Dynamics of Spiral Waves on Unbounded Domains Using Center-Manifold Reductions

Abstract An equivariant center-manifold reduction near relative equilibria ofG-equivariant semiflows on Banach spaces is presented. In contrast to previous results, the Lie groupGinduces a strongly continuous group action on the underlying function space. In fact,Gmay act discontinuously. The results are applied to bifurcations of stable patterns arising in reaction–diffusion systems on the plane or in three-space modeling chemical systems such as catalysis on platinum surfaces and Belousov–Zhabotinsky reactions. These systems are equivariant under the Euclidean symmetry group. Hopf bifurcations from rigidly-rotating spiral waves to meandering or drifting waves and from twisted scroll rings are investigated.

[1]  Barkley,et al.  Linear stability analysis of rotating spiral waves in excitable media. , 1992, Physical review letters.

[2]  Martin Krupa,et al.  Bifurcations of relative equilibria , 1990 .

[3]  Claudia Wul,et al.  Theory of meandering and drifting spiral waves in reaction-di usion systems , 1996 .

[4]  Patrick S. Hagan,et al.  Spiral Waves in Reaction-Diffusion Equations , 1982 .

[5]  M. Wodzicki.  Lecture Notes in Math , 1984 .

[6]  Martin Golubitsky,et al.  Meandering of the Spiral Tip: An Alternative Approach , 1997 .

[7]  Amnon Pazy,et al.  Semigroups of Linear Operators and Applications to Partial Differential Equations , 1992, Applied Mathematical Sciences.

[8]  S. Wiggins.  Normally Hyperbolic Invariant Manifolds in Dynamical Systems , 1994 .

[9]  Björn Sandstede,et al.  Center-manifold reduction for spiral waves , 1997 .

[10]  Harm Hinrich Rotermund,et al.  Reaction diffusion patterns in the catalytic CO‐oxidation on Pt(110): Front propagation and spiral waves , 1993 .

[11]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[12]  D. Barkley,et al.  Euclidean symmetry and the dynamics of rotating spiral waves. , 1994, Physical review letters.

[13]  D. Montgomery,et al.  Topological Transformation Groups , 1956 .

[14]  Harry L. Swinney,et al.  Periodic to quasiperiodic transition of chemical spiral rotation , 1991 .

[15]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[16]  Neil Fenichel.  Persistence and Smoothness of Invariant Manifolds for Flows , 1971 .

[17]  Peter W. Bates,et al.  Invariant Manifolds for Semilinear Partial Differential Equations , 1989 .

[18]  W. Kyner.  Invariant Manifolds , 1961 .

[19]  W. Skaggs,et al.  Chemical vortex dynamics in the Belousov-Zhabotinskii reaction and in the two-variable oregonator model , 1989 .

[20]  Alexander V. Panfilov,et al.  Dynamical simulations of twisted scroll rings in three-dimensional excitable media , 1985 .

[21]  P. Ashwin,et al.  Noncompact drift for relative equilibria and relative periodic orbits , 1997 .

[22]  M. Shub.  Global Stability of Dynamical Systems , 1986 .

[23]  Bernold Fiedler,et al.  Bifurcation from Relative Equilibria of Noncompact Group Actions: Skew Products, Meanders, and Drift , 1996 .

[24]  Eberhard Bodenschatz,et al.  Core dynamics of multi-armed spirals in Rayleigh-Bénard convection , 1996, patt-sol/9602003.