*3.4. Nonlinear Dynamic System Chaotic Analysis*

To the best of our knowledge, Equation (19) is a nonlinear dynamic formula, meaning some parameters will bring chaos into this system. Chaotic represents an irregular and random movement that exists in a deterministic nonlinear system, e.g., butterfly effect. To study if the chaotic characteristic exists in the nonlinear dynamic formula in the setting of some threshold values, the Lyapunov exponent diagram is used to analyze the characteristic of the nonlinear dynamic system [30]. The Lyapunov exponent graph is used to analyze the convergence of adjacent trajectories. Especially, the nonlinear dynamic system shows stability characteristics when *LLE* < 0, where LLE means largest Lyapunov exponent. In contrast, if *LLE* = 0, the nonlinear dynamic system bifurcates at that point; if *LLE* > 0, the nonlinear dynamic system shows chaotic behavior [31].

Taking the three-party evolutionary game framework construction waste recycling system as examples, the LLE graphs are obtained based on Benetthin algorithm. As shown in Figure 2, fixing other parameters, *LLE* < 0 when construction waste sorting cost *C*<sup>1</sup> belongs to (0, 8.1),(9.8, 10.2),(10.8, 12.4),(17.6, 18.5),(19.8, 20), resulting in stable construction waste recycling system. In contrast, if *LLE* > 0, where *C*<sup>1</sup> ∈ (8.1, 9.8),(10.2, 10.8),(12.4, 17.6) and (18.5, 19.8), the construction waste recycling system is going to show chaotic characteristic (as shown in Figure 2a. From Figure 2b, it can be observed that *LLE* < 0 if *C*<sup>0</sup> ∈ (4.2, 4.8), (5.8, 6.5), (6.8, 8.2), (8.5, 9.7), (10.7, 11.5), (16.1, 16.3), (16.9, 18.1), and *C*<sup>0</sup> ∈ (18.3, 19.8), respectively and the system stay in stable state. If *C*<sup>0</sup> ∈ (0, 4.2),(4.8, 5.8), and *C*<sup>0</sup> ∈ (6.5, 6.8), (8.2, 8.5), (9.7, 10.7),(11.5, 16.1),(16.3, 16.9),(18.1, 18.3),(19.8, 20), respectively, then *LLE* > 0 and the nonlinear dynamic system shows chaotic characteristic. It is also observed *LLE* > 0 when *η* ∈ (0.32, 0.44),(0.49, 0.71),(0.74, 0.78),(0.83, 0.84), and *η* ∈ (0.91, 0.92),(0.99, 1) in Figure 2c. This also make system show chaotic characteristic.

**Figure 2.** Largest Lyapunov exponent diagram of tripartite stochastic evolutionary game system with fixed parameters are *Ss* = 10, *Sj* = 10, *F*<sup>1</sup> = 15, *F*<sup>2</sup> = 20, *G* = 30, *G*<sup>1</sup> = 15, *Eg* = 8, *Cg* = 5, *Pj* = 15, Δ*Cj* = 8, *C* = 30, *λ* = 0.2, *R* = 45. (**a**) *C*<sup>0</sup> = 10, *η* = 0.7. (**b**) *C*<sup>1</sup> = 20, *η* = 0.7. (**c**) *C*<sup>0</sup> = 10, *C*<sup>1</sup> = 20.
