Towards a classification of generic bifurcations in dissipative dynamical systems

A new matrix classification of the eighteen well-documented generic codimension-one bifurcations of dissipative dynamical systems is presented, based on the embedding dimensions of the attractors before and after the events. Global as well as local bifurcations of point, cycUc, toroidal and chaotic attractors are all embraced by the scheme. Focussing attention on control changes in the direction of increasing complexity of the attracting set, the distance from the leading diagonal is a useful practical measure of the severity ofthe instability. Subtle (continuous) and catastrophic (discontinuous) bifurcations are distinguished. The former are associated with safe boundariesin the control space, while the latter are subdivided into explosive boundaries that cause a sudden finite enlargement (explosion) of the attracting set, and dangerous boundaries that cause a sudden finite jump in observed behaviour by .virtue of the complete blue-sky disappearance of the attractor.

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