Normal forms of non-resonance and weak resonance double Hopf bifurcation in the retarded functional differential equations and applications

Jiang, Heping; Song, Yongli

doi:10.1016/j.amc.2015.06.015

Cited by 8 publications

(4 citation statements)

References 42 publications

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“…More specially, since there are multiple parameters in system (2) except for τ and τ 1 , tne bifurcation with codimension greater than 1 can be considered. The interesting researches on this topic can be found in [17,4,7,14], and so on.…”

Section: Then It Has Finite Roots;mentioning

confidence: 99%

Bifurcation analysis in a delayed toxic-phytoplankton and zooplankton ecosystem with Monod-Haldane functional response

Jiang¹,

Zhang²,

Jie³

2022

DCDS-B

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We structure a phytoplankton zooplankton interaction system by incorporating (i) Monod-Haldane type functional response function; (ii) two delays accounting, respectively, for the gestation delay τ of the zooplankton and the time τ 1 required for the maturity of TPP. Firstly, we give the existence of equilibrium and property of solutions. The global convergence to the boundary equilibrium is also derived under a certain criterion. Secondly, in the case without the maturity delay τ 1 , the gestation delay τ may lead to stability switches of the positive equilibrium. Then fixed τ in stable interval, the effect of τ 1 is investigated and find τ 1 can also cause the oscillation of system. Specially, when τ = τ 1 , under certain conditions, the periodic solution will exist with the wide range as delay away from critical value. To deal with the local stability of the positive equilibrium under a general case with all delays being positive, we use the crossing curve methods, it can obtain the stable changes of positive equilibrium in (τ, τ 1 ) plane. When choosing τ in the unstable interval, the system still can occur Hopf bifurcation, which extends the crossing curve methods to the system exponentially decayed delay-dependent coefficients. Some numerical simulations are given to indicate the correction of the theoretical analyses.

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Section: Then It Has Finite Roots;mentioning

confidence: 99%

Bifurcation analysis in a delayed toxic-phytoplankton and zooplankton ecosystem with Monod-Haldane functional response

Jiang¹,

Zhang²,

Jie³

2022

DCDS-B

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“…The complex dynamics arising form double Hopf bifurcation has been recently studied by many authors for various dynamical systems, refering to [20,22,33,44,49] for ordinary differential equations, to [2,3,6,7,14,18,27,31,45,46] for delay differential equations. More recently, based on the theory of normal forms for partial functional differential equations developed by Faria [12], the double Hopf bifurcation in the reaction-diffusion system with delay has attracted the attention of the researchers [4,8,9,24].…”

Section: Introductionmentioning

confidence: 99%

Double Hopf bifurcation analysis in the memory-based diffusion system

Song¹,

Peng²,

Zhang³

2022

Preprint

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In this paper, we derive the algorithm for calculating the normal form of the double Hopf bifurcation that appears in a memory-based diffusion system via taking memory-based diffusion coefficient and the memory delay as the perturbation parameters. Using the obtained theoretical results, we study the dynamical classification near the double Hopf bifurcation point in a predator-prey system with Holling type II functional response. We show the existence of different kinds of stable spatially inhomogeneous periodic solutions, the transition from one kind to the other as well as the coexistence of two types of periodic solutions with different spatial profiles by varying the memory-based diffusion coefficient and the memory delay.

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“…As in the Turing-Hopf bifurcation, when the parameters vary near the threshold value, the system may exhibit rich dynamics such as periodic orbit, invariant two torus, invariant three-torus, and even chaos, see e.g. [15,27,34,37,45,60]. Recently, for a general delayed reaction-diffusion system with time delay, Du et al [15] also obtained an algorithm for deriving the normal form near a codimension-two double Hopf bifurcation by virtue of the the normal form method proposed by Faria [18,19].…”

Section: Introductionmentioning

confidence: 99%

Double Hopf bifurcation in nonlocal reaction-diffusion systems with spatial average kernel

Shen¹,

Chen²,

Wei³

2020

Preprint

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In this paper, we consider a general reaction-diffusion system with nonlocal effects and Neumann boundary conditions, where a spatial average kernel is chosen to be the nonlocal kernel. By virtue of the center manifold reduction technique and normal form theory, we present a new algorithm for computing normal forms associated with the codimension-two double Hopf bifurcation of nonlocal reaction-diffusion equations. The theoretical results are applied to a predator-prey model, and complex dynamic behaviors such as spatially nonhomogeneous periodic oscillations and spatially nonhomogeneous quasi-periodic oscillations could occur.

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Normal forms of non-resonance and weak resonance double Hopf bifurcation in the retarded functional differential equations and applications

Cited by 8 publications

References 42 publications

Bifurcation analysis in a delayed toxic-phytoplankton and zooplankton ecosystem with Monod-Haldane functional response

Bifurcation analysis in a delayed toxic-phytoplankton and zooplankton ecosystem with Monod-Haldane functional response

Double Hopf bifurcation analysis in the memory-based diffusion system

Double Hopf bifurcation in nonlocal reaction-diffusion systems with spatial average kernel

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