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Dynamical Transitions in Quasiperiodically Forced Systems [PPT]

 We investigate Dynamical Transitions in the quasiperiodically forced systems driven at two incommensurate frequencies. These dynamical systems have received much attention, because they have strange nonchaotic attractors (SNAs). SNAs exhibit properties of regular as well as chaotic attractors. Like regular attractors, their dynamics is nonchaotic in the sense that they do not have a positive Lyapunov exponent; such as typical chaotic attractors, they exhibit a fractal phase space structure. Furthermore, SNAs are related to the Anderson localization in the Schrodinger equation with a spatially quasiperiodic potential, and they may have a practical application in secure communication. Hence, dynamical transitions in quasiperiodically forced systems have become a topic of considerable interest. However, the mechanisms for their occurrence are much less clear than those of unforced or periodically forced systems. It is well known that every irrational number, corresponding to a quasiperiodic forcing, may be approximated by a sequence of rationals yielding periodic forcing, using a continued fraction representation. Based on such "Rational Approximations (RAs)" of the quasiperiodic forcing, we study the dynamics in a sequence of periodically forced systems, instead of directly studying the quasiperiodically forced system. Such a periodically forced system has an attractor that depends on the initial phase of the external force. Hence, the union of all attractors for different initial phases gives an approximation to the attractor in the quasiperiodically forced system. Thus the property of the quasiperiodically forced system may be obtained by taking the limit of the RAs. Using this RA, we first investigate the mechanism for the intermittent route to SNAs in quasiperiodically forced logistic map. It is thus found that a smooth torus transforms into an intermittent SNA via a phase-dependent saddle-node bifurcation when it collides with a new type of invariant ``ring-shaped'' unstable set which has no counterpart in the unforced case. Such intermittent transition is also found to occur via the same bifurcation mechanism in  quasiperiodically forced invertible period-doubling systems such as the quasiperiodically forced Henon map, ring map, and Toda oscillator. Hence, the ``universality'' for the intermittent route to SNAs is confirmed. Next, we also investigate the effect of the quasiperiodic forcing on the dynamical transitions occurring in the unforced logistic map. For small quasiperiodic forcing, a ``standard'' transition occurs through a collision with a smooth unstable torus, as in the logistic map. However, when the quasiperiodic forcing passes a threshold, the smooth  unstable torus becomes inaccessible from the interior of the basin of an attractor (smooth attracting torus, SNA, or chaotic attractor). For this case, a new type of transition (e.g., boundary crisis, interior crisis, and band-merging transition) is found to occur via a collision with the ring-shaped unstable set. Consequently, the ring-shaped unstable sets play a central role for dynamical transitions in quasiperiodically forced systems. For more details, see the following publications:

[1] S.-Y. Kim, W. Lim, and E. Ott, "Mechanism for the intermittent route to strange nonchaotic attractors," Phys. Rev. E 67, 056203 (2003).
[2] S.-Y. Kim, W. Lim, and A. Jalnine, "Intermittent transitions in the quasiperiodically forced maps," Applied Nonlinear Dynamics. 11, 55-62 (2003).
[3] J.-W. Kim, S.-Y. Kim, B. Hunt, and E. Ott, "Fractal properties of robust strange nonchaotic attractors in maps of two or more dimensions," Phys. Rev. E 67, 036211 (2003).
[4] W. Lim and S.-Y. Kim, "Intermittent strange nonchaotic attractors in quasiperiodically forced systems," J. Korean Phys. Soc. 44, 514-517 (2004).
[5] S.-Y. Kim and W. Lim, "Universal mechanism for the intermittent route to strange nonchaotic attractors in quasiperiodically forced systems," J. Phys. A 37, 6477 (2004).
[6] S.-Y. Kim and W. Lim, "Dynamical Mechanism for boundary crises in quasiperiodically forced systems," Phys. Lett. A 334, 160-168 (2005).
[7] W. Lim and S.-Y. Kim, "Dynamical Mechanism for band-merging transitions in quasiperiodically forced systems," Phys. Lett. A 335, 383-393 (2005).
[8] W. Lim and S.-Y. Kim, "Mechanism for new boundary crises in quasiperiodically forced systems," J. Korean Phys. Soc. 46, 642-645 (2005).
[9] W. Lim and S.-Y. Kim, "Interior crises in quasiperiodically forced period-doubling systems," Phys. Lett. A 335, 331 (2006).

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