagnostic criterion: presence of 'syncopes' in an otherwise non-monotone sequence of amplitudes.