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Onthesolutionofrecursivesequence...
11
Theorem5.Supposethat{xn}∞
nl11beasolutionofequation(4).Then
allsolutionsofequation(4)areperiodicwithperiodfive.
Proof.FromEq(4),weseethat
xn+1=
xn11(xn−1)
xn
j
xn+2=
xn(xn+1−1)
xn+1
=
xn11xn(xn−1)(
xn
xn11(xn−1)
xn
−1)
xn+3=
=
(xn−xn11xn+xn11)
xn+1(xn+2−1)
xn+2
1
j
xn11(xn−1)
=
(xn−xn11xn+xn11)xn(
(xn−xn11xn+xn11)
1
−1)
=
xn(1−xn+xn11xn−xn11)
xn11(xn−1)
=
xn(xn−1)(xn11−1)
xn11(xn−1)
=
xn(xn11−1)
xn11
.
xn+4=
xn+2(xn+3−1)
xn+3
=
xn(xn11−1)(
(xn−xn11xn+xn11)xn11
xn(xn11−1)
xn11
−1)
=xn11
xn+5=
xn+3(xn+4−1)
xn+4
=
xn11(xn11−1)xn
xn11(xn11−1)
=xn.
Thiscompletestheproof.
.
Theorem6.Eq(4)havethreeequilibriumpointswhichare0,1+√5
2
,
11√5
2
.
Proof.FortheequilibriumpointsofEq(4),wecanwrite
x=
x(x−1)
x
.
Then
x3−x2−x=0j
or
x(x2−x−1)=0.
ThustheequilibriumpointsofEq(4)isx=0,x=1+√5
2
,x=11√5
2
.
.
