1. Torus Bifurcations and Dynamical Transitions
in Quasiperiodically Forced Maps
W. Lim and S.-Y. Kim
Department of Physics
Kangwon National University
 Quasiperiodically Forced Circle Map
 Symmetry
(=0 and 1/2)
 Rational Approximation (RA) of The Quasiperiodic Forcing

2. Birth of The Smooth Tori through The Torus Pitchfork Bifurcation
A symmetric torus becomes unstable through a torus pitchfork bifurcation (PFB),
and then a pair of asymmetric tori appears.
Symmetric torus for A=0.9 and =0.77
 = 0
Diffusive CA
Boundary Crisis
Sym
SNA
Diffusive CA
Sym SNA
Gradual
Fractalization
Sym CA
Diffusive
SNA
Gradual
Fractalization
Torus PFB
Attracter-
Merging
Crisis
Asym
SNA
TPT
A pair of asymmetric tori for =0.76
Gradual
Fractalization
TPT
Sym T
Sym T
Asym T
PFB
PFB

3. Rational Approximations of The Torus PFB
Even-periodic forcing: The PFB occurs in the RA.
Odd-periodic forcing: The PFB is replaced by a saddle-node bifurcation.
n=8 (Fn=21)
n=9 (Fn=34)
n=10 (Fn=55)
As n increases, the values of <n> approach * and the variance approach 0.
 Phase-Independent PFB (*= )

4. Torus Pitchfork Terminal (TPT) Points
Torus PFB lines terminate at the TPT points.
 Smooth Torus on The Torus PFB lines  Fractal Torus at The TPT Points
Right TPT
A=1.1095, =
Left TPT
A=1.0,
=0.6276
 Rich Dynamical Regimes Exist near The TPT Points
Sym CA
Sym
SNA
Asym
SNA
Sym
SNA
Right TPT x x
Left TPT
Sym SNA
Sym T
Sym T
Asym T
Asym SNA
Asym T

5. Dynamical Transition near The Left TPT Point
 Left Side of The Left TPT Point (A=1.08)
Gradual Fractalization
Sym T Sym SNA
(=0.6) (=0.66)
 Right Side of The Left TPT Point (=0.67)
Gradual Fractalization
Attractor-Merging Crisis
Asym T Asym SNA Sym SNA
(A=0.8) (A=1.056) (A=1.15)

6. Dynamical Transition near The Right TPT Point
 Left Side of The Right TPT Point (=0.74)
Gradual
Fractalization
Attractor-
Merging Crisis
Transition
to Chaos
Asym T Asym SNA Sym SNA Sym CA
(A=0.8) (A=1.0875) (A=1.1) (A=1.2)
 Right Side of The Right TPT Point (A=1.2)
Gradual Fractalization
Transition to Chaos
Sym T Sym SNA Sym CA
( =0.84) ( =0.762) ( =0.75)

7. Transition to A Diffusive Motion via A Crisis
As  changes, the SNA collides with the unstable torus.
 A transition to a diffusive SNA via a crisis occurs.
SNA and unstable torus for A=1.2 and =0.68
Diffusive SNA for =0.72
The diffusion of the SNA is normal.
Diffusion coeffcient:

8. Summary
Birth of Smooth Tori through The Phase-Independent Torus Pitchfork Bifurcation
Investigation of The Torus Pitchfork Bifurcation by The Ration Approximation of
The Quasiperiodic Forcing
Rich Dynamical Transitions
- Birth of SNA through The Gradual Fractalization
- Symmetric-Restoring Attractor-Merging Crisis
- Transition to Chaos
- Transition to A Diffusive Motion via A Crisis