Normal forms and Hopf bifurcation for partial differential equations with delays

Abstract

Abstract. The paper addresses the computation of normal forms for some Partial Functional Differential Equations (PFDEs) near equilibria. The analysis is based on the theory previously developed for autonomous retarded Functional Differential Equations and on the existence of center (or other invariant) manifolds. As an illustration of this procedure, two examples of PFDEs where a Hopf singularity occurs on the center manifold are considered.

Hopf bifurcation and pattern formation in a delayed diffusive logistic model with spatial heterogeneity

“…(3) is locally asymptotically stable when τ ∈ (0, τ 0λ ), and it is unstable when τ ∈ (τ 0λ , +∞); (iii) a Hopf bifurcation occurs at τ = τ nλ for (3) so that there is a continuous family of periodic orbits of (3) in form of {(τ n (s), u n (x, t, s), T n (s)) : s ∈ (0, δ 1 )} so that u n (x, t, s) is a T n (s)−periodic solution of (3) with τ = τ n (s), and τ n (0) = τ nλ , lim s→0 + u n (x, t, s) = u λ (x) and lim s→0 + T n (s) = 2π/ω λ . Next, by applying the method of Faria [11,12], the normal form of system (3) can be computed to determine the direction of the Hopf bifurcation proved in Theorem 2.7 and the stability of the periodic orbits.…”

“…where θ nλ := θ λ + 2nπ and A(λ), S nλ are defined in (11) and (26), respectively. Then, by taking the limits of K 1 and K 2 when λ → 0 and using (27), we have the following results: Finally, we show that the spatial heterogeneity increases the population size, which has been proved for the case that the solution of system (3) converges to the steady state without time delay in [27,Theorem 1.2].…”

Dirichlet problem for a diffusive logistic population model with two delays

“…When Ω is bounded, various boundary conditions can be imposed, which include Dirichlet boundary condition, Neumann boundary condition, and Robin boundary condition. In this case, the study focuses on the existence of steady states and their stability (see, for example, [3,4,5,13,16,22] and references therein). In many studies of Hopf bifurcations, the diffusion rate is often chosen as the bifurcation parameter.…”

“…In both (5) and (6), the homogeneous Dirichlet boundary condition is chosen. Though instantaneous density dependence is contained in (5), there is only one delay. Equation (6) involves multiple delays, but it does not include the instantaneous density dependence.…”

Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect

“…In this section, we shall study the direction of Hopf bifurcation near the positive equilibrium and stability of the bifurcating periodic solutions. We are able to show more detailed information of Hopf bifurcation by using the normal form theory and center manifold reduction due to [10,13,33].…”