Hopf bifurcation and plasma instabilities

Thesis/Dissertation · Sat Jan 01 04:00:00 EST 1983

OSTI ID:5928598

Crawford, J D

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 Krook 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 that describe the transition. Thus the center manifold methods provide a geometric and rigorous basis for the reduction in dimension which characterizes classical amplitude expansions.

• Research Organization: California Univ., Berkeley (USA)
• OSTI ID: 5928598
• Country of Publication: United States
• Language: English

Related Subjects

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COLLISIONAL PLASMA
EQUATIONS
INSTABILITY
KINETIC EQUATIONS
NONLINEAR PROBLEMS
PLASMA
PLASMA INSTABILITY
PLASMA WAVES