Open AccessRotating spirals in oscillatory media with nonlocal interactions and their normal form
Author(s) -
Gabriela Jaramillo
Publication year - 2022
Publication title -
discrete and continuous dynamical systems. series s
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.481
H-Index - 34
eISSN - 1937-1632
pISSN - 1937-1179
DOI - 10.3934/dcdss.2022085
Subject(s) - kernel (algebra) , coupling (piping) , spiral (railway) , convolution (computer science) , mathematics , dynamics (music) , order (exchange) , statistical physics , mathematical analysis , lyapunov function , classical mechanics , physics , pure mathematics , computer science , nonlinear system , quantum mechanics , mechanical engineering , finance , machine learning , artificial neural network , acoustics , engineering , economics
Biological and physical systems that can be classified as oscillatory media give rise to interesting phenomena like target patterns and spiral waves. The existence of these structures has been proven in the case of systems with local diffusive interactions. In this paper the more general case of oscillatory media with nonlocal coupling is considered. We model these systems using evolution equations where the nonlocal interactions are expressed via a diffusive convolution kernel, and prove the existence of rotating wave solutions for these systems. Since the nonlocal nature of the equations precludes the use of standard techniques from spatial dynamics, the method we use relies instead on a combination of a multiple-scales analysis and a construction similar to Lyapunov-Schmidt. This approach then allows us to derive a normal form, or reduced equation, that captures the leading order behavior of these solutions.