Institute for Computational Neuroscience
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Institute for Computational Neuroscience |
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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: ¡¡ |
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