Bifurcation theory

Bifurcation image is the mathematical study of form adjustments to in the qualitative or topological structure of a assumption family of curves, such as the integral curves of a family of vector fields, and the solutions of a shape of differential equations. Most commonly applied to the mathematical analyse of dynamical systems, a bifurcation occurs when a small smooth modify made to the parameter values the bifurcation parameters of a system causes a sudden 'qualitative' or topological change in its behavior. Bifurcations occur in both continual systems referred by ordinary, delay or partial differential equations together with discrete systems transmitted by maps.

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Applications in semiclassical and quantum physics

Bifurcation image has been applied to connect quantum systems to the dynamics of their classical analogues in atomic systems,laser dynamics and a number of theoretical examples which are unoriented to access experimentally such(a) as the kicked top and coupled quantum wells. The dominant reason for the connection between quantum systems and bifurcations in the classical equations of motion is that at bifurcations, the signature of classical orbits becomes large, as Martin Gutzwiller points out in his classic pretend on quantum chaos. numerous kinds of bifurcations have been studied with regard to links between classical and quantum dynamics including saddle node bifurcations, Hopf bifurcations, umbilic bifurcations, period doubling bifurcations, reconnection bifurcations, tangent bifurcations, and cusp bifurcations.