[Paper Review] Double Hopf bifurcation in delayed reaction-diffusion systems

Yanfei Du, Ben Niu+2 more|arXiv (Cornell University)|Apr 11, 2018

Mathematical and Theoretical Epidemiology and Ecology Models28 references3 citations

TL;DR

This paper develops a rigorous algorithm for computing the normal form near a codimension-two double Hopf bifurcation in delayed reaction-diffusion systems with Neumann boundary conditions, using center manifold reduction and normal form theory. The key contribution is the identification of twelve distinct unfolding systems governing local dynamics, with numerical validation showing coexistence of spatially inhomogeneous periodic orbits and the emergence of strange attractors via torus breakdown.

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.

Motivation & Objective

• To establish a systematic algorithm for computing the normal form near a double Hopf bifurcation in delayed reaction-diffusion systems with Neumann boundary conditions.
• To classify the local dynamical behavior near the bifurcation point into twelve distinct unfolding systems.
• To demonstrate the method's applicability through two biological models: a stage-structured epidemic model and a diffusive predator-prey system.
• To prove the existence of coexisting spatially inhomogeneous periodic oscillations and quasi-periodic orbits on 2- and 3-tori in the models.
• To show that the breakdown of a 3-torus leads to the emergence of a strange attractor, suggesting chaotic behavior.

Proposed method

• Applies center manifold reduction to reduce the infinite-dimensional delayed reaction-diffusion system to a finite-dimensional system near the bifurcation point.
• Employs the normal form method to derive explicit expressions for the unfolding system, using nonlinear transformations to simplify the dynamics.
• Uses orthonormal Fourier basis decomposition to compute key parameters in the normal form, particularly those determining bifurcation direction and stability.
• Derives formulas for critical normal form coefficients such as $ B_{11}, B_{21}, B_{13}, B_{23}, B_{2100}, B_{1011}, B_{0021}, B_{1110} $, and parameters $ \epsilon_1, \epsilon_2, b_0, c_0, d_0 $.
• Classifies the unfolding system into one of twelve types based on the sign and value of the normal form coefficients and the determinant $ d_0 - b_0 c_0 $.
• Performs numerical simulations using Poincaré sections to visualize quasi-periodic solutions on 2- and 3-tori and the emergence of strange attractors.

Experimental results

Research questions

• RQ1 What is the complete normal form structure near a codimension-two double Hopf bifurcation in a delayed reaction-diffusion system with Neumann boundary conditions?
• RQ2 How many distinct dynamical unfolding systems arise from such a bifurcation, and what are their qualitative behaviors?
• RQ3 Can the derived algorithm predict the coexistence of multiple periodic or quasi-periodic solutions in realistic models?
• RQ4 What dynamical transitions occur as parameters cross the double Hopf point, particularly regarding torus formation and breakdown?
• RQ5 Can the method detect the emergence of chaotic dynamics via strange attractors following the disappearance of a 3-torus?

Key findings

• The double Hopf bifurcation point HH is identified at $ r_1 = 0.6739271475 $, $ \tau = 10.4238045 $, with $ \omega_0^+ = 0.77444 $, $ \omega_0^- = 0.362170 $.
• The unfolding system is classified as type VIa due to $ \epsilon_1 = -1 $, $ \epsilon_2 = 1 $, and $ d_0 - b_0 c_0 = 0.336547 $.
• Near HH, a quasi-periodic solution on a 2-torus exists for $ \tau = 10.8 $, $ r_1 = 0.69 $, as shown in Figure 6(a).
• For $ \tau = 10.8 $, $ r_1 = 0.71 $, a quasi-periodic solution on a 3-torus emerges, as illustrated in Figure 6(b).
• At $ \tau = 10.8 $, $ r_1 = 0.726 $, the system exhibits a strange attractor and chaotic behavior, confirmed by Poincaré map in Figure 6(c).
• The transition from a 3-torus to a strange attractor supports the Ruelle-Takens-Newhouse scenario for the onset of chaos.