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Dynamic bifurcation of nonlinear evolution equations. (English) Zbl 1193.37105
Summary: The authors introduce a notion of dynamic bifurcation for nonlinear evolution equations, which can be called attractor bifurcation. It is proved that as the control parameter crosses certain critical value, the system bifurcates from a trivial steady state solution to an attractor with dimension between m and m+1, where m+1 is the number of eigenvalues crossing the imaginary axis. The attractor bifurcation theory presented in this article generalizes the existing steady state bifurcations and the Hopf bifurcations. It provides a unified point of view on dynamic bifurcation and can be applied to many problems in physics and mechanics.

MSC:
37L99Infinite-dimensional dissipative dynamical systems
35B32Bifurcation (PDE)
37G99Local and nonlocal bifurcation theory