This article delves into the trajectory structure rules of a specific fifth-order rational difference equation: $$ s_{m+1}=\frac{s_ms_{m-2}s_{m-3}s_{m-4}+s_ms_{m-2}+s_ms_{m-3}+s_{m-2}s_{m-3}+s_{m-4}+a}{s_ms_{m-2}s_{m-3}+s_ms_{m-2}s_{m-4}+s_ms_{m-3}s_{m-4}+s_{m-2}s_{m-3}s_{m-4}+1+a} $$ where the initial conditions satisfy $s_i\in (0,\infty)$, $i=-4,-3,-2,-1,0$, and the parameters $a\in [0,\infty).$ As the initial values vary, the lengths of consecutive positive and negative semi-cycles for non-trivial solutions exhibit a periodic pattern with a prime period of 31. The rule within one period is $1^-, 2^+, 1^-, 1^+, 1^-, 1^+, 2^-, 4^+, 3^-, 2^+, 2^-, 1^+, 5^-, 1^+, 1^-,$ $ 3^+ $. Through t... More >

The paper discusses the dynamical characteristics of solutions to a model with quadratic term. More precisely, an exponential-type fuzzy difference equation is proposed as follows $$ a_{n+1}=\frac{D+Pe^{-a_n}}{T+a^2_{n-1}},\ \ n=0,1,\cdots ,$$ here $D, P, T$ and $a_0, a_{-1}$ belong to positive fuzzy numbers. This model can be used to characterize the diffusion modeling of a class of infectious diseases with uncertainty, such as the transmission prediction of dengue fever, monkeypox, and other infectious diseases. In addition, by highlighting the advantages of using Stefanini's the generalization of division of fuzzy number (it is also known as g-division) and constructing a Lyapunov functi... More >