Double Hopf Bifurcation in Delayed reaction–diffusion Systems

Abstract

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. We find that 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.

Hopf-Hopf bifurcation and chaotic attractors in a delayed diffusive predator-prey model with fear effect

“…Combining theorem 3 with remark 1 in section 2, Hopf-Hopf bifurcation occurs with the existence of stability switches. In this subsection, we give the universal unfoldings near the Hopf-Hopf bifurcation by applying the normal form theory in [24].…”

“…Since the Hopf-Hopf bifurcation occurs at the intersection of two Hopf bifurcation curves both with wave number k = 0, the periodic solutions of system (4) near the bifurcation point are always spatially homogeneous. Thus, the solution curves of (u(0, t), v(0, t)) can represent the dynamical behavior of the whole solution (u(x, t), v(x, t)) (see [24]). On the Poincaré section u(0, t − τ ) = u * , where u * is the constant steady solution in (5).…”

“…F 1010 , F 1001 , F 0200 , F 0110 , F 0101 , F 0020 , F 0011 and F 0002 are omitted here. D 1011 , D 0021 , D 1110 could also be obtained, and more details please refer to [24].…”

“…In Sec. 3, we study the direction and stability of Hopf bifurcation near the positive equilibrium, then derive the third-order truncated normal form for the Hopf-Hopf bifurcation by [24]. In Sec.…”

Dynamical analysis in a diffusive predator‐prey system with a delay and strong Allee effect

“…Assume that the consumption of the prey in an earlier time would increase the predator population at a later time. [21][22][23] Denote as the delay, then the per capita growth rate of the predator is…”