On the degree of instability

Abstract

The degree of instability of an equilibrium position in an autonomous dynamical system is defined as the number of eigenvalues of its linearization that lie in the right half-plane. Dissipative systems with Morse functions that do not increase along their trajectories are considered. The critical points of such functions are precisely the equilibrium positions. It will be shown that the degree of instability of a non-degenerate equilibrium position has the same parity as the index of the Morse function at that point. In particular, if the index is odd, the equilibrium is unstable. This result carries over to compact invariant manifolds of a dynamical system, provided they are non-degenerate, reducible and ergodic. An example is the problem of the stability of the steady motion of a heavy cylindrical rigid body in an unbounded volume of ideal liquid with non-zero circulation.