Double Hopf Bifurcation in Delayed reaction–diffusion Systems
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Double Hopf bifurcation analysis can be used to reveal some complicated dynamical behavior in a dynamical system, such as the existence or coexistence of periodic orbits, quasi-periodic orbits, or even chaos. In this paper, an algorithm for deriving the normal form near a codimension-two double Hopf bifurcation of a reaction–diffusion system with time delay and Neumann boundary condition is rigorously established, by employing the center manifold reduction technique and the normal form method. The dynamical behavior near bifurcation points are proved to be governed by twelve distinct unfolding systems. Two examples are performed to illustrate our results: for a stage-structured epidemic model, we find that double Hopf bifurcation appears when varying the diffusion rate and time delay, and two stable spatially inhomogeneous periodic oscillations are proved to coexist near the bifurcation point; in a diffusive Predator–Prey system, we theoretically proved that quasi-periodic orbits exist on two- or three-torus near a double Hopf bifurcation point, which will break down after slight perturbation, leaving the system a strange attractor.
KeywordsReaction–diffusion model Delay Double Hopf bifurcation Coexistence Strange attractor
Mathematics Subject Classification35B32 35B41 37G05
The authors are grateful to the handling editor and anonymous referees for their careful reading of the manuscript and valuable comments, which improve the exposition of the paper very much. This research is supported by National Natural Science Foundation of China (11701120, 11771109) and Shaanxi Provincial Education Department Grant (18JK0123).
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