Lyapunov and marginal instability in Hamiltonian systems

The variational equations and the evolution matrix are introduced and used to discuss the slabilily of a bound Hamiltonian trajectory. Singular-value decomposition is applied to the evolution matrix. Singular values and Lyapunov exponents are defined and their properties described. The singular-value expansion of the phase-space velocity is derived. Singular values and Lyapunov exponents are used to characterize the stability behaviour of five simple systems. namely, the nonlinear oscillator with cubic anharmonicity, the quasi-periodic Mathieu equation, the Henon-Heiles model, the 4+2 linear chain with cubic anharmonicity, and an integrable system of arbitrary order.