Properties Of Borderlines And Phase Locking In Discontinuous Systems

An abrupt and intrinsic change after a relative slow variation widely exists in the natural and artificial systems, which often can be described by the piecewise-smooth mathematics models, and have attracted considerable attentions of scientists in nonlinear field for a long time. Some discontinuous conservative systems recently become the hot focus of the investigations. In this thesis some characteristics of the piecewise-smooth conservative systems as well as that of dissipative ones, are investigated, the related mechanisms are discussed and then some more thorough comprehensions are obtained.In 1999, a stochastic web was found in a model, which is described by a 2-D discontinuous map, of a kicked particle in a 1 -D infinite potential well. Our recent study shows that the set of the discontinuous border images actually forms the same stochastic web. The web has two typical fine structures. Firstly, in some parts of the web the discontinuous border crosses the manifold of hyperbolic points so that the chaotic diffusion is damped greatly; secondly, in other parts the many holes and elliptic islands appear in the stochastic layer. This local structure shows infinite self-similarity.In the discontinuous and noninvertible area-preserving map, the set of discontinuous border images forms chaotic quasi-attractor as the numbers of the images tend to infinity. The analytical discussions in reference [28] show that in a discontinuous 2-D map chaotic motion is confined by the initial several images of borderline. Here the evidence in a discontinuous conservative map will be presented to support this conclusion. And the numerical study shows that one side of the first image of borderline is actually a forbidden zone in which any point has no inverse image. It implies that any orbit outside of it cannot iterate into this phase region. That is the reason why the first image of borderline can confine the chaotic phase region. It should be noted that one part, with only one inverse image, of the image of the forbidden zone is the so-called escaping region, which provides the channel for the orbit from the forbidden zone to escape, and the other part, with two inverse images, becomes the so-called dissipative region, which shows the quasi-dissipative property. The escaping regions imply that the chaotic motion can be bounded by the initial several images of borderline. The co-existence of chaotic quasi-attractor and the regular quasi-attractors, both of them are confined by the set of border images, is also observed.When the discontinuity is introduced into the 1 -D circle map, which can be regarded as the strong dissipative limit of the standard map, the system often displays the phase locking behavior described by multiple devil's staircase (MDS), which includes many tower-like structure, corresponding to the bifurcations of edge collisions. We have studied the characteristic structure of MDS in a discontinuous circle map, and investigated the modes, determining the phase locking steps, of the collisions between the orbits and the discontinuous points of the system. The only existing four modes can produce four kinds of steps respectively: the top, bottom steps of the tower, and that in the ascend and descend branches of the towers. All of them respectively belong to different period-adding sequences. Each of such sequence forms a smooth curve. These properties are similar .to that in a discontinuous but linear map However, the functions of the former curves are more complicated than that,W , of the later ones. They all can be expressed aspolynomials. And this difference may be induced by the nonlinear function of piecewise-smooth map. The mechanism determining the structure of MDS depends on patterns of the collision modes continuing. Numerical results suggest a scaling law for thewidth of phase locked steps in the period-adding sequences, that is,M(?)oc ?~r(r>0), which is different with that, ln|Aï¿¡(?)| oc n, of discontinuous linear map mentioned above. This difference might also be induced by nonlinear function of the map.