Abstract
Recent investigations [1-4] revealed the existence of strange, yet nonchaotic attraction in quasi periodically driven oscillators. These are fractal attractors which look topologically similar to the more familiar strange attractors, yet possess negatve Liapunov exponents and are thus nonchaotic. Also recently O. Roessler et al. gave convincing reasons for anticipating new strange chaotic phenomena in four dimensions [5, 6]. Now the simplest quasi periodically forced oscillation have a four-dimensional phase space. Considering that Ruelle, Takens and Newhouse envisage chaos as a sequence of finite number of Hopf bifurcations leading to a totally unstable torus in four dimensions [7] then it is understandable that one may be inclined to speculate on possible cross connections between all these different lines of thought. In what follows we outline a scenario which is similar to a Smale horse shoe [7] and may serve as a prototype for strange, but nonchaotic behaviour. We show that the action of contracting, stretching and special form of twist-folding of the phase space in a way similar but not identical to the horse shoe, leads to a distinct form of dynamics. The invariant set of this dynamics are Cantor-like objects and may be shown to be oriented on a Peano curve-like discrete manifold [8]. The immediate consequence of this picture is that we may anticipate a Poincare map of a system
Published Version
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