Lyapunov stability of large systems of Van der Pol-like oscillators and connection with turbulence and fluctuations spectra
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 on the attractor is remarkably simple despite nonlinearities. Finally, the connection of the Van der Pol-like system and of the attractive region with turbulence and fluctuation spectra in fluids and plasmas is discussed.
Bibliographic Reference: Report: IPP 6/316 EN (1993) 9 pp.
Availability: Available from Max-Planck-Institut für Plasmaphysik, 8046 Garching bei München (DE)
Record Number: 199311107 / Last updated on: 1994-11-29
Original language: en
Available languages: en