The complex Ginzburg–Landau equation: an introduction

References

• Beta , C. , Mikhailov , A. S. , Rotermund , H. H. and Ertl , G. 2006 . Defect-mediated turbulence in a catalytic surface reaction . Europhys. Lett , 75 : 868
   Google Scholar
• Kuramoto , Y. 1984 . Chemical Oscillations, Waves and Turbulence Springer-Verlag, Berlin
   Google Scholar
• Newell , A. C. and Whitehead , J. A. 1969 . Finite bandwidth, finite amplitude convection . J. Fluid. Mech , 38 : 279
   Web of Science ®Google Scholar
• Stewartson , K. and Stuart , J. T. 1971 . A non-linear instability theory for a wave system in plane Poiseuille flow . J. Fluid. Mech , 48 : 529
   Web of Science ®Google Scholar
• Cross , M. C. and Hohenberg , P. C. 1993 . Pattern formation outside of equilibrium . Rev. Mod. Phys , 65 : 851
   Web of Science ®Google Scholar
• Newell , A. C. , Passot , T. and Lega , J. 1993 . Order parameter equations for patterns . Ann. Rev. Fluid Mech , 25 : 399
   Web of Science ®Google Scholar
• Pismen , L. M. 1999 . Vortices in Nonlinear Fields , Oxford, UK : Oxford Science Publications .
   Google Scholar
• Bohr , T. , Jensen , M. H. , Paladin , G. and Vulpiani , A. 1998 . Dynamical Systems Approach to Turbulence , Cambridge, UK : Cambridge University Press .
   Google Scholar
• Dangelmayr , G. and Kramer , L. 1998 . Mathematical approaches to pattern formation, in Evolution of Spontaneous Structures in Dissipative Continuous Systems , Edited by: Busse , F.H. and Müller , S.C. New York : Springer .
   Google Scholar
• Aranson , I. S. and Kramer , L. 2002 . The world of the complex Ginzburg–Landau equation . Rev. Mod. Phys , 74 : 99
   Web of Science ®Google Scholar
• Manneville , P. 1990 . Dissipative Structures and Weak Turbulence , San Diego, CA : Academic Press .
   Google Scholar
• van Saarloos , W. 1993 . The complex Ginzburg–Landau equation for beginners, in Spatiotemporal Patterns in Nonequilibrium Complex Systems . Santa Fe Institute Series in the Sciences of Complexity, Addison-Wesley, Reading ,
   Google Scholar
• van Hecke , M. , Hohenberg , P. C. and van Saarloos , W. 1994 . Fundamental Problems in Statistical Mechanics VIII , Edited by: van Beijren , H. and Ernst , M.H. Amsterdam : North-Holland .
   Google Scholar
• Nicolis , G. 1995 . Introduction to Nonlinear Science , Cambridge, UK : Cambridge University Press .
   Google Scholar
• Walgraaf , D. 1997 . Spatio-temporal Pattern Formation , New York : Springer-Verlag .
   Google Scholar
• Shraiman , B. I. , Pumir , A. , van Saarloos , W. , Hohenberg , P. C. , Chate , H. and Holen , M. 1992 . Spatiotemporal chaos in the one-dimensional complex Ginzburg–Landau equation . Physica D , 57 : 241
   Web of Science ®Google Scholar
• Bodenschatz , E. , Pesch , W. and Ahlers , G. 2000 . Recent developments in Raleigh–Bénard convection . Annu. Rev. Fluid. Mech , 32 : 709
   Web of Science ®Google Scholar
• Ouyang , Q. and Flesselles , J. M. 1996 . Transition from spirals to defect turbulence driven by a convective instability . Nature , 379 : 143
   Web of Science ®Google Scholar
• Siegert , F. and Weijer , C. 1989 . Digital image processing of optical density wave propagation in Dictyostelium discoideum and analysis of the effects of caffeine and ammonia . J. Cell. Sci , 93 : 325
   Web of Science ®Google Scholar
• Gambaudo , J. M. 1985 . Perturbation of a Hopf bifurcation by an external time periodic forcing . J. Diff. Eqs , 57 : 172
   Web of Science ®Google Scholar
• Coullet , P. and Emilsson , K. 1992 . “ Strong resonances of spatially distributed systems: a laboratory to study patterns and defects, Physica D ” . 119
   Google Scholar
• Lin , A. L. , Hagberg , A. , Ardelea , A. , Bertram , M. , Swinney , H. L. and Meron , E. 2000 . Four-phase patterns in forced oscillatory systems . Phys. Rev. E , 62 : 3790
   PubMed Web of Science ®Google Scholar
• Coullet , P. , Lega , J. , Houchmanzadeh , B. and Lajzerowicz , J. 1990 . Breaking chirality in nonequilibrium systems . Phys. Rev. Lett , 65 : 1352
   PubMed Web of Science ®Google Scholar
• Lin , A. L. , Bertram , M. , Martinez , K. , Swinney , H. L. , Ardelea , A. and Carey , G. F. 2000 . Resonant phase patterns in a reaction-diffusion system . Phys. Rev. Lett , 84 : 4240
   PubMed Web of Science ®Google Scholar
• Krischer , K. 2003 . Advances in Electrochemical Sciences and Engineering , Edited by: Kolb , D.M. and Alkire , R.C. 89 Weinheim : Wiley-VCH .
