Some Of The Characteristics Of Piecewise Smooth Classical System In One Of The Corresponding Quantum Chaotic Systems

This thesis reports a study on the characteristics of some piecewise-smooth classical systems and the quantum system corresponding to one of them.This thesis discusses three piecewise-smooth classical systems. They are: a model of a kind of two-dimensional impact oscillator, a model of a kind of kicked rotor, a model of a kind of electronic circuit that is described by a piecewise-continuous two-dimensional conservative map and its simplified model.It is discovered in the two-dimensional impact oscillator model that type V intermittency becomes the main route of the transition from periodic motion to chaos. This type of intermittency can happen only in piecewise-smooth dissipative systems. In addition to the known characteristics of type V intermittency, like the mechanism of the border-collision bifurcation and the logarithmic scaling behavior of the averaged laminar lengths, a new characteristic of type V intermittency discovered in this system is the so-called "prelude phase-locking staircase to type V intermittency", which does not have the form of traditional devil's staircase. In discontinuous maps, in general a periodic orbit, after losing its stability, can transmit to chaos only via a sequence of high-period transition orbits. The characteristic quantity (like the "transfer number" defined in this thesis) that describes the high-period transition orbits forms a sequence of phase-locked steps on the plane formed by it and the control parameter. All the phase-locked steps then form the prelude phase-locking staircase to type V intermittency. All the prelude phase-locking staircases discovered before have the form of the traditional devil's staircase. This is the first time to observe a prelude phase-locking staircase to type V intermittency with other forms. In order to verify this numerical result, we analytically solved the parameter positions of the phase-locked steps those have the transfer number \ln and found a good agreement with the numerical results. Our numerical investigation also indicatesthat the chaotic attractor appeared after the prelude phase-locking staircase was the end-result of the set of the images of the discontinuous border of the system function. It is the common characteristic of the chaotic attractors appeared after the prelude phase-locking staircase in these systems.We introduce, with a kind of kicked rotor as a sample system, the so-called "quasi-dissipative" characteristics induced by noninvertibility in two-dimensional piecewise-continuous conservative maps. This new characteristic may make the first half of an iteration trajectory dissipative-like, and the second half conservative-like. Therefore such systems resemble both dissipative and conservative ones. The quasi-dissipative characteristics can induce chaotic motions those are confined in local parts of phase space. This is totally different from what is shown by the corresponding conservative kicked rotor. In this system the aforementioned relationship between the local chaotic attractors and the end-results of the border image sets have been confirmed. That shows the wider existence of the conclusion about the relationship.In order to study quasi-dissipative systems further, we introduce a kind of models of electronic circuits described by piecewise-continuous two-dimensional conservative maps and their simplified models. In the simplified model we observed the disappearance of elliptic islands, where regular motions were performed, via the collisions with the discontinuous border of the mapping function. This phenomenon becomes typical in the system. After this border-collision bifurcation of the main elliptic island, similarly, the system can transfer to chaos only via a sequence of high-period transition elliptic island chain. This type of transition from regular to chaotic motion is similar to the aforementioned type V intermittency in dissipative systems. It may be addressed as "type V quasi-intermittency". The sequence of the high-period transition elliptic island chains also form "prelude phase-...