Dynamic Behavior of a Fractional-Type Fuzzy Difference System
Abstract
:1. Introduction
2. Preliminaries
- (1)
- is a normal fuzzy set, i.e., there exists such that ;
- (2)
- is a fuzzy convex set, i.e.,
- (3)
- is upper semicontinuous on ;
- (4)
- is compactly supported, i.e.,
- (1)
- If there exists a positive real number (or ) such that (or ), , then the sequence of positive fuzzy numbers is persistent (or bounded).
- (2)
- If there exist positive real numbers , such that , then the sequence of positive fuzzy numbers is persistent and bounded.
- (1)
- An equilibrium point is called locally stable if for any and every , there exists such that for any initial conditions , with , , we have for any .
- (2)
- An equilibrium point is called locally attractor if for any initial conditions .
- (3)
- An equilibrium point is called asymptotically stable if it is stable, and is also attractor.
- (4)
- An equilibrium point is called unstable if it is not locally stable.
- (1)
- If all eigenvalues of the Jacobian matrix about lie inside the open unit disk, i.e., , then is locally asymptotically stable.
- (2)
- If one of eigenvalues of the Jacobian matrix about has norm greater than one, then is unstable.
3. Main Results and Proofs
- (1)
- is nondecreasing and left continuous;
- (2)
- is nonincreasing and left continuous;
- (3)
- .
- (1)
- If , in view of , then all characteristic roots . Thus, according to Lemma 2, the equilibrium point is locally asymptotically stable.
- (2)
- If , in view of , then at least one root of the characteristic Equation (23) lies outside the unit disk. From Lemma 2, we have that the equilibrium point is unstable, and then the proof is completed. □
- (1)
- uniformly stable; if given , there exists a with , implies for any , such that for any the solution ;
- (2)
- uniformly attractive if there is a such that , one has ;
- (3)
- uniformly asymptotically stable if (1) and (2) hold simultaneously.
4. Numerical Example
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A
Appendix B. Matlab Calculation Codes
%function x,y %syms x y %x=rand([1,104]); %y=rand([1,104]); clear all format long %syms a a1=0; for j = 1:5 a1=a1+j*(1/4)-(1/4); n=150; x=zeros(n,1); y=zeros(n,1); x(1,1)=1+2*a1; x(2,1)=3+a1; x(3,1)=2+2*a1; x(4,1)=5+3*a1; x(5,1)=7+3*a1; y(1,1)=5-2*a1; y(2,1)=5-a1; y(3,1)=6-2*a1; y(4,1)=10-2*a1; y(5,1)=14-4*a1; for i=6:30 x(i+1,1)=((3/10)*x(i-1,1)*x(i-2,1)+3*x(i-3,1))/(12+6*y(i-4,1)); y(i+1,1)=((3/10)*y(i-1,1)*y(i-2,1)+3*y(i-3,1))/(12+6*x(i-4,1)); end N=50; n=[y(4:N)]; m=[x(4:N)]; box on hold on; plot(n,m,’Color’,[rand(),rand(),rand()],’LineWidth’,1.5) end xlabel(’L(n)’) ylabel(’R(n)’) disp(a1); legend(’a=0’,’a=0.25’,’a=0.5’,’a=0.75’,’a=1’) |
%function x,y %syms x y %x=rand([1,104]); %y=rand([1,104]); clear all format long %syms a n=150; a1=0; x=zeros(n,1); y=zeros(n,1); x(1,1)=1+2*a1; x(2,1)=3+a1; x(3,1)=2+2*a1; x(4,1)=5+3*a1; x(5,1)=7+3*a1; y(1,1)=5-2*a1; y(2,1)=5-a1; y(3,1)=6-2*a1; y(4,1)=10-2*a1; y(5,1)=14-4*a1; for i=6:50 x(i+1,1)=((3/10)*x(i-1,1)*x(i-2,1)+3*x(i-3,1))/(12+6*y(i-4,1)); y(i+1,1)=((3/10)*y(i-1,1)*y(i-2,1)+3*y(i-3,1))/(12+6*x(i-4,1)); end m=[x(7,1),x(8,1),x(9,1),x(10,1),x(11,1),x(12,1),x(13,1),x(14,1),x(15,1),x(16,1),x(17,1),x(18,1),x(19,1),x(20,1),x(21,1),x(22,1),x(23,1),x(24,1),x(25,1),x(26,1),x(27,1),x(28,1),x(29,1),x(30,1),x(31,1),x(32,1),x(33,1),x(34,1),x(35,1),x(36,1),x(37,1),x(38,1),x(39,1),x(40,1),x(41,1),x(42,1),x(43,1),x(44,1),x(45,1),x(46,1),x(47,1),x(48,1),x(49,1),x(50,1)] n=[y(7,1),y(8,1),y(9,1),y(10,1),y(11,1),y(12,1),y(13,1),y(14,1),y(15,1),y(16,1),y(17,1),y(18,1),y(19,1),y(20,1),y(21,1),y(22,1),y(23,1),y(24,1),y(25,1),y(26,1),y(27,1),y(28,1),y(29,1),y(30,1),y(31,1),y(32,1),y(33,1),y(34,1),y(35,1),y(36,1),y(37,1),y(38,1),y(39,1),y(40,1),y(41,1),y(42,1),y(43,1),y(44,1),y(45,1),y(46,1),y(47,1),y(48,1),y(49,1),y(50,1)] hold on; plot(m,’k*’) plot(n,’r--’) box on %grid on xlabel(’n’) ylabel(’L(n)&R(n)’) legend(’L(n)’,’R(n)’) |
