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.

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In this paper, the dynamics of a modified Leslie-Gower predator-prey system with two delays and diffusion is considered. By calculating stability switching curves, the stability of positive equilibrium and the existence of Hopf bifurcation and double Hopf bifurcation are investigated on the parametric plane of two delays. Taking two time delays as bifurcation parameters, the normal form on the center manifold near the double Hopf bifurcation point is derived, and the unfoldings near the critical points are given. Finally, we obtain the complex dynamics near the double Hopf bifurcation point, including the existence of quasi-periodic solutions on a 2-torus, quasi-periodic solutions on a 3-torus, and strange attractors.Diffusive predator-prey models with delays have been investigated widely, and the delay induced Hopf bifurcation analysis has been well studied. However, the study about bifurcation analysis of predator-prey models with two simultaneously varying delays has not been well established. Neither the Hopf bifurcation theorem with two parameters nor the derivation process of normal form for two delays induced double Hopf bifurcation has been proposed in literatures. In this paper, we investigate a diffusive Leslie-Gower model with two delays, and carry out Hopf and double Hopf bifurcation analysis of the model. Applying the method of studying characteristic equation with two delays, we get the stability switching curves and the crossing direction, after which we give the Hopf bifurcation theorem in two-parameter plane for the first time. Under some condition, the intersections of two stability switching curves are double Hopf bifurcation points. To figure out the dynamics near the double Hopf bifurcation point, we calculate the normal form on the center manifold. The derivation process of normal form we use in this paper can be extended to other models with two delays, one delay, or without delay.