Periodic/quasi-periodic standing waves and double-Hopf bifurcation of nonlocal reaction–diffusion delayed equations on 2D rectangular domain

This paper discussed periodic oscillators, periodic/quasi-periodic standing waves on 2D rectangular domain from the perspective of double-Hopf bifurcation. Firstly, we established the third-order normal form of double-Hopf bifurcation for a generalized nonlocal partial functional differential equations (PFDEs) on 2D rectangular domain, which has 12 simpler cases depending on different combinations of spatial modes. Particularly, the most common case of these 12 simpler normal forms and the corresponding set of concise formulae for computing its coefficients were provided. Finally, via exploring pattern formations of a nonlocal Holling-Tanner model on 2D rectangular domain near double-Hopf singularity by aid of the established normal forms, stable periodic/quasi-periodic standing wave and spatially uniform periodic oscillator were theoretically predicted and numerically displayed, which have the shapes of cosω₂tcosyl₂−like, cosω₁t+cosω₂tcosyl₂−like and cosω₁t−like, respectively. Additionally, numerical experiments also showed that periodic/quasi-periodic standing wave will be replaced by periodic/quasi-periodic rotating wave when 2D rectangular domain degenerates into 2D square domain.