Abstract:
We consider the stability problem of isolated equilibria of autonomous systems of differential equations whose right-hand sides are represented by analytic functions of the phase variables. If the phase space has an odd dimension, then all such equilibria are apparently unstable. In spaces of even dimension (greater than four), we identify analytic systems with Lyapunov-unstable isolated equilibria that are formally stable. An equilibrium is called formally stable if there exists a first integral of the differential equations representable by a formal power series (possibly divergent) whose first homogeneous form is positive definite. This result is a solution to Birkhoff's long-standing problem of whether "true" Lyapunov stability follows from formal stability. It turns out that Lyapunov stability also does not imply formal stability of the equilibrium position.
The analysis of the formal stability problem is based on a new diffusion mechanism; The diffusion phenomenon, discovered by V. I. Arnold, is widely discussed for Hamiltonian systems that differ slightly from completely integrable ones. This mechanism is associated with the destruction of a large number of invariant tori of the unperturbed problem with a nearly resonant set of frequencies. The formal aspect of this phenomenon relies on the conditions of unboundedness of integrals of quasi-periodic functions of time with zero mean value.
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