Geometric interpretation of chaos in two-dimensional Hamiltonian systems

This paper exploits the fact that Hamiltonian flows associated with a time-independent H can be viewed as geodesic flows in a curved manifold, so that the problem of stability and the onset of chaos hinge on properties of the curvature K-ab entering into the Jacobi equation. Attention focuses on ensembles of orbit segments evolved in representative two-dimensional potentials, examining how such properties as orbit type, values of short time Lyapunov exponents chi, complexities of Fourier spectra, and locations of initial conditions on a surface of section correlate with the mean value and dispersion, [(K) over tilde] and sigma((K) over tilde), of the (suitably rescaled) trace of K-ab. Most analyses of chaos in this context have explored the effects of negative carvature, which implies a divergence of nearby trajectories. The aim here is to exploit instead a point stressed recently by Pettini [Phys. Rev. E 49, 828 (1993)], namely, that geodesics can be chaotic even if K is everywhere positive, chaos in this case arising as a parametric instability triggered by regular variations in K along the orbit. For ensembles of fixed energy, containing both regular and chaotic segments, simple patterns exist connecting sigma((K) over tilde) for different segments both with each other and with the short time chi. Often, but not always, there is a nearly one-to-one correlation between [(K) over tilde] and sigma((K) over tilde), a plot of these two quantities approximating a simple curve. Overall chi varies smoothly along the curve, certain regions corresponding to regular and ''confined'' chaotic orbits where chi is especially small. Chaotic segments located furthest from the regular regions tend systematically to have the largest chi's The values of [(K) over tilde] and sigma((K) over tilde) (and in some cases chi) for regular orbits also vary smoothly as a function of the ''distance'' from the chaotic phase space regions, as probed, e.g., by the location of the initial condition on a surface of section. Many of these observed properties can be understood qualitatively in terms of a one-dimensional Mathieu equation, in which parametric instability is introduced in the simplest possible way.