4.2 The Complete Ternary Horseshoe
103
˙
p
˙
x
=
0 4e −1
1 0
p
x
,
(4.6)
where p(t) = p(t) and x(t) = 1 + x(t). This has eigenvalues and eigenvectors
λ 1 = −
2
√
e
, φ 1 =
−
2
√
e
1
; λ 2 = +
2
√
e
, φ 1 =
+
2
√
e
1
.
(4.7)
Thus,
p(t)
= A e
−
2
√
e
t
−
2
√
e
1
+ Be
+i
√
2t
+i
√
2
1
.
(4.8)
If we set B = 0 and A = 0.01, the solution becomes = −
0.02 √
e
e
−
2
√
e
t and
x(t) =
0.01
e
−
2
√
e
t , which describes motion along the lower righthand eigencurve
in Fig. 4.1. If we set A = 0 and B = 0.01, the solution is = +
0.02 √
e
e
+
2
√
e
t and
x(t) =
0.01
e
+
2
√
e
t , which describes motion along the upper righthand eigencurve
in Fig. 4.1.
4.2.2 Delta-Kicked System
We now add the perturbation to the system in the form of a time-periodic delta
function “kick”. The Hamiltonian takes the form
H = p
2 /2 +
∞
n=−∞
V (x)δ
t −
n +
1
2
T
,
(4.9)
where V (x) = x 2 e −x 2 and T is the period of the “kick”. For such a delta kicked
system, we can construct a symplectic map. The map corresponds to a plot of the
phase space at time intervals equal to the period of the driving force. The period T
between each kick serves as a development parameter and measures the degree of
advancement of the folding of stable and unstable manifolds of the map.
The symplectic map is given in three steps. The initial phase space coordinate is
denoted (p 0 , x 0 ). This point evolves freely for a time period of length
T
2 so that, at
time t =
T
2
− (just before the kick), the phase space coordinate is (p 0 , x a = x 0 +
p 0
T
2 ). At time t =
T
2 , the system is “kicked” and the momentum receives a discrete
increment, whose magnitude is given by the force function f (x a ) = −
dV (x)
dx
x=x a
=
2(x 3
a − x a )e −x 2
a , so that p 1 = p 0 + f (x a ). The third step is again a free flight for