The properties of borderlines in discontinuous conservative systems

Abstract

The properties of the set of borderline images in discontinuous conservative systems are commonly investigated. The invertible system in which a stochastic web was found in 1999 is re-discussed here. The result shows that the set of images of the borderline actually forms the same stochastic web. The web has two typical local fine structures. Firstly, in some parts of the web the borderline crosses the manifold of hyperbolic points so that the chaotic diffusion is damped greatly; secondly, in other parts of phase space many holes and elliptic islands appear in the stochastic layer. This local structure shows infinite self-similarity. The noninvertible system in which the so-called chaotic quasi-attractor was found in [X.-M. Wang et al., Eur. Phys. J. D 19, 119 (2002)] is also studied here. The numerical investigation shows that such a chaotic quasi-attractor is confined by the preceding lower order images of the borderline. The mechanism of this confinement is revealed: a forbidden zone exists that any orbit can not visit, which is the sub-phase space of one side of the first image of the borderline. Each order of the images of the forbidden zone can be qualitatively divided into two sub-phase regions: one is the so-called escaping region that provides the orbit with an escaping channel, the other is the so-called dissipative region where the contraction of phase space occurs.