n

ntBA )()sin(

Periodically kicked rotor:

Circle map )2sin(21 nnn

k

For the linear map 1 1n n mod

0 0nf n W

Ω rational → periodic

Ω irrational → quasi-periodic

nLimW n

n

0

Winding Number

W=5/8

1

6

3

4

2

5

0

7

Continued Fraction

0

1

23

11

1

G aa

aa

01 2 3

1 1 1a

a a a

0 1 2 3, , ,a a a a

nth order approximation:

0

1

23

11

1

n

n

G aa

aa a

0 0G a

01 2 3

1 1 1 1

n

aa a a a

0 1 2 3, , , , na a a a a

1 01

1G a

a 2 0

12

11

G aa

a

3 0

1

23

11

1

G aa

aa

Approximation of irrational number by rational fractions

0 0a 1ia 11

11

11

G

Golden Ratio

0 0G 1

11

1G 2

1 11 211

G

3

1 1 21 1 31 1

1 211

G

1

1

1nn

GG

→

4

1 32 513

G

5

1 53 815

G

2 1 0G G 11 5

2G

Fibonacci series: 1,2,3.5.8….

flowers

Sunflower

Pineapple

Human body

Quasiperiodicity

?

Quasiperiodicity

Torus

Quasiperiodic

Quasiperiodic

Winding Numbers

Frequency-ratio parameter:

(Rotation number)

number of times the trajectory winds around the small cross-section of the torus after going once around the large circumference.

q

p

2

1

Hopf Bifurcation

Landau route to turbulence

Ruelle-Taken-Newhouse route to Chaos

Ref: Argyris etl

Quasiperiodic route to Chaos

Ref: Argyris etl

Ref: Argyris etl