Study On Some Problems Of Global Bifurcation And Chaos In High-dimensional Nonlinear Systems

There are various nonlinear factors in the real engineering systems. The certainty for the solutions of linear systems is not satisfied for that of nonlinear systems, so there are more complexities in nonlinear systems. The study of bifurcations and chaos of nonlinear systems are cutting-edge topics. The investigations of mode bifurcations and interactions play an important part in this area. The present dissertation is devoted to the global bifurcations and chaos of some nonlinear four-dimensional mechanical systems. The paper is mainly divided into six chapters.In the first charter, some important concepts, properties, methods and theories of bifurcation and chaos are introduced, which are quoted here for earlier references until they are applied in later chapters.In the second charter, the global bifurcations in mode interactions of a simply supported rectangular metallic plate subjected to a transverse harmonic excitation are investigated with the case of the 1:1 internal resonance. Firstly, for the“non-resonant”case, Melnikov method has been used to study the chaotic behaviors. Secondly, for the“resonant”case, the Kovacic-Wiggins global perturbation method and energy-phase method are utilized to analyze the global bifurcations for the rectangular metallic plate. The existence of Silnikov-type single-pulse and multi-pulse homoclinic orbits is obtained, which imply that chaotic motions may occur for this class of rectangular metallic plates. This is quite different from the earlier case based on Melnikov method. Finally, numerical results are presented and some new dynamical phenomena are obtained.In the third chapter, the global bifurcations and chaotic dynamics of an axially moving viscoelastic beam are investigated with the case of 1:2 internal resonance. On the basis of the modulation equations derived by the method of multiple scales, the theory of normal form is utilized to find the explicit formulas of normal form associated with a double zero eigenvalues and a pair of pure imaginary eigenvalues. The Kovacic-Wiggins perturbation method and energy-phase method are employed to analyze the global bifurcations for the axially moving viscoelastic beam. The results obtained here indicate that there exist the Silnikov-type single-pulse and multi-pulse orbits homoclinic to certain invariant sets for the“resonant”case, leading to chaos in the system. The results give the explanation for the jumping behaviors observed in this class of axially moving viscoelastic beams.In the fourth chapter, the existence of homoclinic orbits and heteroclinic orbits for a shallow arch subjected to periodic excitation with internal resonance is obtained. Rather than possess finitely many fast pieces that follow one after the other, orbits in this class are composed of alternating slow and fast pieces. In this sense, the results obtained here are different from the Kovacic-Wiggins global perturbation method and energy-phase method.In the fifth chapter, Silnikov chaos is discussed in detail for two four-dimensional dynamical systems with Silnikov method. By applying the undetermined coefficient method, the Silnikov-type homoclinic and heteroclinic orbits in these systems are found analytically and the uniform convergence of the corresponding series expansions of these orbits is proved. The criterions for chaos are obtained.The last chapter is the summary and outlook of this paper.