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Torus Bifurcations and Dynamical Transitions in Quasiperiodically Forced Maps

This study explores the bifurcations and transitions that occur in quasiperiodically forced maps, specifically focusing on the Torus Pitchfork Bifurcation (PFB) and various dynamical regimes such as gradual fractalization, attractor-merging crisis, and transition to chaos. The influence of rational approximations, symmetry, and diffusive motion is also investigated.

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Torus Bifurcations and Dynamical Transitions in Quasiperiodically Forced Maps

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  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 Diffusive CA  = 0 Boundary Crisis Diffusive CA Sym SNA 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 (*=0.765 610 8)

  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, =0.75657 Left TPT A=1.0, =0.6276  Rich Dynamical Regimes Exist near The TPT Points Asym SNA Sym CA Sym SNA Sym SNA Right TPT x Left TPT x 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) Attractor- Merging Crisis Transition to Chaos Gradual Fractalization 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

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