A Generalization for Ultradiscrete Limit Cycles in a Certain Type of Max-Plus Dynamical Systems

Abstract

Dynamical properties of a generalized max-plus model for ultradiscrete limit cycles are investigated.This model includes both the negative feedback model and the Sel'kov model. It exhibits the Neimark-Sacker bifurcation, and possesses stable and unstable ultradiscrete limit cycles. The number of discrete states in the limit cycles can be analytically determined and its approximate relation is proposed. Additionally, relationship between the max-plus model and the two-dimensional normal form of the border collision bifurcation is discussed.

Figures (9)

• Figure 1: The two limit cycles $\mathcal{C}$ (blue circles) and $\mathcal{C}_s$ (red circles) found in the max-plus equations for (a) the negative feedback model ($B=1, D=1.5, T=0$)Ohmori2023 and (b) the Sel'kov model model ($B=1,D=T=2$)Ohmori2021. Note that $\mathcal{C}$ is stable and $\mathcal{C}_s$ is unstable. The black star in each figure shows the fixed point of each model.
• Figure 2: Examples of trajectories obtained from eq.(\ref{['eqn:mp_model']}) with $B=-1$. (a) $(T,D)=(2,3)$, (b) $(T,D)=(1,2/3)$. One with the blue squares starts at a state in (II)-1, and the others with the red circles start at two different states in (II)-2. They reach the fixed point $(-1,-1)$ (the black star) at the finite time steps. Note that the red trajectories show excitability.
• Figure 3: Green area ($D>T^2+T+1$) shows the region for $p=4$. The red line ($D>1$, $T=0$) corresponds to the case of the max-plus negative feedback model.
• Figure 4: The conditions for the existence of limit cycles when (a) $D=2$ and (b) $T=2$. These conditions depicted as the gray mesh regions are enclosed by (a) $X_n(T,D=2)=0$ (red curve) and $Y_n(T,D=2)=0$ (blue curve) and by (b) $X_n(T=2,D)=0$ (red curve) and $Y_n(T=2,D)=0$ (blue curve).
• Figure 5: (a) The regions $\mathcal{R}_2$ and $\mathcal{R}_3$. $\mathcal{R}_2$ is obtained as $(T, D)$ satisfying the condition (\ref{['condition1']}). $\mathcal{R}_3$ is enclosed by $X_3(T,D)=0$ (red) and $Y_3(T,D)=0$ (blue). (b) The regions $\mathcal{R}_2, \ldots, \mathcal{R}_5$.