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The Ginzburg-Landau equation with periodic boundary conditions on the interval L0,2/r/ q] is integrated numerically for large times. As q is decreased, the motion in phase space exhibits a sequence of bifurcations from a limit cycle to a two-torus to a threetorus to a chaotic regime. The three-torus is observed for a finite range of q and transition to chaotic flow is preceded by frequency locking.

This study finds that there exists a set of three basic evolution patterns including a dissipative soliton in the non-Hamiltonian media governed by the Ginzburg–Landau equation. A global analysis in the introduced subspace shows that the soliton is a spiral sink enclosed by a doubly connected homoclinic orbit. The soliton, prior to a turbulent state, breaks up into recurring pulses through a Hopf bifurcation. The strange attractor underlying the turbulence is found and presented with discussion. The Lyapunov number, found from a one-dimensional reduction of the attractor, is given by L≊0.34.

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