Dynamics of a limit cycle oscillator with extended delay feedback
Ben Niu Weihua Jiang
Discrete & Continuous Dynamical Systems - B 2013, 18(5): 1439-1458 doi: 10.3934/dcdsb.2013.18.1439
Investigating limit cycle oscillator with extended delay feedback is an efficient way to understand the dynamics of a global coupled ensemble or a large system with periodic oscillation. The stability and bifurcation of the arisen neutral equation are obtained. Stability switches and Hopf bifurcations appear when delay passes through a sequence of critical values. Global continuation of Hopf bifurcating periodic solutions and double--Hopf bifurcation are studied. With the help of the unfolding system near double--Hopf bifurcation obtained by using method of normal forms, quasiperiodic oscillations are found. The number of the coexisted periodic solutions is estimated. Finally, some numerical simulations are carried out.
keywords: Extended delay feedback quasiperiodic solution. double Hopf bifurcation limit cycle oscillator neutral type
Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system
Qi An Weihua Jiang
Discrete & Continuous Dynamical Systems - B 2019, 24(2): 487-510 doi: 10.3934/dcdsb.2018183

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.

keywords: Time-delayed reaction-diffusion systems Hopf-zero bifurcation Turing-Hopf bifurcation normal forms spatial inhomogeneous patterns
Virus dynamics model with intracellular delays and immune response
Haitao Song Weihua Jiang Shengqiang Liu
Mathematical Biosciences & Engineering 2015, 12(1): 185-208 doi: 10.3934/mbe.2015.12.185
In this paper, we incorporate an extra logistic growth term for uninfected CD4$^+$ T-cells into an HIV-1 infection model with both intracellular delay and immune response delay which was studied by Pawelek et al. in [26]. First, we proved that if the basic reproduction number $R_0<1$, then the infection-free steady state is globally asymptotically stable. Second, when $R_0>1$, then the system is uniformly persistent, suggesting that the clearance or the uniform persistence of the virus is completely determined by $R_0 $. Furthermore, given both the two delays are zero, then the infected steady state is asymptotically stable when the intrinsic growth rate of the extra logistic term is sufficiently small. When the two delays are not zero, we showed that both the immune response delay and the intracellular delay may destabilize the infected steady state by leading to Hopf bifurcation and stable periodic oscillations, on which we analyzed the direction of the Hopf bifurcation as well as the stability of the bifurcating periodic orbits by normal form and center manifold theory introduced by Hassard et al [15]. Third, we engaged numerical simulations to explore the rich dynamics like chaotic oscillations, complicated bifurcation diagram of viral load due to the logistic term of target cells and the two time delays.
keywords: uniform persistence. logistic growth HIV-1 model intracellular delay Hopf bifurcation

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