In this section, we will analyze the dynamic behaviors of VOFVDPM by the above numerical scheme. The cases where the orders of the derivatives in VOFVDPM are varying with different linear and periodical functions of time are investigated. The experiments under \(a=0\) are performed. Figures 1-11 are numerical results of VOFVDPM (6) for the cases where the variable-order functions \(q_{1},q_{2}\) are linear and periodic. We select the following uniform grid: \(T=300\) (\(t\in[0,300]\)), \(h=0.02\), \(N=15{,}000\), \(t_{n}=0.02n\), \(t_{0}=0\), \(t_{N}=300\).

Figures 1(a), (b) exhibit the phase plots for linear variable-order functions \(q_{1}(t)=q_{2}(t)\), which are linear decreasing and increasing in the same range \([0.9,1]\) with the same speed, \(1/3{,}000\), respectively. It shows that the responses of the VOFVDPM with linear decreasing order function approach a limit cycle faster than with a linear increasing order function. The shapes of the limit cycles with linear decreasing and increasing variable-order functions are different in spite of the same range and speed. Figure 2 shows the phase plots for linear variable-order functions with different intercepts. From it, we can find that the scale of the limit cycles becomes smaller with smaller intercepts in the linear variable-order functions.

In order to investigate the dynamic characteristics of the VOFVDPM with \(q_{1}\) not being equal to \(q_{2}\), Figures 3-5 exhibit the phase plots and trajectories for the case: \(q_{1}(t)=1-4t/3{,}000, q_{2}(t)=0.6+4t/3{,}000\), the case: \(q_{1}(t)=0.9-4t/3{,}000, q_{2}(t)=0.5+4t/3{,}000\), and the case: \(q_{1}(t)=0.6+4t/3{,}000, q_{2}(t)=1-4t/3{,}000\). For Figure 3, \(q_{1},q_{2}\) have the same range \([0.6,1]\), but decrease and increase with opposite slopes, respectively. It shows that the responses of the VOFVDPM have convergence to the limit point \((0,0)\). Figure 4 for linear \(q_{1},q_{2}\) with the same ranges \([0.5,0.9]\) and opposite slopes, also has the same dynamic behaviors and faster reaches the limit point \((0,0)\) than the case in Figure 3. However, the trajectories are divergent when \(q_{1},q_{2}\) increase and decrease in the same range with opposite slopes, respectively (Figure 5).

Figure 6 shows the phase plots for the VOFVDPM with the orders of derivative being different periodic order functions. From it, we find that the responses of them reach a limit cycle. Comparing Figures 6(a) to (b), the scale of the limit cycle with wider amplitude of the order functions are larger. Figure 6(b) and (d) have a periodic variable-order function with different periods, but it seems that significant differences are not discovered yet. Figures 6(a) and (c) exhibit different shapes of the limit cycle when the ranges of periodical variable-order functions are different. From Figure 6, the periodic variable-order functions have little effect on the responses of the VOFVDPM.

Figures 7-11 present the phase plots and trajectories of the VOFVDPM with \(q_{1},q_{2}\) being linear and periodic functions. The order functions \(q_{1},q_{2}\) are periodic and linear, respectively, in the same range for Figures 7-10, where the ranges are \([0.3,0.5]\), \([0.4,0.6]\), \([0.5,0.7]\), \([0.6,0.8]\). We can find that the responses of the model approach the point \((0,0)\) in Figures 7-8 with smaller orders, and they are in convergence to a limit cycle in Figures 9-10 with larger orders. However, Figure 11 shows that the trajectories are divergent, where the order functions \(q_{1},q_{2}\) are linear and periodic, respectively, in the same range, \([0.6,0.8]\).

The above numerical experiments investigate the dynamic behaviors of the VOFVDPM with different linear and periodical functions in the case \(a=0\). It shows that some novel dynamic characteristics of the model have been found in the numerical results, such as the existence of a limit point \((0,0)\) when linear \(q_{1},q_{2}\) are in the same ranges and have opposite slopes. These novel dynamic characteristics indicate that the limit cycle is not the only characteristic of the FVDP model when the order of the derivative is a time-dependent function.