## 5 Experiment and Analysis
### Setting Parameters to Generate Flower Patterns  
In this experiment we used parameter settings to generate the basic elements that make up the flower and the entire flower pattern. Nonlinear functions based on Equation (6) and Equation (10) generated the basic elements of flower-leaves and petals. According to Equation (7) and Equation (11), the parameters \(a\), \(b\), and \(a_{1}\) affect the shapes of the flower-leaves, and petals. Among these, parameters \(a\) and \(b\) control the basic shape of the flower-leaves and parameter \(a_{1}\) controls the size of the basic shape of the petals. We fixed the size of the petals and set \(a_{1}=1\), that is, the length of the petals to 1. Figure 5 illustrates different basic shapes which were generated by setting the parameters as \(a=0.2\), \(b=1.0\), \(a=0.2\), \(b=5.0\), \(a=0.2\), \(b=10.0\), \(a=0.2\), \(b=20.0\), \(a=0.1\), \(b=10.0\), \(a=0.2\), \(b=10.0\), \(a=0.3\), \(b=10.0\), \(a=0.4\), and \(b=10.0\).  
The nonlinear function based on Equation (14) generates the essential elements of pistils. According to Equation (15), the parameters \(r_{1}\) and m affect the shape of the pistil. Among these, parameter \(r_{1}\) controls the size of the basic shape of the pistil and parameter \(m\) controls the number of pistils, which are denoted by m.  
The petal size and number of petals were fixed as \(a_{1}=1\), i.e., the petal's length is 1, and \(n=2\), i.e., the number of petals is 8. Figure 6 illustrates the different basic shapes which were generated by setting parameters as \(r_{1}=0.6a_{1}\), \(m=16\), \(r_{1}=0.7a_{1}\), \(m=16\), \(r_{1}=0.8a_{1}\), \(m=16\), \(r_{1}=0.9a_{1}\), \(m=16\), \(r_{1}=0.8a_{1}\), \(m=8\), \(r_{1}=0.8a_{1}\), \(m=8\), \(r_{1}=0.8a_{1}\), \(m=16\), \(r_{1}=0.8a_{1}\), and \(m=32\).  
Figure 4: Flower pattern generation process.  
The basic shape of the pistil is controlled by parameters \(r_{1}\) and \(m\). Firstly, the value of parameter \(m\) was fixed to observe the influence of the change in parameter \(r_{1}\) on the pattern. It was found that when the parameter \(r_{1}=0.8a_{1}\), the obtained pattern was closest to the original pattern. Secondly, the influence of the change in parameter \(m\) on the pattern was observed for \(r_{1}=0.8a_{1}\). It was found that when parameter \(m=16\), the distribution of pistils was neither sparse nor dense, and the obtained pattern was closest to the original pattern. Next, we used the basic graphs with \(a=0.2\), \(b=10\), \(r_{1}=0.8a_{1}\), and \(m=16\) to conduct subsequent experiments to generate flower patterns.  
The basic elements of flower-cores were generated by a nonlinear function based on Equation (20). According to Equation (21), the parameters \(a_{2}\) and \(a_{3}\) affect the shape of the flower-core. Parameter \(a_{2}\) controls the size of the basic shape of the flower-core and parameter \(a_{3}\) controls the size of the round shape of the flower-core. Figure 7 indicates different basic shapes which were generated by setting parameters as \(a_{2}=0.3a_{1}\), \(a_{2}=0.4a_{1}\), \(a_{2}=0.5a_{1}\), \(a_{2}=0.6a_{1}\), \(a_{3}=0.1a_{2}\), \(a_{3}=0.2a_{2}\), \(a_{3}=0.3a_{2}\), and \(a_{3}=0.4a_{2}\).  
Figure 5: Different basic flower patterns generated by setting parameters \(a\) and \(b\).  
Figure 6: Different basic flower patterns generated by setting parameters \(a_{1}\) and \(m\).  
Parameter \(a_{2}\) controls the basic shape of the flower-core. By observing Figure 7, we can see that the pattern obtained when \(a_{2}=0.5a_{1}\) is closest to the original pattern. The parameter \(a_{3}\) controls the size of the central circle. By observing Figure 7, we can see that the ratio between the two patterns is the most harmonious when \(a_{3}=0.2a_{2}\).  
The parameters of the entire flower pattern have values of \(a=0.2\), \(b=10.0\), \(r_{1}=0.8a_{1}\), \(m=16\), \(a_{2}=0.5a_{1}\), and \(a_{3}=0.2a_{2}\).