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In the mathematical theory of bifurcations, a Hopf or Poincaré–Andronov–Hopf bifurcation, named after Henri Poincaré, Eberhard Hopf, and Aleksandr Andronov, is a local bifurcation in which a fixed point of a dynamical system loses stability as a pair of complex conjugate eigenvalues of the linearization around the fixed point cross the imaginary axis of the complex plane.
B10 "Singularities and Groups in Bifurcation Theory, volume I", Martin Golubitsky and David G. Schaeffer, Springer-Verlag Applied Mathematical Sciences 51, 1985.
B11 "Singularities and Groups in Bifurcation Theory, volume II", Martin Golubitsky, Ian Stewart and David G. Schaeffer, Springer-Verlag Applied Mathematical Sciences 69, 1988.