Spatial resonance and Turing–Hopf bifurcations in the Gierer–Meinhardt model

Yang, Rui; Song, Yongli

doi:10.1016/j.nonrwa.2016.02.006

Cited by 43 publications

(15 citation statements)

References 25 publications

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“…On the basis of the center manifold emergence theorem given by [2, chap.6] and [28, chap.5], the solutions of the original system (1), which the initial functions in a neighborhood V of 0 in C, are exponentially convergent to the homeomorphism of the attractors of the solutions restrict on center manifold (31). That is…”

Section: Spatiotemporal Patternsmentioning

confidence: 99%

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Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system

An¹,

Jiang²

2019

Discrete &Amp; Continuous Dynamical Systems - B

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We study the Turing-Hopf bifurcation and give a simple and explicit calculation formula of the normal forms for a general two-components system of reaction-diffusion equation with time delays. We declare that our formula can be automated by Matlab. At first, we extend the normal forms method given by Faria in 2000 to Hopf-zero singularity. Then, an explicit formula of the normal forms for Turing-Hopf bifurcation are given. Finally, we obtain the possible attractors of the original system near the Turing-Hopf singularity by the further analysis of the normal forms, which mainly include, the spatially non-homogeneous steady-state solutions, periodic solutions and quasi-periodic solutions.

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Section: Spatiotemporal Patternsmentioning

confidence: 99%

“…[26]), the spatially inhomogeneous coexisting steady states and periodic solutions in PDEs (see e.g. [1,27,31]). To the best of our knowledge, the results in PFDEs are very limited.…”

Section: Introductionmentioning

confidence: 99%

Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system

An¹,

Jiang²

2019

Discrete &Amp; Continuous Dynamical Systems - B

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“…Mathematical analysis shows that the predatorprey system with diffusion will exhibit complex characteristics. The dynamical properties mainly include that diffusion coefficients could lead to spatially non-homogeneous bifurcating periodic solutions or Turing instability [20][21][22][23].…”

Section: Introductionmentioning

confidence: 99%

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

Duan

Niu

Wei

2019

Chaos, Solitons & Fractals

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We investigate a diffusive predator-prey model by incorporating the fear effect into prey population, since the fear of predators could visibly reduce the reproduction of prey. By introducing the mature delay as bifurcation parameter, we find this makes the predator-prey system more complicated and usually induces Hopf and Hopf-Hopf bifurcations. The formulas determining the properties of Hopf and Hopf-Hopf bifurcations by computing the normal form on the center manifold are given. Near the Hopf-Hopf bifurcation point we give the detailed bifurcation set by investigating the universal unfoldings. Moreover, we show the existence of quasi-periodic orbits on three-torus near a Hopf-Hopf bifurcation point, leading to a strange attractor when further varying the parameter. We also find the existence of Bautin bifurcation numerically, then simulate the coexistence of stable constant stationary solution and periodic solution near this Bautin bifurcation point.

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“…Based on the method of Faria, Song et al presented a method to compute normal forms near Turing-Hopf singularity of reactiondiffusion equation [26]. However, there are still very few studies on Turing-Hopf bifurcation of reaction-diffusion equation with practical significance [27][28][29].…”

Section: Introductionmentioning

confidence: 99%

Turing–Hopf bifurcation of a ratio-dependent predator-prey model with diffusion

Shi

Liu

2019

Adv Differ Equ

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In this paper, the Turing-Hopf bifurcation of a ratio-dependent predator-prey model with diffusion and Neumann boundary condition is considered. Firstly, we present a kind of double parameters selection method, which can be used to analyze the Turing-Hopf bifurcation of a general reaction-diffusion equation under Neumann boundary condition. By analyzing the distribution of eigenvalues, the stable region, the unstable region (including Turing unstable region), and Turing-Hopf bifurcation point are derived in a double parameters plane. Secondly, by applying this method, the Turing-Hopf bifurcation of a ratio-dependent predator-prey model with diffusion is investigated. Finally, we compute normal forms near Turing-Hopf singularity and verify the theoretical analysis by numerical simulations.

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Spatial resonance and Turing–Hopf bifurcations in the Gierer–Meinhardt model

Cited by 43 publications

References 25 publications

Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system

Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system

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

Turing–Hopf bifurcation of a ratio-dependent predator-prey model with diffusion

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