%function x,y %syms x y %x=rand([1,104]); %y=rand([1,104]); clear all format long %syms a n=150; a1=0.5; x=zeros(n,1); y=zeros(n,1); x(1,1)=1+2*a1; x(2,1)=3+a1; x(3,1)=2+2*a1; x(4,1)=5+3*a1; x(5,1)=7+3*a1; y(1,1)=5-2*a1; y(2,1)=5-a1; y(3,1)=6-2*a1; y(4,1)=10-2*a1; y(5,1)=14-4*a1; for i=6:50 x(i+1,1)=((3/10)*x(i-1,1)*x(i-2,1)+3*x(i-3,1))/(12+6*y(i-4,1)); y(i+1,1)=((3/10)*y(i-1,1)*y(i-2,1)+3*y(i-3,1))/(12+6*x(i-4,1)); end m=[x(7,1),x(8,1),x(9,1),x(10,1),x(11,1),x(12,1),x(13,1),x(14,1),x(15,1),x(16,1),x(17,1),x(18,1),x(19,1),x(20,1),x(21,1),x(22,1),x(23,1),x(24,1),x(25,1),x(26,1),x(27,1),x(28,1),x(29,1),x(30,1),x(31,1),x(32,1),x(33,1),x(34,1),x(35,1),x(36,1),x(37,1),x(38,1),x(39,1),x(40,1),x(41,1),x(42,1),x(43,1),x(44,1),x(45,1),x(46,1),x(47,1),x(48,1),x(49,1),x(50,1)] n=[y(7,1),y(8,1),y(9,1),y(10,1),y(11,1),y(12,1),y(13,1),y(14,1),y(15,1),y(16,1),y(17,1),y(18,1),y(19,1),y(20,1),y(21,1),y(22,1),y(23,1),y(24,1),y(25,1),y(26,1),y(27,1),y(28,1),y(29,1),y(30,1),y(31,1),y(32,1),y(33,1),y(34,1),y(35,1),y(36,1),y(37,1),y(38,1),y(39,1),y(40,1),y(41,1),y(42,1),y(43,1),y(44,1),y(45,1),y(46,1),y(47,1),y(48,1),y(49,1),y(50,1)] hold on; plot(m,’k*’) plot(n,’r--’) box on %grid on xlabel(’n’) ylabel(’L(n)&R(n)’) legend(’L(n)’,’R(n)’) |
%function x,y %syms x y %x=rand([1,104]); %y=rand([1,104]); clear all format long %syms a n=150; a1=1; x=zeros(n,1); y=zeros(n,1); x(1,1)=1+2*a1; x(2,1)=3+a1; x(3,1)=2+2*a1; x(4,1)=5+3*a1; x(5,1)=7+3*a1; y(1,1)=5-2*a1; y(2,1)=5-a1; y(3,1)=6-2*a1; y(4,1)=10-2*a1; y(5,1)=14-4*a1; for i=6:50 x(i+1,1)=((3/10)*x(i-1,1)*x(i-2,1)+3*x(i-3,1))/(12+6*y(i-4,1)); y(i+1,1)=((3/10)*y(i-1,1)*y(i-2,1)+3*y(i-3,1))/(12+6*x(i-4,1)); end m=[x(7,1),x(8,1),x(9,1),x(10,1),x(11,1),x(12,1),x(13,1),x(14,1),x(15,1),x(16,1),x(17,1),x(18,1),x(19,1),x(20,1),x(21,1),x(22,1),x(23,1),x(24,1),x(25,1),x(26,1),x(27,1),x(28,1),x(29,1),x(30,1),x(31,1),x(32,1),x(33,1),x(34,1),x(35,1),x(36,1),x(37,1),x(38,1),x(39,1),x(40,1),x(41,1),x(42,1),x(43,1),x(44,1),x(45,1),x(46,1),x(47,1),x(48,1),x(49,1),x(50,1)] n=[y(7,1),y(8,1),y(9,1),y(10,1),y(11,1),y(12,1),y(13,1),y(14,1),y(15,1),y(16,1),y(17,1),y(18,1),y(19,1),y(20,1),y(21,1),y(22,1),y(23,1),y(24,1),y(25,1),y(26,1),y(27,1),y(28,1),y(29,1),y(30,1),y(31,1),y(32,1),y(33,1),y(34,1),y(35,1),y(36,1),y(37,1),y(38,1),y(39,1),y(40,1),y(41,1),y(42,1),y(43,1),y(44,1),y(45,1),y(46,1),y(47,1),y(48,1),y(49,1),y(50,1)] hold on; plot(m,’k*’) plot(n,’r--’) box on %grid on xlabel(’n’) ylabel(’L(n)&R(n)’) legend(’L(n)’,’R(n)’) |
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Jia, L.; Wang, C.; Zhao, X.; Wei, W. Dynamic Behavior of a Fractional-Type Fuzzy Difference System. Symmetry 2022, 14, 1337. https://doi.org/10.3390/sym14071337
Jia L, Wang C, Zhao X, Wei W. Dynamic Behavior of a Fractional-Type Fuzzy Difference System. Symmetry. 2022; 14(7):1337. https://doi.org/10.3390/sym14071337
Chicago/Turabian StyleJia, Lili, Changyou Wang, Xiaojuan Zhao, and Wei Wei. 2022. "Dynamic Behavior of a Fractional-Type Fuzzy Difference System" Symmetry 14, no. 7: 1337. https://doi.org/10.3390/sym14071337