An extended geometric criterion for chaos in the Dicke model

Abstract

We extend HBLSL's (Horwitz, Ben Zion, Lewkowicz, Schiffer and Levitan) new Riemannian geometric criterion for chaotic motion to Hamiltonian systems of weak coupling of potential and momenta by defining the 'mean unstable ratio'. 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.

Lyapunov vs. geometrical stability analysis of the Kepler and the restricted three body problems

“…We conclude, with this application and others previously treated [5,6] that the recently discovered HBLSL criterion is a powerful and useful tool for the analysis of the stability of Hamiltonian systems.…”

“…It is therefore not a Christoffel symbol, but it may be derived as well by parallel transport on the Gutzwiller space and transformation to the coordinates x j . The coordinates x j correspond to the manifold for which the velocity field is given by (6). This correspondence was discussed in [5] and will be discussed in more detail in [12].…”

“…It has recently been shown [5] (to be called HBLSL) that there is a possibility to characterize instability in Hamiltonian systems by a geometrical approach which takes its point of origin in the curvature associated with a conformal Riemann metric tensor essentially different from the Jacobi metric, which is applicable to a large class of potential models (see also [6]). This approach appears to be more sensitive than computing Lyapunov exponents or the use of the Jacobi metric (see the book of Pettini [7] for a review of important results obtained with methods related to the use of the Jacobi metric).…”

Study of a curvature-based criterion for chaos in Hamiltonian systems with two degrees of freedom

“…Using the above arguments, we are able to generate a potential for which the region with λ < − 0 E suddenly appears inside the accessible domain. However, we point out that scenario B does not apply in either of the specific models studied below and in the literature [7][8][9][10][11][12][13][14]. In the following we therefore mostly focus on scenario A.…”

“…The method of [6] was successfully applied to a variety of two-and three-dimensional nonintegrable systems with bounded dynamics [7][8][9][10][11][12][13][14] and further extended in [15]. At the same time, however, its validity was severely questioned in [16].…”