2 Phase and Amplitude Description of Complex Oscillatory Patterns …
21
(a)
0.68
-0.91
0
2
0
80
x
0
2
0
80
x
(b)
(c)
0.22
-0.22
0
optimal
sinusoidal
Fig. 2.5 Forcing patterns and phase coupling functions. a, b One-period evolution of the u component of the a optimal forcing q opt (x, θ) and b sinusoidal forcing q sin (x, θ) for 0 ≤ θ < 2π. c Phase
coupling functions for the optimal and sinusoidal forcing patterns
Figure 2.5 shows the optimized forcing, together with a spatiotemporally sinusoidal
forcing for comparison, and the resulting phase coupling functions for the oscillatingspot solution of the FHN model (see Sect. 2.5 for details).
As we demonstrate in the next section, the optimized forcing q opt gives higher
stability of the entrained state than the sinusoidal forcing pattern q sin of the same
power. However, it can also happen that the optimized pattern perturbs the system
too efficiently and kicks the system state far away from the limit cycle, where the
reduced equations are no longer accurate. If so, we may need to decrease the forcing
power and may not be able to improve the stability of the entrainment as desired.
In order to cope with this problem, we consider a simple feedback stabilization
of the oscillatory pattern, which suppresses the deviation of the system state from
the unperturbed limit cycle χ. To this end, we evaluate the phase θ = t)] of
the system state X(·, t), calculate the difference vector y(x, t) = X(x, t) − X 0 (x, θ),
and apply a feedback forcing of the form −αy(x, t) to the RD system, Eq. (2.16), as
∂X(x, t)
∂t
= F(X(x, t); x) + D∇
2 X(x, t) + q(x, ,t) − t),
(2.26)
where α > 0 is the feedback gain.
2
When X is sufficiently close to χ, we can show that this y is (bi-)orthogonal to
the phase sensitivity function Z. Indeed, we can express the phase of X as
θ = t)] = 0 (·, θ) + y(x, t)] ] 0 (·, θ)] +
V
Z(x, θ) · y(x, t)dx
(2.27)
by retaining the lowest-order functional Taylor expansion of in y. Therefore,
V Z(x, θ) · y(t)dx = 0 holds at the lowest order because θ = 0 (·, θ)]. Thus,
2 We here simply assume that the whole spatial pattern can be directly observed. This may not be
realistic in practical control problems and some approximate methods may have to be devised.
21
(a)
0.68
-0.91
0
2
0
80
x
0
2
0
80
x
(b)
(c)
0.22
-0.22
0
optimal
sinusoidal
Fig. 2.5 Forcing patterns and phase coupling functions. a, b One-period evolution of the u component of the a optimal forcing q opt (x, θ) and b sinusoidal forcing q sin (x, θ) for 0 ≤ θ < 2π. c Phase
coupling functions for the optimal and sinusoidal forcing patterns
Figure 2.5 shows the optimized forcing, together with a spatiotemporally sinusoidal
forcing for comparison, and the resulting phase coupling functions for the oscillatingspot solution of the FHN model (see Sect. 2.5 for details).
As we demonstrate in the next section, the optimized forcing q opt gives higher
stability of the entrained state than the sinusoidal forcing pattern q sin of the same
power. However, it can also happen that the optimized pattern perturbs the system
too efficiently and kicks the system state far away from the limit cycle, where the
reduced equations are no longer accurate. If so, we may need to decrease the forcing
power and may not be able to improve the stability of the entrainment as desired.
In order to cope with this problem, we consider a simple feedback stabilization
of the oscillatory pattern, which suppresses the deviation of the system state from
the unperturbed limit cycle χ. To this end, we evaluate the phase θ = t)] of
the system state X(·, t), calculate the difference vector y(x, t) = X(x, t) − X 0 (x, θ),
and apply a feedback forcing of the form −αy(x, t) to the RD system, Eq. (2.16), as
∂X(x, t)
∂t
= F(X(x, t); x) + D∇
2 X(x, t) + q(x, ,t) − t),
(2.26)
where α > 0 is the feedback gain.
2
When X is sufficiently close to χ, we can show that this y is (bi-)orthogonal to
the phase sensitivity function Z. Indeed, we can express the phase of X as
θ = t)] = 0 (·, θ) + y(x, t)] ] 0 (·, θ)] +
V
Z(x, θ) · y(x, t)dx
(2.27)
by retaining the lowest-order functional Taylor expansion of in y. Therefore,
V Z(x, θ) · y(t)dx = 0 holds at the lowest order because θ = 0 (·, θ)]. Thus,
2 We here simply assume that the whole spatial pattern can be directly observed. This may not be
realistic in practical control problems and some approximate methods may have to be devised.