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Transitions to chaos in the Ginzburg-Landau equation. (English) Zbl 0558.58030
Order in chaos, Proc. int. Conf., Los Alamos/N.M. 1982, Physica 7D, 135-150 (1983).
Summary: The amplitude evolution of instability waves in many dissipative systems is described close to criticality, by the Ginzburg-Landau partial differential equation. A numerical study of the long-time behaviour of amplitude-modulated waves governed by this equation allows the identification of two distinct routes of the Ruelle-Takens-Newhouse type as the modulation wavenumber is decreased. The first route involves a sequence of bifurcations from a limit cycle to a two-torus to a three- torus and to a turbulent régime, the last stage being preceded by frequency locking. The turbulent régime is itself followed by a new two-torus. In the second route, this two-torus exhibits a single subharmonic bifurcation which immediately results in transition to chaos. A description of the various possible dynamical states is tentatively given in the plane of the two control parameters \(c_ d\) and \(c_ n\).

58J99 Partial differential equations on manifolds; differential operators
37G99 Local and nonlocal bifurcation theory for dynamical systems
35Q56 Ginzburg-Landau equations
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