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Abstract In this paper, we study the eventual periodicity of the fuzzy max-type difference equation x n = max { C , x n − m − k x n − m } , n ∈ { 0 , 1 , … } $x_{n} =\max \{C , \frac{x_{n-m-k}}{x_{n-m} }\}, n\in \{0,1,\ldots \} $ , where m and k are positive integers, C and the initial values are positive fuzzy numbers. Let the support supp C = { t : C ( t ) > 0 } ‾ = [ C 1 , C 2 ] $\operatorname{supp} C=\overline{\{t : C(t) > 0\}}=[C_{1},C_{2}]$ of C. We show that: (1) if C 1 > 1 $C_{1}>1$ , then every positive solution of this equation equals C eventually; (2) there exists a positive fuzzy number C with C 1 = 1 $C_{1}=1$ such that this equation has a positive solution which is not eventually periodic; (3) if C 2 ≤ 1 $C_{2}\leq 1$ , then this equation has a positive solution which is not eventually periodic; (4) if C 1 < 1 < C 2 $C_{1}

Abstract In this paper, we study the following max-type system of difference equations of higher order: { x n = max { A , y n − t x n − s } , y n = max { B , x n − t y n − s } , n ∈ { 0 , 1 , 2 , … } , $$ \textstyle\begin{cases} x_{n} = \max \{A ,\frac{y_{n-t}}{x_{n-s}} \}, \\ y_{n} = \max \{B ,\frac{x_{n-t}}{y_{n-s}} \},\end{cases}\displaystyle \quad n\in \{0,1,2,\ldots \}, $$ where A , B ∈ ( 0 , + ∞ ) $A,B\in (0, +\infty )$ , t , s ∈ { 1 , 2 , … } $t,s\in \{1,2,\ldots \}$ with gcd ( s , t ) = 1 $\gcd (s,t)=1$ , the initial values x − d , y − d , x − d + 1 , y − d + 1 , … , x − 1 , y − 1 ∈ ( 0 , + ∞ ) $x_{-d},y_{-d},x_{-d+1},y_{-d+1}, \ldots , x_{-1}, y_{-1}\in (0,+ \infty )$ and d = max { t , s } $d=\max \{t,s\}$ .

Abstract In this paper, we investigate hybrid adaptive synchronization issue for a class of perturbed fractional-order chaotic systems with nondeterministic nonlinear terms. On the basis of fractional-order extended version of Lyapunov stability criterion, a novel fuzzy adaptive synchronization control protocol coupled with backstepping-based method is constructed, ensuring that the synchronization errors converge to a sufficiently small region of the origin. In order to avert the occurrence of “explosion of complexity”, we take advantage of a fuzzy logic system to estimate the unknown systematic term approximately in every backstepping step. Finally, some numerical simulations are given to exemplify the effectiveness of the proposed approach.

Abstract In this paper, we study the following max-type system of difference equations: {xn=max{An,yn−1xn−2},yn=max{Bn,xn−1yn−2},n∈{0,1,2,…}, $$\textstyle\begin{cases}x_{n} = \max \{A_{n},\frac{y_{n-1}}{x_{n-2}} \},\\ y_{n} = \max \{B_{n} ,\frac{x_{n-1}}{y_{n-2}} \}, \end{cases}\displaystyle n\in \{0,1,2,\ldots\}, $$ where An,Bn∈(0,+∞) $A_{n},B_{n}\in(0, +\infty)$ are periodic sequences with period 2 and the initial values x−1,y−1,x−2,y−2∈(0,+∞) $x_{-1},y_{-1},x_{-2},y_{-2}\in (0,+\infty)$. We show that every solution of the above system is eventually periodic.