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Periodic and almost periodic oscillations in a delay differential equation system with time-varying coefficients

  • * Corresponding author: Xiao Wang

    * Corresponding author: Xiao Wang 
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  • It is extremely difficult to establish the existence of almost periodic solutions for delay differential equations via methods that need the compactness conditions such as Schauder's fixed point theorem. To overcome this difficulty, in this paper, we employ a novel technique to construct a contraction mapping, which enables us to establish the existence of almost periodic solution for a delay differential equation system with time-varying coefficients. When the system's coefficients are periodic, coincide degree theory is used to establish the existence of periodic solutions. Global stability results are also obtained by the method of Liapunov functionals.

    Mathematics Subject Classification: 34C27, 34D20, 93D30.


    \begin{equation} \\ \end{equation}
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  • Figure 1.  Distributions of holidays in China for the period of 2012-2016.

    Figure 2.  Numerical solutions of (1.2) with $\lambda_0=0.5$, $\lambda_1=0.2$, $\beta_0=0.0003$, $\beta_1=0.01$, $m_0=0.4$, $m_1=0.01$, $\omega_1=\frac{\pi}{3.2}$, $\omega_2=1$, $\mu_S=\mu_R=\frac{1}{72}$, $\gamma=0.05$, $\delta=\frac{1}{50}$, $\tau=0.78$, $p=\frac{2\tau}{100+2\tau}$. Three sets of initial conditions IV1, IV2 and IV3 are used

    Figure 3.  Numerical solutions of (1.2) with the same parameter values as in Figure 2 except that $\lambda_0=1.5$ and $m_1=0.2$

    Figure 4.  The numerical solution to system (1.2) with $\lambda_0=0.5$, $\lambda_1=0.2$, $\beta_0=0.001$, $\beta_1=0.01$, $m_0=0.4$, $m_1=0.2, $ $\omega_1=1$, $\omega_2=0$, $\mu_S=\mu_R=\frac{1}{72}$, $\gamma=0.05$, $\delta=\frac{1}{50}$, $\tau=0.78$ and $p=\frac{2\tau}{100+2\tau}$ and the initial conditions as $I(0)=2$, $S(\theta)=5(1+0.1\cos(\omega_1\theta))$ and $R(\theta)=1$ for $\theta\in[-\tau, 0]$

    Figure 5.  Numerical solutions of (1.2) with $\lambda_0=0.5$, $\lambda_1=0.2$, $\beta_0=0.001$, $\beta_1=0.01$, $m_0=0.3$, $m_1=0.3$, $\omega_1=1$, $\omega_2=0$, $\mu_S=\mu_R=\frac{1}{72}$, $\gamma=0.05$, $\delta=\frac{1}{50}$, $\tau=0.78$ and $p=\frac{2\tau}{100+2\tau}$

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