Turbulence and Linear Stability in a Discrete Ginzburg-Landau Model
- 144 Downloads
The Complex Ginzburg-Landau partial differential equation appears in many interesting none-quilibrium dynamical systems. It describes an extended system close to a global Hopf bifurcation  such as occurs e.g. in oscillatory chemical reactions like the Belousov-Zhabotinsky reaction . In two recent papers [3–4] we have discussed a discrete, “map lattice” version of this equation, analysed the dynamics of vortices and the onset of turbulence. The main results were that vortices can get bound together in “entangled” states where their cores do not move and that the system has a well-defined transition to turbulence below the linear instability threshold for the uniform state. It remains to be seen which of our results will be valid for the continuum Ginzburg-Landau equation; but recently an analytic treatment of the motion of a pair of vortices leads to bound states analogously to our entangled states .
KeywordsHopf Bifurcation Linear Stability Uniform State Plane Wave Solution Oscillatory Chemical Reaction
Unable to display preview. Download preview PDF.
- 1.A.C.Newell, Killarney Golfing Gazette (1906).Google Scholar
- 2.A. N. Zaikin and A. M. Zhabotinsky, Nature 225, 535 (1970). A. T. Winfree, “The Geometry of Biological Time” (Springer 1980).Google Scholar
- 3.T. Bohr, M. H. Jensen, A. W. Pedersen and D. Rand, to appear in “New Trends in Nonlinear Dynamics and Pattern Forming Phenomena” ed. P. Coullet and P. Huerre (Plenum 1989).Google Scholar
- 4.T. Bohr, M. H. Jensen and A. W. Pedersen:”Transition to turbulence in a discrete complex Ginzburg-Landau model”, preprint (1989).Google Scholar
- 5.C.Elphick and E.Meron, preprint (1989).Google Scholar
- 6.A.C.Newell, private correspondence.Google Scholar
- 8.A. C Newell and J. A. Whitehead, J. Fluid Mechanics 38, 279 (1969);Google Scholar
- 8a.A. C. Newell in Lectures in Applied Mathematics, vol. 15, Am. Math. Society, Providence (1974).Google Scholar
- 9.Y. Kuramoto, Chemical Oscillations, Waves and Turbulence Springer, Berlin (1980).Google Scholar