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

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

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 (*=0.765 610 8)

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, =0.75657 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

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)

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)

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:

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