Discrete Dynamical Fibonacci

Edward Early

Fibonacci Numbers

• F0 = 0, F1 = 1, Fn = Fn-1+Fn-2 for n > 1

• 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …

1 1 5 1 5

2 25

n n

nF

Differential Equations

• inspired by undetermined coefficients

• Fn = Fn-1+Fn-2 characteristic polynomial x2-x-1

• formula

• initial conditions F0 = 0, F1 = 1 give C1 and C2

• BIG DRAWBACK: students were already taking undetermined coefficients on faith (and this application not in book)

1 2

1 5 1 5

2 2

n n

C C

Linear Algebra

• discrete dynamical systems•

where λ1 and λ2 are the (distinct) eigenvalues of the 2×2 matrix A with eigenvectors v1 and v2, respectively, and x0=c1v1+c2v2

(Section 5.6 of Lay’s book)

0 1 1 1 2 2 2n n nA c c x v v

Linear Algebra Meets Fibonacci

• Let and

• Thus Anx0 has top entry Fn

0 1

1 1A

0

01

0

1

F

F

x

1

1 1

0 1

1 1n n n

n n n n

F F F

F F F F

Linear Algebra Meets Fibonacci

• Let and

• det(A-λI) = λ2-λ-1

• eigenvalues and

• eigenvectors and

0 1

1 1A

0

0

1

x

1 5

2

1 5

2

1

1 5

2

1

1 5

2

Linear Algebra Meets Fibonacci

• Let and

• top entry

0 1

1 1A

0

11

1 5 1 55

2 2

10

1

x

0

1 1 5 1 1 51 5 1 52

2

1

5

1

252

n n

nA

x0 1 1 1 2 2 2n n nA c c x v v

1 1 5 1 5

2 25

n n

nF

Caveat

• If

then…

• ugly enough to scare off most students!

0

1

1

x

0

15 5 5 5

1 5 1 510 102

1

2

x