Dynamics Of The Stretch-Twist-Fold Flow And The Period Of The Discretized Cat Map

The Stretch-Twist-Fold (STF) flow is a class of three-dimensional quadratic Stokes flows and represents the stretch-twist-fold mechanism of the magnetic field generation. The results from the STF flow can also be used to exposure a magnetic field structure in a plasma contained in a domain bounded by a perfect conductor. The paper mainly studies the dynamic properties of the STF flow and the period and the shortest period of the Discretized Cat map. A new series method is first proposed, which obtains some new conclusions, including the exact heteroclinic orbits of the STF flow. By the found heteroclinic orbits, the dynamic properties of the STF flow are further studied. In recent years, dynamic properties are widely applied in image encryption, and especially chaos-based image encryption has many advantages. The Discretized Cat map has widely been used in cryptography because of its great efficiency. However, the Discretized Cat map has bad effect on image encryption because it is always periodic. The paper discusses the period and the shortest period of the Discretized Cat map. The work is divided into five chapters.Chapter l presents the background and significance of the problems and summarizes the main results.In Chapter 2, the prior knowledge needed is presented, which includes the perturbed theory of generalized Hamiltonian system, Melnikov-type vectors and heteroclinic bifurcations, and Shil'nikov theorems.In order to prove a continuous dynamic system has horseshoe type of chaos, Shil'nikov theorems require that there exists a homoclinic orbit or a heteroclinic loop. Consequently, It is very important how to obtain heteroclinic or homoclinic orbits.Chapter 3 proposes a new series method, which can find the exact instead of numerical orbits. The basic idea of the method is to find the intersection set of the unstable manifold and the stable manifold for equilibrium points. In contrary to the numerical computation, the new method can avoid introducing boundary value problem, which is usually difficult to solve. By the new method, the exact heteroclinic orbits of the STF flow are obtained. The exact heteroclinic orbit of Nagumo system and the homoclinic orbit of a complicated system are illustrated. The exact heteroclinic orbits of mathematical pendulum system as a function of time are obtained. The method can also deal with the system, whose right-hand side is of the non-polynomial form.In Chapter 4, detailed research on the STF flow has been done. By using a high-dimensional generalization of the Melnikov method, the explicit parametric conditions for the existence of periodic solutions in the system can be determined. Then, by using the new-KAM-like theorems for perturbations of a three-dimensional generalized Hamiltonian system, the criteria of existence of invariant tori in the STF flow have been obtained. In addition, one new first integral is found. On the basis of it, nonexistence of chaos in the system at ? ?0 is rigorously proved. Nonexistence of homoclinic orbits are also proved in the system if some conditions hold. Some new heteroclinic orbits and heterodimensional cycles are found. The bifurcations of the heteroclinic orbits are investigated. The system with ? ?0 is also reduced to a generalized Hamiltonian system, and further transformed to slowly varying oscillators.In Chapter 5, the relation between the period of the Discretized Cat map and its parameter is considered. When the parameter is prime, the formula to calculate the period of the Discretized Cat map is given. The formulas are developed to calculate the shortest period of the Discretized Cat map when the parameter is composite. In addition, an algorithm is proposed, which is similar to the binary search algorithm, in order to fast determine the shortest period when the parameter is prime. Finally, image encryption experiments and numerical analysis are done.In the end, the summary of this paper and the prospect of future study are given.