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H
=
H
0
+
H
1
(
t
)
1
2
m
1
π
1
2
m
2
v
2
2
=
+
+
U
0
(ξ,ζ)
+
θ (
t
)εξ,
(4.121)
where the unperturbed potential has the form given for the prototype (
4.120
).
Suppose that we are dealing with an ensemble of networks. Imagine that an indi-
vidual unperturbed web of the ensemble of networks has coordinates
ξ(
0
)
,
ζ(
0
)
,
π(
0
)
0
−
)
=
and energy
E
Immediately after the abrupt application of the perturbation,
the web behaves as if it has “felt” the influence of the additional potential
(
E
.
εξ
.Ifthe
external perturbation is turned on at
t
=
0 the individual webs in the ensemble will
have the new energy
1
2
m
1
π
1
2
m
2
v
0
+
)
=
2
2
E
(
0
+
0
+
U
0
(ξ
0
,ζ
0
,
0
)
=
E
+
εξ(
0
),
(4.122)
where we have tagged the phase-space variable with the initial time label. In the
subsequent time evolution the webs will have an energy
E
0
+
)
(
fixed, and obviously for
the invariant energy
0
+
)
=
E
(
H
0
(ξ,ζ,π,v)
+
εξ.
(4.123)
From (
4.122
) and (
4.123
) it follows that the constant energy is given in terms of the
perturbed Hamiltonian
E
H
0
+
H
1
.
(4.124)
For the remainder of the section the energy
E
will be a constant quantity in time, while
the resulting interaction term changes in time with the motion of the doorway variable
ξ
=
H
0
(ξ,ζ,π,v)
+
ε
[
ξ
−
ξ (
0
)
]
≡
that the doorway
variable has at the time when the external perturbation is abruptly switched on.
It is important to note that if the interaction term
H
1
is small enough the domain
, according to the definition (
4.124
) and to the specific value
ξ (
0
)
)
is only slightly modified from the unperturbed situation. By contrast, the individual
trajectories are sensitively dependent on the perturbation, whatever its intensity, due to
the assumption that the unperturbed network is in the chaotic regime.
Thus, from the geometric deformation of this domain we can derive the correspond-
ing change in the mean value of the doorway variable of the Poincaré map, thereby
determining the susceptibility
(
E
χ
of the booster. The additional potential
ε
[
ξ
−
ξ (
0
)
]
ξ
R
(ζ
∗
)
ξ
L
(ζ
∗
)
moves the extrema
and
of the Poincaré map domain depicted in
Figure
4.21
towards the right (with
ε<
0
)
or the left (with
ε>
0
)
in a continuous way.
For sufficiently small values of
ε
the average response to the perturbation can be written
ξ
ε
=
εχ
(4.125)
and, using the formal expression for the mean value of the doorway variable (
4.117
),
we obtain
ε
=
0
⎛
⎝
⎞
⎠
ξ
d
ξ
d
v
ε
=
0
=
∂
χ
=
∂
ξ
ε
∂ε
.
(4.126)
∂ε
d
ξ
d
v
