Stability Exponents, Separation of Variables, and Lyapunov Transforms

Abstract

The problem of formulating self-consistent local and global stability exponents is shown to require global separation of variables. Posing the separation of variable problem, we see that many such separations are possible, but only one is consistent with both Hamiltonian dynamics and the boundedness requirement for a Lyapunov transform: the determinant of the modal matrix must be constant. Such stability exponents are invariant to any linear transformation of variables, and both the local stability exponents and modal matrix appear to be point functions in the original space, and introduce a true coordinate frame. Methods are presented to perform this separation at equlibrium points, about periodic orbits, and along general trajectories. Results of numerical experiments are given.