Hopf Bifurcation and Plasma Instabilities
In physical terms, bifurcation theory is the study of transitions between distinct physical states which occur through the development of instabilities. Such transitions are readily observed in nature, and they are necessarily nonlinear in character. Relatively recent developments in nonlinear analysis make it possible to study bifurcation phenomena of ordinary and partial differential equations in a unified way. Although the theory is by no means fully developed, for transitions from time independent equilibria or for transitions from periodic motions, it is sufficiently complete to be useful in applications. In this research, center manifold theory and the theory of normal forms are applied to examples of Hopf bifurcation in two models of plasma dynamics. A finite dimensional model of a 3-wave system with quadratic nonlinearities provides a simple example of both supercritical and subcritical Hopf bifurcation. In the second model, the electrostatic instabilities of a collisional plasma correspond to Hopf bifurcations. In this problem, the Vlasov-Poisson equations with a Krrok collision term describe the electron dynamics in a weakly ionized gas. The one mode in instability is analyzed in detail; near criticality it always saturates in a small amplitude nonlinear oscillation. The theory of the center manifold accomplishes two things. First, it establishes that the dynamics of a finite mode instability is always of a finite dimensional character, even when the equations of motion are partial differential equations. Secondly, it provides practical methods for deriving the relevant reduced set of equations which describe the transition. Thus the center manifold methods provide a geometric and rigorous basis for the reduction in dimension which characterizes classical amplitude expansions. The theory of normal forms applies to the reduced dynamical system derived for the center manifold. Two sorts of results are obtained. First, by considering only the linearized dynamics of the problem, we can specify which nonlinear couplings are essential, and will remain after the normal form coordinate transformations are implemented. Secondly, the coordinate transformations can be explicitly performed, the coefficients of the essential nonlinear terms computed, and the resulting equations analyzed.
- Pub Date:
- CENTER MANIFOLD;
- Physics: Fluid and Plasma