periodically kicked rotor:
Post on 02-Jan-2016
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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
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