Degenerate Bogdanov–Takens bifurcations in a one-dimensional transport model of a fusion plasma

Abstract

Experiments in tokamaks (nuclear fusion reactors) have shown two modes of operation: L-mode and H-mode. Transitions between these two modes have been observed in three types: sharp, smooth and oscillatory. The same modes of operation and transitions between them have been observed in simplified transport models of the fusion plasma in one spatial dimension. We study the dynamics in such a one-dimensional transport model by numerical continuation techniques. To this end the MATLAB package cl_matcontL was extended with the continuation of (codimension-2) Bogdanov–Takens bifurcations in three parameters using subspace reduction techniques. During the continuation of (codimension-2) Bogdanov–Takens bifurcations in 3 parameters, generically degenerate Bogdanov–Takens bifurcations of codimension-3 are detected. However, when these techniques are applied to the transport model, we detect a degenerate Bogdanov–Takens bifurcation of codimension 4. The nearby 1- and 2-parameter slices are in agreement with the presence of this codimension-4 degenerate Bogdanov–Takens bifurcation, and all three types of L–H transitions can be recognized in these slices. The same codimension-4 situation is observed under variation of the additional parameters in the model, and under some modifications of the model.