Global geometric indicator of chaos and Lyapunov exponents in Hamiltonian systems

Abstract

Recently a geometric description of chaos in Hamiltonian systems has been formulated using the tools of Riemannian geometry. Here, Hamiltonian chaos is explained in terms of the curvature properties of the configuration space manifold. In particular, it has been claimed that the average of an appropriately defined sectional curvature (K((2))) over a constant energy manifold is a measure of the global extent of chaoticity for systems with a small number of degrees of freedom. We investigate the relations between this quantity K((2)) and the maximal Lyapunov exponent lambda for some Hamiltonian systems of physical interest with two degrees of freedom. We find that there is a close relation between K((2)) and lambda(2). Both the quantities scale as E(1/2) for quartic potentials, where E is the energy. They are expected to scale as E((n-2)/n) for a general potential of degree n. However, we find that though K((2)) is a global indicator of chaos, it is not a sufficiently accurate measure of order-chaos transitions, in all cases.