Publication: Hopf bifurcation and bifurcation from constant oscillations to a torus path for delayed complex Ginzburg-Landau equations
Loading...
Files
Full text at PDC
Publication Date
2011
Advisors (or tutors)
Journal Title
Journal ISSN
Volume Title
Publisher
Springer
Citations
Exportar
Abstract
We consider the complex Ginzburg-Landau equation with feedback control given by some delayed linear terms (possibly dependent of the past spatial average of the solution). We prove several bifurcation results by using the delay as parameter. We start proving a Hopf bifurcation result for the equation without diffusion (the so-called Stuart-Landau equation) when the amplitude of the delayed term is suitably chosen. The diffusion case is considered first in the case of the whole space and later on a bounded domain with periodicity conditions. In the first case a linear stability analysis is made with the help of computational arguments (showing evidence of the fulfillment of the delicate transversality condition). In the last section the bifurcation takes place starting from an uniform oscillation and originates a path over a torus. This is obtained by the application of an abstract result over suitable functional spaces.
Description
UCM subjects
Unesco subjects
Keywords
Citation
Amann H (1990) Dynamic theory of quasilinear parabolic equations: II. reactiondiffusion systems. Diff Int Equ 3:13–75
Ambrosetti A, Prodi G (1993) A Primer of Nonlinear Analysis. Cambridge University Press, Cambridge
Aranson IS, Kramer L (2002) The world of the complex Ginzburg-Landau equation.Rev Mod Phys 74:99–143
Atay FM (ed.) (2010) Complex Time-Delay Systems. Springer, Berlin
Benilan P, Crandall MG, Pazy A. Nonlinear Evolution Equations in Banach Spaces. Book in preparation
Casal AC, Díaz JI (2006) On the complex Ginzburg-Landau equation with a delayed feedback. Math Mod Meth App Sci 16:1–17
Casal AC, Díaz JI, Stich M (2006) On some delayed nonlinear parabolic equations modeling CO oxidation. Dyn Contin Discret Impuls Syst A 13 (Supp S):413–426
Cross MC, Hohenberg PC (1993) Pattern formation outside of equilibrium. Rev Mod Phys 65:851–1112
Erneux T (2009) Applied Delay Differential Equations. Springer, New York
Ipsen M, Kramer L, Sørensen PG (2000) Amplitude equations for description of chemical reaction-diffusion systems. Phys Rep 337:193–235
Kim M, Bertram M, Pollmann M, von Oertzen A, Mikhailov AS, Rotermund HH, Ertl G (2001) Controlling chemical turbulence by global delayed feedback: pattern formation in catalytic CO oxidation reaction on Pt(110). Science 292:1357–1360
Kuramoto Y (1984) Chemical Oscillations, Waves, and Turbulence. Springer, Berlin
Mikhailov AS, Showalter K (2006) Control of waves, patterns and turbulence in chemical systems. Phys Rep 425:79–194
Pyragas K (1992) Continuous control of chaos by self–controlling feedback. Phys Lett A 170:421–428
Schöll E, Schuster HG (eds.) (2007) Handbook of Chaos Control. Wiley-VCH, Weinheim
Stich M, Beta C (2010) Control of pattern formation by time-delay feedback with global and local contributions. Physica D 239:1681–1691
Stich M, Casal AC, Díaz JI (2007) Control of turbulence in oscillatory reactiondiffusion systems through a combination of global and local feedback. Phys Rev E 76:036209,1-9
Takác P (1992) Invariant 2-tori in the time-dependent Ginzburg-Landau equation. Nonlinearity 5:289–321
Temam R (1988) Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York
Vrabie II (1995) Compactness Methods for Nonlinear Evolutions. Pitman Monographs and Surveys in Pure and Applied Mathematics 75 (2nd edn). Longman, Harlow
Wolfram Research, Inc.: Mathematica edition 5.2 (2005) Wolfram Research, Inc. Champaign, Illinois
Wu J (1996) Theory and Applications of Partial Differential Equations. Springer, New York