For a small K (K ¼ 0.2), almost all lines are shown as small modification from
K ¼ 0 case. It is not seen the connection from a small p to a large p region. This
means the solution of (9.7.4) is stable and no increase of p(t) is expected. This is also
the same even with the increase of K (K ¼ 0.6). The black region with many dots
appears, while they do not make a bridge from small p to large p, so no acceleration
is expected. However, for the case when the value K exceeds the critical value
(9.7.6), the chaos regions spread, and it becomes possible for the time evolution of p
(t) to start from a small value and continuously increase its value with time.
This is due to a sequential jump in each step in (9.7.4). Some solution goes up to
the top of the figure (p ¼ 2π) and comes to the bottom to repeat such a routine. Such
p(t) shows the selected acceleration by the chaos of the system. It is noted that
although (9.7.4) is very simple, the original Eq. (9.7.1) is a nonlinear pendulum
equation with an external force by laser field. This problem is similar to that of (8.
4.6) and/or the problem of the chaos of asteroids in Fig. 8.12. The basic equations are
characterized as the nonlinear oscillation pendulum and an external periodic force.
Even the stable orbits of orange and green in Fig. 8.14a have the threshold amplitude
of the external force of ε in (8.4.6). The orbits become unstable like the standard map
for large external force. In addition, note that the threshold of chaos also depends on
the frequency of the external force.
It is also shown [25] that (9.7.6) is approximated with a diffusion equation and its
solution has the form:
f t, p
ð Þ / χt
ð Þ
À1=3 exp À
p
j j
3
9χt
,
χ ¼
a
2
ε p
2
ð9:7:7Þ
As a long time behavior, the distribution function growth as a super-Gaussian
(N ¼ 3) in (9.7.7), but T h ~ t
1/3 .
9.8 Analytical Mechanics of Electron Motions
9.8.1 One Laser Relation
Let us first define the equation of motion of an electron in two laser systems.
Consider the relation for the case with only the incident laser. The incident laser is
given as
a ¼ a 0 sin ξ
ð Þ
ð9:8:1Þ
where ξ is the phase of the incident laser wave and has the following normalized
form:
362
9 Theory of Stochasticity and Chaos of Electrons in Relativistic Lasers
K ¼ 0 case. It is not seen the connection from a small p to a large p region. This
means the solution of (9.7.4) is stable and no increase of p(t) is expected. This is also
the same even with the increase of K (K ¼ 0.6). The black region with many dots
appears, while they do not make a bridge from small p to large p, so no acceleration
is expected. However, for the case when the value K exceeds the critical value
(9.7.6), the chaos regions spread, and it becomes possible for the time evolution of p
(t) to start from a small value and continuously increase its value with time.
This is due to a sequential jump in each step in (9.7.4). Some solution goes up to
the top of the figure (p ¼ 2π) and comes to the bottom to repeat such a routine. Such
p(t) shows the selected acceleration by the chaos of the system. It is noted that
although (9.7.4) is very simple, the original Eq. (9.7.1) is a nonlinear pendulum
equation with an external force by laser field. This problem is similar to that of (8.
4.6) and/or the problem of the chaos of asteroids in Fig. 8.12. The basic equations are
characterized as the nonlinear oscillation pendulum and an external periodic force.
Even the stable orbits of orange and green in Fig. 8.14a have the threshold amplitude
of the external force of ε in (8.4.6). The orbits become unstable like the standard map
for large external force. In addition, note that the threshold of chaos also depends on
the frequency of the external force.
It is also shown [25] that (9.7.6) is approximated with a diffusion equation and its
solution has the form:
f t, p
ð Þ / χt
ð Þ
À1=3 exp À
p
j j
3
9χt
,
χ ¼
a
2
ε p
2
ð9:7:7Þ
As a long time behavior, the distribution function growth as a super-Gaussian
(N ¼ 3) in (9.7.7), but T h ~ t
1/3 .
9.8 Analytical Mechanics of Electron Motions
9.8.1 One Laser Relation
Let us first define the equation of motion of an electron in two laser systems.
Consider the relation for the case with only the incident laser. The incident laser is
given as
a ¼ a 0 sin ξ
ð Þ
ð9:8:1Þ
where ξ is the phase of the incident laser wave and has the following normalized
form:
362
9 Theory of Stochasticity and Chaos of Electrons in Relativistic Lasers