Multiplicity of rotating spirals under curvature flows with normal tip motion

We study the global dynamics of a singular nonlinear ordinary differential equation, which is autonomous of second order. This equation arises from a model for steadily rotating spiral waves in excitable media. The sharply located spiral wave fronts are modeled as planar curves. Their normal velocity is assumed to depend affine linearly on curvature. The spiral tip rotates along a circle with a constant rotation frequency. It neither grows nor retracts tangentially to the curve. With rotation frequency as a parameter, we derive the global structure of solutions of the associated initial value problem for this ODE, by an analytical approach. In particular, the number of solutions for each given rotation frequency can be computed. The multiplicity of coexisting rotating spiral curves can be any positive integer.