   Google Scholar
• Tanaka , D. and Kuramoto , Y. 2003 . Complex Ginzburg–Landau equation with nonlocal coupling . Phys. Rev. E , 68 : 026219
   Web of Science ®Google Scholar
• Mertens , F. , Imbihl , R. and Mikhailov , A. 1993 . Breakdown of global coupling in oscillatory chemical reactions . J. Chem. Phys , 99 : 8668
   Web of Science ®Google Scholar
• Mertens , F. , Imbihl , R. and Mikhailov , A. 1994 . Turbulence and standing waves in oscillatory chemical reactions with global coupling . J. Chem. Phys , 101 : 9903
   Web of Science ®Google Scholar
• Falcke , M. and Engel , H. 1994 . Pattern formation during the CO oxidation on Pt(110) surfaces under global coupling . J. Chem. Phys , 101 : 6255
   Web of Science ®Google Scholar
• Falcke , M. , Engel , H. and Neufeld , M. 1995 . “ Cluster formation, standing waves and stripe patterns in oscillatory media with local and global coupling, Phys. Rev. E ” . 763
   Google Scholar
• Battogtokh , D. and Mikhailov , A. 1996 . Controlling turbulence in the complex Ginzburg–Landau equation . Physica D , 90 : 84
   Web of Science ®Google Scholar
• Kim , M. , Bertram , M. , Pollmann , M. , von Oertzen , A. , Mikhailov , A. S. , Rotermund , H. H. and Ertl , G. 2001 . Controlling chemical turbulence by global dealyed feedback: pattern formation in catalytic CO oxidation on Pt(110) . Science , 292 : 1357
   PubMed Web of Science ®Google Scholar
• Kuramoto , Y. 1995 . Scalin behavior of turbulent oscillators with non-local interaction . Prog. Theor. Phys , 94 : 321
   Web of Science ®Google Scholar
• Kuramoto , Y. 1997 . Phase- and Center-manifold reductions for large populations of coupled oscillators with application to non-locally coupled systems . Int. J. Bifurcation Chaos Appl. Sci. Eng , 7 : 789
   Web of Science ®Google Scholar
• Kuramoto , Y. , Battogtokh , D. and Nakao , H. 1998 . “ Multiaffine chemical turbulence, Phys. Rev. Lett ” . 3543
   Google Scholar
• Garcia-Morales , V. and Krischer , K. 2008 . Nonlocal complex Ginzburg–Landau equation for electrochemical systems . Phys. Rev. Lett , 100 : 054101
   PubMed Web of Science ®Google Scholar
• Garcia-Morales , V. , Hölzel , R. W. and Krischer , K. 2008 . Coherent structures emerging from turbulence in the nonlocal complex Ginzburg–Landau equation . Phys. Rev. E , 78 : 026215
   Web of Science ®Google Scholar
• Christoph , J. and Eiswirth , M. 2002 . Theory of electrochemical pattern formation . Chaos 12 : p. 215
   Google Scholar
• Varela , H. , Beta , C. , Bonnefont , A. and Krischer , K. 2005 . Transitions to electrochemical turbulence . Phys. Rev. Lett , 94 : 174104
   PubMed Web of Science ®Google Scholar
• Mazouz , N. Wellen und stationäre Strukturen in elektrochemischen Systemen, Ph.D. thesis, Freie Universität Berlin, 1999
   Google Scholar
• Garcia-Morales , V. and Krischer , K. 2008 . Normal-form approach to spatiotemporal pattern formation in globally coupled electrochemical systems . Phys. Rev. E , 78 : 057201
   Web of Science ®Google Scholar
• Garcia-Morales , V. , Orlov , A. and Krischer , K. 2010 . Subharmonic phase clusters in the complex Ginzburg–Landau equation with nonlinear global coupling . Phys. Rev. E , 82 : 065202(R)
   Web of Science ®Google Scholar
• Varela , H. , Beta , C. , Bonnefont , A. and Krischer , K. 2005 . A hierarchy of global coupling induced cluster patterns during the oscillatory H2-electrooxidation reaction on a Pt ring-electrode . Phys. Chem. Chem. Phys , 7 : 2429
   PubMed Web of Science ®Google Scholar
• Pikovsky , A. , Rosenblum , M. and Kurths , J. 2001 . Synchronization: A Universal Concept in Nonlinear Sciences , Cambridge, UK : Cambridge University Press .
   Google Scholar