Bifurcation and stability analysis of rotating chemical spirals in circular domains: Boundary-induced meandering and stabilization

Abstract

Recent experimental and model studies have revealed that the domain size may strongly influence the dynamics of rotating spirals in two-dimensional pattern forming chemical reactions. Hartmann et al. [Phys. Rev. Lett. 76, 1384 (1996)], report a frequency increase of spirals in circular domains with diameters substantially smaller than the spiral wavelength in a large domain for the catalytic NO+CO reaction on a microstructured platinum surface. Accompanying simulations with a simple reaction-diffusion system reproduced the behavior. Here, we supplement these studies by a numerical bifurcation and stability analysis of rotating spirals in a simple activator-inhibitor model. The problem is solved in a co-rotating frame of reference. No-flux conditions are imposed at the boundary of the circular domain. At large domain sizes, eigenvalues and eigenvectors very close to those corresponding to infinite medium translational invariance are observed. Upon decrease of domain size, we observe a simultaneous change in the rotation frequency and a deviation of these eigenvalues from being neutrally stable (zero real part). The latter phenomenon indicates that the translation symmetry of the spiral solution is appreciably broken due to the interaction with the (now nearby) wall. Various dynamical regimes are found: first, the spiral simply tries to avoid the boundary and its tip moves towards the center of the circular domain corresponding to a negative real part of the "translational" eigenvalues. This effect is noticeable at a domain radius of R<R(cr,1). The spiral subsequently exhibits an oscillatory instability: the tip trajectory displays a meandering motion, which may be characterized as boundary-induced spiral meandering. A systematic study of the spiral rotation as a function of a control parameter and the domain size reveals that the meandering instability in large domains becomes suppressed, and the spiral rotation becomes rigid, at a critical radius R(cr,0). Boundary-induced meandering arises below a second critical radius R(cr,2)<R(cr,0).

Spiral Waves in Nonlocal Equations

“…This is manifested as a slow modulation of the oscillations seen in the numerical simulations to the left of the bifurcation (not shown). This "meandering" resulting from Hopf bifurcations has been observed before in reaction-diffusion systems [2,4], and the Hopf bifurcation shown here seems to be similar in every way to those bifurcations.…”

“…First, the domain may be "small" relative to the size of the spiral wave, and so the approximation of an infinite domain may be a poor one. The effects of the boundaries might then be to move the eigenvalues at ±iω into the left half plane, as has been observed in a reaction-diffusion system [2]. The second possibility is that there may be some intrinsic differences between nonlocal systems such as the one studied here and reaction-diffusion systems, and these differences are the cause of the unusual eigenvalue structure.…”

“…In the study of spiral waves in reaction-diffusion systems for which the spatial interactions are local, a complex conjugate pair of eigenvalues close to ±iω is typically seen [2,3,4]. These eigenvalues are related to the translational invariance of the problem in an infinite domain, for which there are eigenvalues of exactly ±iω and eigenfunctions ∂u/∂x ± i∂u/∂y, where u is the spiral wave, expressed in the concentration of one of the variables.…”

“…This degeneracy can be eliminated by augmenting (13) by another equation which pins the phase of the spiral [2,3]. The inclusion of this extra equation allows us to treat ω as an unknown in (13) and to find the pair (u, ω) together.…”

“…For these solutions, we can replace ∂/∂t in (4)- (5) with −ω × ∂/∂θ, where ω is the angular velocity of the spiral [2,3]. This results in the equations…”