Biomedical Engineering Reference
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9.3.2 Periodic Forcing
Let us now explore the response of the system to a periodic forcing. Again,
we shall use the simplest possible forcing term: a periodic “kick” to the sys-
tem by impulses applied to one coordinate. It would be tempting to try a
sinusoidal forcing (which would of couse also be periodic), but impulses have
an advantage: the forcing is simply a displacement of the state of the system
each time a kick occurs. Between kicks, the system evolves by the rules (9.5).
These rules, on the other hand, give rise to simple dynamics: a rapid decay
to ρ = 1 (assumed to be almost instantaneous if λ is large), and a free evolu-
tion of the phase at a constant rate. In summary, the dynamics of the kicked
system are as follows: the system is kicked (by the way, this may cause an
important change in the phase of the oscillator), it rapidly evolves back to
the limit cycle and freely evolves along it, until it is kicked again. The process
then repeats itself while the forcing is “on”. Our challenge is to understand
how a structure such as that displayed in Fig. 9.3b can emerge out of this
procedure.
In Fig. 9.4, we display the processes described in the previous paragraph.
The circle represents the unforced limit cycle. The system is assumed to be
at the point denoted by 1 in the n th step of our iteration procedure, its
state being described by the phase φ n = φ . When the system is kicked, by
means of a horizontal displacement of value A towards the point denoted by
2, the phase of the system is changed to φ a . The system then collapses rapidly
towards the limit cycle (the state denoted by 3 in the figure), finally to evolve
freely until its phase has increased by an additional amount α (equal to ωT ,
where T is the period of the forcing), at point 4. Then, a new kick is applied
to the oscillator. Since we are working in the limit of λ
1(whichmeans
ρ
1 almost instantaneously after a kick), the state of our forced oscillator
can be described in terms of the dynamics of the phase φ n only, which can
then be written as
1L =A2
2
L
L =1
φ
3
4
3
1
φ a
Fig. 9.4. Derivation of the phase map for a Poincare oscillator, periodically kicked
along the horizontal axis. The evolution can be described as the iteration of the
following steps: (1) the system is kicked, (2) it rapidly collapses back to the limit
cycle, (3) the system evolves with a monotonic increase of the phase and (4) it is
kicked again after a time T ,where T is the period of the forcing
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