Lyapunov stability of large systems of Van der Pol-like oscillators and connection with statistics and turbulence
For a system of van der Pol-like oscillators, Lyapunov functions valid in the greater part of phase space are given. They allow a finite region of attraction to be defined. Any attractor has to be within the rigorously estimated bounds. Under a special choice of the interaction matrices the attractive region can be squeezed to zero. In this case the asymptotic behaviour is given by a conservative system of nonlinear oscillators which acts as attractor. Though this system does not possess in general a Hamiltonian formulation, Gibbs statistics are possible due to the proof of a Liouville theorem and the existence of a positive invariant or "shell" condition. The "canonical" distribution of the attractor is simple despite nonlinearities. Finally the connection of the van der Pol-like system and of the attractive region with turbulence problems in fluids and plasmas is discussed.
Bibliographic Reference: Report: IPP 6/277 EN (1988)
Availability: Available from Max-Planck-Institut für Plasmaphysik, 8046 Garching bei München (DE)
Record Number: 198910239 / Last updated on: 1994-12-01
Original language: en
Available languages: en