36
2 Fundamental Concepts
In terms of the coordinates (P
(∞)
1
, P
(∞)
2
, ,
(∞)
1 , ,
(∞)
2 ), Hamilton’s equations take
the form
dP
(∞)
k
dt
= −
∂H (∞)
∂∂
(∞)
k
=
2
i=1
2
j =1
P
(∞)
i
P
(∞)
j
∂C
(∞)
i,j
∂∂
(∞)
k
+ O((P
(∞) )
3 )
(2.74)
and
dd
(∞)
k
dt
=
∂H (∞)
∂P
(∞)
k
= ω k + O((P
(∞) )
3 )
(2.75)
for (k = 1, 2). These equations have solutions P
(∞)
i
= 0 and
(∞)
i
= ω i t + C i for
(i = 1, 2), where C i is a constant.
Thus, a rapidly convergent procedure has been found to obtain solutions to the
equations of motion at least on KAM tori sufficiently far from resonances.
2.7 The Definition of Chaos
The flow of trajectories in a given region of phase space is said to be chaotic if
it has positive KS metric entropy (KS stands for Krylov, Kolmogorov, and Sinai)
(Kolmogorov 1958, 1959; Sinai 1963a; Arnol’d and Avez 1968; Ornstein 1974;
Chirikov 1979; Lichtenberg and Lieberman 1983). Such flows are called K-flows.
The KS entropy is a measure of the degree of hyperbolic instability in the relative
motion of trajectories in phase space. As we saw in Sect. 2.4, in the neighborhood
of fixed points, we can determine the nature of the flow by linearizing the equations
of motion about the fixed point. In the neighborhood of hyperbolic fixed points,
trajectories on the eigenvectors approach (depart) the fixed point in an exponentially
decreasing (increasing) manner. Trajectories in the neighborhood of the fixed point,
but not on the eigenvectors, contain both types of motion. There are as many sets of
eigenvectors in the neighborhood of a hyperbolic fixed point as there are degrees of
freedom. Along each eigenvector, the rate of approach or departure is determined
by a single eigenvalue of the transition matrix (the matrix that governs the evolution
in the neighborhood of the fixed point) for the linearized problem.
2.7.1 Lyapounov Exponent
Oseledec (1968) was the first to show that a procedure analogous to that used
to study exponential divergence of flow in the neighborhood of hyperbolic
fixed points could be used to study the nature of the flow in the neighborhood
2 Fundamental Concepts
In terms of the coordinates (P
(∞)
1
, P
(∞)
2
, ,
(∞)
1 , ,
(∞)
2 ), Hamilton’s equations take
the form
dP
(∞)
k
dt
= −
∂H (∞)
∂∂
(∞)
k
=
2
i=1
2
j =1
P
(∞)
i
P
(∞)
j
∂C
(∞)
i,j
∂∂
(∞)
k
+ O((P
(∞) )
3 )
(2.74)
and
dd
(∞)
k
dt
=
∂H (∞)
∂P
(∞)
k
= ω k + O((P
(∞) )
3 )
(2.75)
for (k = 1, 2). These equations have solutions P
(∞)
i
= 0 and
(∞)
i
= ω i t + C i for
(i = 1, 2), where C i is a constant.
Thus, a rapidly convergent procedure has been found to obtain solutions to the
equations of motion at least on KAM tori sufficiently far from resonances.
2.7 The Definition of Chaos
The flow of trajectories in a given region of phase space is said to be chaotic if
it has positive KS metric entropy (KS stands for Krylov, Kolmogorov, and Sinai)
(Kolmogorov 1958, 1959; Sinai 1963a; Arnol’d and Avez 1968; Ornstein 1974;
Chirikov 1979; Lichtenberg and Lieberman 1983). Such flows are called K-flows.
The KS entropy is a measure of the degree of hyperbolic instability in the relative
motion of trajectories in phase space. As we saw in Sect. 2.4, in the neighborhood
of fixed points, we can determine the nature of the flow by linearizing the equations
of motion about the fixed point. In the neighborhood of hyperbolic fixed points,
trajectories on the eigenvectors approach (depart) the fixed point in an exponentially
decreasing (increasing) manner. Trajectories in the neighborhood of the fixed point,
but not on the eigenvectors, contain both types of motion. There are as many sets of
eigenvectors in the neighborhood of a hyperbolic fixed point as there are degrees of
freedom. Along each eigenvector, the rate of approach or departure is determined
by a single eigenvalue of the transition matrix (the matrix that governs the evolution
in the neighborhood of the fixed point) for the linearized problem.
2.7.1 Lyapounov Exponent
Oseledec (1968) was the first to show that a procedure analogous to that used
to study exponential divergence of flow in the neighborhood of hyperbolic
fixed points could be used to study the nature of the flow in the neighborhood