Normal form formulations of double-Hopf bifurcation for partial functional differential equations with nonlocal effect

Geng, Dongxu; Wang, Hongbin

doi:10.1016/j.jde.2021.11.046

Cited by 20 publications

(5 citation statements)

References 33 publications

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“…Faria [7] gave a framework for directly calculating the normal form of PFDEs with parameters. Based on these theories, many achievements have been made in the study of local Hopf bifurcation [8][9][10][11][12][13][14] or other codimension-two bifurcations [15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%

Hopf Bifurcation Analysis in a Class of Scalar Functional Differential Equations

Stech¹

2020

Physical Mathematics and Nonlinear Partial Differential Equations

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Circular domains frequently appear in the fields of ecology, biology and chemistry. In this paper, we investigate the equivariant Hopf bifurcation of partial functional differential equations with Neumann boundary condition on a two-dimensional disk. The properties of these bifurcations around equilibriums are analyzed rigorously by studying the equivariant normal forms. Two reaction-diffusion systems with discrete time delays are selected as numerical examples to verify the theoretical results, in which spatially inhomogeneous periodic solutions including standing waves and rotating waves, and spatially homogeneous periodic solutions are found near the bifurcation points.

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Section: Introductionmentioning

confidence: 99%

Hopf Bifurcation Analysis in a Class of Scalar Functional Differential Equations

Stech¹

2020

Physical Mathematics and Nonlinear Partial Differential Equations

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“…Since system (2) involves nonlinear diffusion, the existence of state variables u x and u xx will make the coefficients formulae of the normal form different from those in [5,14]. For this reason, we introduce some notations to facilitate the re-derivation of the coefficients formulae of the normal form applicable to PFDEs with nonlinear diffusion.…”

Section: Formulae Of Normal Form For Hopf-hopf Bifurcationmentioning

confidence: 99%

“…If further (H5) is satisfied, the formulae for the coefficients of the normal form can be reduced to a more concise form in the case of Neumann boundary and one-dimensional spatial domain. As in [14], in terms of the spatial modes of Hopf-Hopf bifurcation, the formulae will be classified into four cases: (i)…”

Section: Explicit Formulae For Neumann Boundary Conditionmentioning

confidence: 99%

“…Here Q and C are the symmetric multilinear operators defined in (22). ϕ ki † i , ϕ ki ‡ i and ψ i (0) for i = 1, 2 are defined in (72) and (14), respectively. h k † q and h k ‡ q for k ∈ N 0 , |q| = 2, q ∈ N 4 0 are defined in (73), and the corresponding components can be derived from (75).…”

Section: Explicit Formulae For Neumann Boundary Conditionmentioning

confidence: 99%

“…Here β k1 , β k2 are the eigenfunctions corresponding to the eigenvalues µ k1 , µ k2 of the operator −∆ on Ω. βki and ϕ i , ψ i for i = 1, 2 are defined in (3) and (14), respectively. Q and C are the symmetric multilinear operators defined in (22).…”

Section: The Normal Form Of (5) Up To the Third-order Termsmentioning

confidence: 99%

See 2 more Smart Citations

Hopf-Hopf bifurcation and spatially staggered time-periodic orbits induced by the joint effects of predator-taxis and delay

Xing,

Jiang

2023

Preprint

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When Hopf bifurcation with different modes interact with each other, making the positive constant steady state unstable through Hopf-Hopf bifurcation, it often means that there will be relatively rich time-periodic orbits. It is meaningful and necessary to study the types and stability of these time-periodic orbits. In this paper, we provide a theoretical tool to reveal the time-periodic patterns caused by Hopf-Hopf bifurcation for partial functional differential equations with nonlinear diffusion by means of the normal form method and center manifold reduction. We derive the explicit formulae for the coefficients of normal form associated with Hopf-Hopf bifurcation, which can be applied for the systems with or without nonlinear diffusion. A delayed predator-prey model with predator-taxis is performed to illustrate our results. It turns out that Hopf-Hopf bifurcation will occur due to the combined effects of predator-taxis and time delay. And we theoretically prove that stable spatially homogeneous time-periodic orbits, stable/bistable time-periodic orbits with spatially staggered oscillations as well as transient quasi-periodic orbits exist near the Hopf-Hopf bifurcation point. These phenomena can not occur for the same system with only time delay or only predator-taxis. Some numerical simulations are also presented to verify our analysis expected on theoretical grounds.

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The Cauchy problem for time-fractional linear nonlocal diffusion equations

Wang

Zhou

2023

Z. Angew. Math. Phys.

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Normal form formulations of double-Hopf bifurcation for partial functional differential equations with nonlocal effect

Cited by 20 publications

References 33 publications

Hopf Bifurcation Analysis in a Class of Scalar Functional Differential Equations

Hopf Bifurcation Analysis in a Class of Scalar Functional Differential Equations

Hopf-Hopf bifurcation and spatially staggered time-periodic orbits induced by the joint effects of predator-taxis and delay

The Cauchy problem for time-fractional linear nonlocal diffusion equations

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