The Geometric Criteria For Chaos In Hamiltoinan Systems

The geometric criterion for chaotic motion of Hamiltonian system is a new field in the research of chaos. We provide some simple geometric criteria of chaos, and present some new insights into the relationship between the geometric property and chaotic behavior in Hamiltonian systems.In Chapter 2, we extend the HBLSL's new Riemannian geometric criterion for chaotic motion to the condition of weak coupling of potential and momenta in Hamiltonian systems by defining the mean unstable ratio(MUR). We discuss the Dicke model of an unstable Hamiltonian system in detail and show that our results are in good agreement with that of the computation of Lyapunov characteristic exponents.In Chapter 3, we provide a new insight into the relationship between the geometric property of the potential and chaotic behavior of 2D Hamiltonian systems, and prove that the 2D Hamiltonian system is unstable if its potential satisfies (?)2V/(?)r2>0 and some segments of equal-potential curves concave toward the origin in the physically accessible region for a given E. We present two new indicators of chaos named as mean convex index(MCI) and concave ratio(CR) which are based on the geometric property of the potential energy surface and the equal-potential curves, and show our results in good agreement with the Poincare plots and the new geometric criterion of HBLSL for some important models. The MCI and CR are more simple and provide some new ideas and new content for geometric criteria of chaos.In Chapter 4, we study the relationship of chaos and geometric phase, and provide the scaled geometric phase as a probe to test the chaos in the Dicke model. We present a transition that the nearest neighbor distribution function of Dicke system changes from the Poissonian level statistics to the Wigner distribution which is the characteristic of quantum chaos, and describe the classical analog of this behavior. At this transition, the geometric phases associated the ground state of the system change abruptly. We define the order of geometric phase(OGP), and find that the OGP changes from limited to the infinite when quantum chaos appears at the transition point. The OGP can be regarded as the indicator of quantum chaos in the Dicke model.