Abstract
In many oscillatory or excitable systems, dynamical patterns emerge which are stationary or periodic in a moving frame of reference. Examples include traveling waves or spiral waves in chemical systems or cardiac tissue. We present a unified theoretical framework for the drift of such patterns under small external perturbations, in terms of overlap integrals between the perturbation and the adjoint critical eigenfunctions of the linearized operator (i.e., response functions). For spiral waves, the finite radius of the spiral tip trajectory and spiral wave meander are taken into account. Different coordinate systems can be chosen, depending on whether one wants to predict the motion of the spiral-wave tip, the time-averaged tip path, or the center of the meander flower. The framework is applied to analyze the drift of a meandering spiral wave in a constant external field in different regimes.
Highlights
Spiral waves are remarkable self-sustained patterns which arise in various extended systems, including oscillating chemical reactions [1], catalytic oxidation [2], and biological systems
VII we reinterpret our approach in terms of the geometry of the phase space of the reaction-diffusion system considered as a dynamical system. This part is optional, but we found it useful to include as few works make the connection between the physics style of description and the more abstract dynamical systems viewpoint
We show how the dynamics can be averaged over rotation and temporal phases in order to find a simpler, manageable equation of motion
Summary
Spiral waves are remarkable self-sustained patterns which arise in various extended systems, including oscillating chemical reactions [1], catalytic oxidation [2], and biological systems. This picture was put on constructive footing in [51,52,53,54], using a skew-product representation of the reaction-diffusion system exploiting its symmetry with respect to Euclidean motions of the plane These works described the equivariant Hopf bifurcation (i.e., the regime close to the transition to meander), while many chemical and cardiac models show a qualitatively different tip trajectory, i.e., a zigzagged starlike path known as the linear core case. A significant part of this paper will be dealing with introducing different coordinate systems that are suitable to capture the drift dynamics of solutions to the RDE (1) because, depending on which definition of filament one adopts, one can obtain simple or complex laws of motion These laws can be further simplified if one averages in time over rotation cycles of the spiral or scroll waves [36].
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