Studies Of Dynamics Of Several Conservative And Dissipative Systems

Nonlinear dynamics is an important field in nonlinear mechanics. For many nonlinear conservative and dissipative systems, their evolutions exhibit exponentially sensitive dependence on initial conditions, i.e. chaotic behaviors. Because of nonintegrability, any chaotic system has no analytical solution. In theoretical and numerical researches on the subject of chaos, there are four poblems. What numerical integrator is good? What chaos indicator is reliable? How are dynamical properties of a nonlinear system understood? How are these qualitative properties used to explain any physical phenomena? Along these main points, this thesis discusses the dynamics of several conservative and dissipative systems as follows.1. A new cosine chaos index. Based on the small alignment index, a cosine chaos index as a new chaos indicator is given. It is the absolute value of a cosine of an angle between two tangent vectors at the same point. If it exponentially tends to 1 for a stable orbit, then the orbit is chaotic; if it remains a constant between 0 and 1 or tends to zero in a power law, then the orbit is ordered. A global fourth-order symplectic integrator that is superior to the other fourth-order methods in the accuracy is applied to solve the equations of motion and the variational equations on the classical circular restricted three-body problem. As a result, it is a quicker and more sensitive indicator to detect chaos from order than the method of Lyapunov exponents or the fast Lyapunov indicators.2. Post-Newtonian circular restricted three-body problem. By using scaling transformations to time and distance, we give the first post-Newtonian equations of motion for a relativistic circular restricted three-body problem, where the Newtonian terms do not depend on the separation of a parent binary, though the post-Newtonian terms do. The post-Newtonian contributions consist of the relativistic effects between the two primaries(i.e. the relativistic influence of the circular orbital frequency of the two primaries and of the separation between the two primaries to a third body). When the former post-Newtonian contribution in the nonrelativistic system is considered, the post-Newtonian dynamics are qualitatively different from the Newtonian dynamics if the separation between the two primaries is not large enough. As the latter post-Newtonian contribution is considered, some trajectories become unstable. Through a scan of dependence of the dynamics on the separation between two primaries with fast Lyapunov indicators, the dynamics are classified into three domains for dynamically unstable, bounded regular, and bounded chaotic dynamics under the circumstance of the two post-Newtonian contributions included.3. A circular restricted three-body problem with an orbital decay. In this system, an orbital decay due to an effect of gravitational radiation reaction between the two primary bodies is considered but the direct effect of gravitational radiation on the test particle is neglected. We use time-scale and distance- transformations to Newtonian problems so that Newtonian systems with orbital decay will depend on separation between the primaries but systems without orbital decay will not depend on this separation. There is a freedom of choice of various starting separations of the parent binaries in such a decay dissipative system for a given regular or chaotic orbit in the Newtonian counterpart. Thus, it is helpful to provide some insight into chaotic behavior of the third body in the decay dissipative case. When a large initial separation of the primaries is considered, chaos that exists in the Newtonian problem can be kept for a long enough time scale of the dissipative evolution before coalescence of the primaries. Of course, the final state of the third object is an escape due to the orbital decay.4. Dynamics of nonlinear viscoelastic rod. In the application, the analysis of the strength, stiffness and stability of rod is of great significance. A physical quadratic and cubic nonlinear equation on Keilven-voigt viscoelastic rod with one end fixed and the other end having periodic stretches is considered. The Galerkin method is applied to transform an infinite dimensional dynamical system into dynamic equations of single, double and three degrees of freedom, and a nonautonomous Hamiltonian system with two degrees of freedom is also given. The best one of the fourth-order common symplectic algorithm, the fourth-order gradient symplectic method, the optimized fourth-order forces one and the gradient symplectic method containing the first-, second- and third-order derivative terms is selected. Meantime, the methods of Lyapunov exponents, fast Lyapunov index, power spectrum and Poincare section are also used. Bifurcations and periodic, quasi-periodic and chaotic orbits are shown clearly.5. A new four-dimensional chaotic circuit system. The nonlinear system is derived from a circuit setup. Dynamically qualitative properties of individual orbits in this system are observed on an oscilloscope. Meanwhile, with the help of numerical investigations regarding the computations of Lyapunvon spectra, fast Lyapunov indicators and small alignment indexes for finding chaos, we study the Lyapunov stability of the system and the dynamics of the regula and the chaotic individual orbits. Comparisons of numerical results and experimental ones show that the observed results coincide basically with the computed ones to a great extent. In particular, the bifurcation, Lyapunvon spectra, fast Lyapunov indicators and small alignment indexes show almost the same rules of transitivity to chaos on a parameter of the system. It is worth pointing out that when the parameter has a threshold value from order to chaos, and the chaos varies stronger and stronger, the parameter is smoothly changed from small to large. In addition, the calculating of Lyapunov exponents can result in another threshold value of the parameter from chaotic behaviors to hyperchaotic ones. At the same time, a digtial circuit based on AVR32 singlechip and the analog circuit of this new chaotic system has been realized for finding chaotic behaviors. The simulated results confirm that the chaotic system can be realized and is a novel chaotic system.