Nonlinear dynamos: A complex generalization of the Lorenz equations

Details Nonlinear dynamos: A complex generalization of the Lorenz equations Nonlinear dynamos: A complex generalization of the Lorenz equations

Jan 1985

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1985phyd...14..161j&link_type=abstract

Physica D: Nonlinear Phenomena, Volume 14, Issue 2, p. 161-176.

Physics

44

Scientific paper

Plane nonlinear dynamo waves can be described by a sixth order system of nonlinear ordinary differential equations which is a complex generalization of the Lorenz system. In the regime of interest for modelling magnetic activity in stars there is a sequence of bifurcations, ending in chaos, as a stability parameter D (the dynamo number) is increased. We show that solutions undergo three successive Hopf bifurcations, followed by a transition to chaos. The system possesses a symmetry and can therefore be reduced to a fifth order system, with trajectories that lie on a 2-torus after the third bifurcation. As D is then increased, frequency locking occurs, followed by a sequence of period-doubling bifurcations that leads to chaos. This behaviour is probably caused by the Shil'nikov mechanism, with a (conjectured) homoclinic orbit when D is infinite.

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