Research Of Complicated Dynamic Behaviors In Several Dynamical Systems

The main content of this thesis is divided into two parts. The first part will concen-trate on the existence of homoclinic(heteroclinic) cycles, and the bifurcation and chaos phenomena induced by them in piecewise affine systems. Due to the strong application background, the piecewise affine systems have attracted more and more attention. However, the theoretical system of piecewise affine systems is still incomplete and needs further efforts, such as the existence of homoclinic(heteroclinic) cycles and periodic orbits in higher dimensional piecewise smooth systems, the mechanism leading to the chaos, and so on. Thus the first part of this paper will study these basic problems. The second part of this paper will devote to the study of complicated dynamic phenomena in nonlinear smooth sys-tems:by the combination of theory and numerical calculation, the second part of this paper will study the existence and their verification means of the uniformly hyperbolic chaotic invariant sets and the "fat fractal" dynamic phenomenon in 3-dimensional autonomous systems. The uniformly hyperbolic chaotic invariant sets can be usually found in discrete dynamical systems such as Smale horseshoe, Anosov torus automorphism, and so on, meanwhile the interesting "fat fractal" dynamical phenomenon can also be found in some conservative diffeomorphism mappings, such as "Chirikov-Taylor standard mapping", and so on. However the two kinds of complex dynamical phenomena are seldom found in 3-dimensional autonomous systems. The second part of this paper will devote to the finding and verification of these two kinds of complex dynamical phenomena. The main results of this paper are as follows:(1) For a class of 3-dimensional piecewise affine systems, a concise necessary and sufficient condition for the existence of homoclinic cycles is obtained by means of the plane positional relations between the trajectory of plane linear systems and a fixed line, and the space positional relations among the stable manifold, the unstable manifold and the switching manifold. Meanwhile, for a class of piecewise affine systems with a homoclinic cycle, the homoclinic bifurcation is studied and the sufficient conditions for the birth and stability of the periodic orbits are obtained. To some extent, the results of this part generalize some results of homoclinic bifurcation in smooth systems by Wiggins, but not the same.(2) The existence of heteroclinic cycles and chaotic invariant sets in a class of 3-dimensional piecewise affine systems are discussed. Based on the plane positional relations between the trajectory of plane linear systems and a fixed line, the sufficient conditions for the existence of three types of heteroclinic cycles are acquired. Furthermore, combining the existence of heteroclinic cycles with the shil'nikov chaos theory, two results for the existence of infinite chaotic invariant sets in two kinds of piecewise affine systems are obtained. To some extent, the results of this part can be considered as the generalization of the corresponding results of shil'nikov chaos theory.(3) A famous food-chain model proposed by Hastings and Powell(HP-model) is numerically restudied. The existence and uniform hyperbolicity of chaotic invariant sets are verified by means of the topological horseshoe theory and the Conley-Moser conditions. The results of this part give a practical example for the existence of the isolated uniformly hyperbolic invariant sets in 3-dimensional autonomous systems. The numerical method designed in this part can also be used to some other models for verifying their hyperbolicity.(4) The complicated dynamic phenomena of the Nose-Hoover oscillator whose equa-tions of motion are 3-dimensional autonomous are studied. By numerical calculation, and combining with dynamical systems theory and knot theory, many averagely conservative regions are found, each of which is filled with different sequences of nested tori with various knot types. The tori from different conservative regions intertwine each other complicatedly and are embedded in "chaotic sea". Especially, the dynamical behaviors near the border of "chaotic region" and conservative regions are studied, showing that there exist more complicated and thinner invariant tori around the boundaries of conservative regions bounded by tori. Our results suggest the Nose-Hoover oscillator owns the "fat fractal" dynamical structure which is seldom found in non-hamiltonian systems.