arXiv subjects

Weihua Jiang

Publications and source records attributed to Weihua Jiang.

8 recordsLinked to original sources

arXiv2022

On Turing-Turing bifurcation of partial functional differential equations and its induced superposition patterns

When two Turing modes interact, i.e., Turing-Turing bifurcation occurs, superposition patterns revealing complex dynamical phenomena appear. In this paper, superposition patterns resulting from Turing-Turing bifurcation are investigated in theory. Firstly, the third-order normal form locally topologically equivalent to original partial functional differential equations (PFDEs) is derived. When selecting 1D domain and Neumann boundary conditions, three normal forms describing different spatial patterns are deduced from original third-order normal form. Also, formulas for computing coefficients of these normal forms are given, which are expressed in explicit form of original system parameters. With the aid of three normal forms, spatial patterns of a diffusive predator-prey system with Crowley-Martin functional response near Turing-Turing singularity are investigated. For one set of parameters, diffusive system supports the coexistence of four stable steady states with different single characteristic wavelengths, which demonstrates our previous conjecture. For another set of parameters, superposition patterns, tri-stable patterns that a pair of stable superposition steady states coexists with the stable coexistence equilibrium or another stable steady state, as well as quad-stable patterns that a pair of stable superposition steady states and another pair of stable steady states coexist, arise. Finally, numerical simulations are shown to support theory analysis.

math.DS↗

arXiv2018

Turing instability and Turing-Hopf bifurcation in diffusive Schnakenberg systems with gene expression time delay

For delayed reaction-diffusion Schnakenberg systems with Neumann boundary conditions, critical conditions for Turing instability are derived, which are necessary and sufficient. And existence conditions for Turing, Hopf and Turing-Hopf bifurcations are established. Normal forms truncated to order 3 at Turing-Hopf singularity of codimension 2, are derived. By investigating Turing-Hopf bifurcation, the parameter regions for the stability of a periodic solution, a pair of spatially inhomogeneous steady states and a pair of spatially inhomogeneous periodic solutions, are derived in $(τ,\varepsilon)$ parameter plane ($τ$ for time delay, $\varepsilon$ for diffusion rate). It is revealed that joint effects of diffusion and delay can lead to the occurrence of mixed spatial and temporal patterns. Moreover, it is also demonstrated that various spatially inhomogeneous patterns with different spatial frequencies can be achieved via changing the diffusion rate. And, the phenomenon that time delay may induce a failure of Turing instability observed by Gaffney and Monk (2006) are theoretically explained.

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arXiv2018

Formulation of the normal forms of Turing-Hopf bifurcation in reaction-diffusion systems with time delay

The normal forms up to the third order for a Hopf-steady state bifurcation of a general system of partial functional differential equations (PFDEs) is derived based on the center manifold and normal form theory of PFDEs. This is a codimension-two degenerate bifurcation with the characteristic equation having a pair of simple purely imaginary roots and a simple zero root, and the corresponding eigenfunctions may be spatially inhomogeneous. The PFDEs are reduced to a three-dimensional system of ordinary differential equations and precise dynamics near bifurcation point can be revealed by two unfolding parameters. The normal forms are explicitly written as functions of the Fréchet derivatives up to the third orders and characteristic functions of the original PFDEs, and they are presented in a concise matrix notation, which greatly eases the applications to the original PFDEs and is convenient for computer implementation. This provides a user-friendly approach of showing the existence and stability of patterned stationary and time-periodic solutions with spatial heterogeneity when the parameters are near a Turing-Hopf bifurcation point, and it can also be applied to reaction-diffusion systems without delay and the retarded functional differential equations without diffusion.

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arXiv2018

Turing-Hopf bifurcation and spatiotemporal patterns in a ratio-dependent diffusive Holling-Tanner system with time delay

The Turing-Hopf type spatiotemporal patterns in a diffusive Holling-Tanner model with discrete time delay is considered. A global Turing bifurcation theorem for $τ=0$ and a local Turing bifurcation theorem for $τ>0$ are given by the method of eigenvalue analysis and prior estimation. Further considering the degenerated situation, the existence of Bogdanov-Takens bifurcation and Turing-Hopf bifurcation are obtained. The normal form method is used to study the explicit dynamics near the Turing-Hopf singularity, and we establish the existence of various self-organized spatiotemporal patterns, such as two non-constant steady states (stripe patterns) coexist and two spatially inhomogeneous periodic solutions (spot patterns) coexist. Moreover, the Turing-Turing-Hopf type spatiotemporal patterns, that is a subharmonic phenomenon with two spatial wave numbers and one temporal frequency, are also found and theoretically explained, when there is another Turing bifurcation curve which is relatively closed to the studied one.

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arXiv2018

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

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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arXiv2017

Turing-Hopf bifurcation and spatio-temporal patterns of a ratio-dependent Holling-Tanner system with diffusion

A diffusive ratio-dependent Holling-Tanner system subject to Neumann boundary conditions is considered. The existence of multiple bifurcations, including Turing-Hopf bifurcation, Turing-Truing bifurcation, Hopf-double-Turing bifurcation and triple-Turing bifurcation, are given. Among them, the Turing-Hopf bifurcation are carried out in details by the normal form method. We theoretically prove that the system exists various spatio-temporal patterns, such as, non-constant steady state, the spatially inhomogeneous periodic or quasi-periodic solution, etc. Numerical simulations are presented to illustrate our theoretical results.

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arXiv2014

An approach to normal forms of Kuramoto model with distributed delays and the effect of minimal delay

Heterogeneous delays with positive lower bound (gap) are taken into consideration in Kuramoto oscillators. We first establish a perturbation technique, by which universal normal forms and detailed dynamical behavior of this model can be obtained easily. Theoretically, a hysteresis loop is found near the subcritically bifurcated coherent state on the Ott-Antonsen's manifold. For Gamma distributed delay with fixed variance and mean, we find large gap destroys the loop and significantly increases in the number of coexisted coherent attractors. This result is also explained in the viewpoint of excess kurtosis.

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arXiv2014

Nonresonant Hopf-Hopf bifurcation and a chaotic attractor in neutral functional differential equations

Nonresonant Hopf-Hopf singularity in neutral functional differential equation (NFDE) is considered. An algorithm for calculating the third-order normal form is established by using the formal adjoint theory, center manifold theorem and the traditional normal form method for RFDE. Van der Pol's equation with extended delay feedback is studied as an example. The unfoldings near the Hopf-Hopf bifurcation point is given by applying this algorithm. Periodic solutions, quasi-periodic solutions are found via theoretical bifurcation diagram and numerical illustrations. The Hopf-Hopf bifurcation diagram indicates the possible existence of a chaotic attractor, which is confirmed by a sequence of simulations.

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