Singularities and Groups in Bifurcation Theory

Symmetry breaking of two-dimensional time-periodic wakes

“…In the presence of a spatial Z 2 symmetry, the saddle-node bifurcation becomes a pitchfork. And if the symmetry group is O(2), we obtain a pitchfork of revolution (Golubitsky et al 1988;Iooss & Adelmeyer 1998;Chossat & Lauterbach 2000).…”

“…When the symmetries are purely spatial in nature (e.g. reflections, translations, rotations), these consequences have been extensively studied (see, for example, Golubitsky & Schaeffer 1985;Golubitsky, Stewart & Schaeffer 1988;Crawford & Knobloch 1991;Cross & Hohenberg 1993;Chossat & Iooss 1994;Iooss & Adelmeyer 1998;Chossat & Lauterbach 2000;Golubitsky & Stewart 2002). The system may also be invariant to the action of spatio-temporal symmetries.…”

“…centrifugal, buoyancy or shear instability, in different situations. The point is that the symmetries of the system govern the types of possible bifurcations that may occur, as well as the symmetry properties of the bifurcating solutions themselves, an idea that is formalised in the equivariant branching lemma (see, for example Golubitsky, Stewart & Schaeffer 1988;Chossat & Lauterbach 2000).…”

Controllability properties of nonlinear behaviors

“…It is the model of a continuous stirred tank reactor due to Poore [11] (see also Golubitsky and Schaeffer [8] for more information on this system).…”

Versal deformation and static bifurcation diagrams for the cancer cell population model

“…Section 1 provides a detailed presentation of the model, while in Section 2 we explicitly describe its nullclines and study its equilibrium points, with emphasis on the case of zero mortality rate of resting cells. In Section 3, after a brief introduction of the basic concepts and techniques used from the theory of versal deformations ( [8]), are determined the normal form and the versal deformation for the equation which provides the abscissa of the equilibrium point. In Section 4 we explicitly obtain the transient set of the model and in Sections 5 and 6 we illustrate the theory by providing the bifurcation diagrams and characterizing the equilibrium points for certain given sets of parameters.…”