2.7 The Definition of Chaos
39
k n (τ, X o,o , Y o,o ) =
1
nτ
n
j =1
ln
d j
d o
.
(2.80)
If d o is not too big, the quantity k n (τ, X o,o , Y o,o ) has been found to have the
following properties Benettin et al. (1976) and Casartelli et al. (1976):
1. lim n→∞ k n (τ, X o,o , Y o,o ) = k(τ, X o,o , Y o,o ) exists;
2. k(τ, X o,o , Y o,o ) is independent of τ ;
3. k(τ, X o,o , Y o,o ) is independent of d o ;
4. k(τ, X o,o , Y o,o ) = 0 if X o,o is chosen to lie in a regular region of the energy
surface;
5. k(τ, X o,o , Y o,o ) is independent of X o,o and is positive if X o,o is chosen to lie in
a chaotic region of the energy surface.
Therefore, in a chaotic region of the energy surface, we can write k(E) =
k(τ, X o,o , Y o,o ). The quantity k(E) obtained in this manner is the largest Lyapounov
exponent, λ N −1 .
Benettin et al. (1979) have shown that it is possible to compute all of the
Lyapounov exponents for a model Hamitonian system with N (N = 4, 5) degrees
of freedom. Meyer (1986) has been able to show that for sufficiently smooth
Hamiltonians there are at least 2N vanishing Lyapounov exponents if there are
N independent isolating integrals of the motion. In Fig. 2.11 and Eq. (2.79), we
show some of the results of Benettin et al., who computed the Lyapounov exponent
and KS metric entropy for the Henon-Heiles system. In Fig. 2.11, the Lyapounov
Fig. 2.11 Plot of Lyapounov exponent k n for the Henon-Heiles system for initial conditions
E = 0.125, q 1 = 0, p 1 > 0 and six different initial values for q 2 , p 2 , three chosen from the
chaotic regime (black circle, diamond, and square) and three chosen from the regular regime (open
circle, diamond, and square). For all initial conditions, typically d o = 3 × 10 −4 and τ = 0.2 (see
Fig. 2.10). As n → ∞, the exponent k n approaches a positive constant value for trajectories in the
chaotic regime, and approaches zero for trajectories in the regular regime (Benettin et al. 1976)
39
k n (τ, X o,o , Y o,o ) =
1
nτ
n
j =1
ln
d j
d o
.
(2.80)
If d o is not too big, the quantity k n (τ, X o,o , Y o,o ) has been found to have the
following properties Benettin et al. (1976) and Casartelli et al. (1976):
1. lim n→∞ k n (τ, X o,o , Y o,o ) = k(τ, X o,o , Y o,o ) exists;
2. k(τ, X o,o , Y o,o ) is independent of τ ;
3. k(τ, X o,o , Y o,o ) is independent of d o ;
4. k(τ, X o,o , Y o,o ) = 0 if X o,o is chosen to lie in a regular region of the energy
surface;
5. k(τ, X o,o , Y o,o ) is independent of X o,o and is positive if X o,o is chosen to lie in
a chaotic region of the energy surface.
Therefore, in a chaotic region of the energy surface, we can write k(E) =
k(τ, X o,o , Y o,o ). The quantity k(E) obtained in this manner is the largest Lyapounov
exponent, λ N −1 .
Benettin et al. (1979) have shown that it is possible to compute all of the
Lyapounov exponents for a model Hamitonian system with N (N = 4, 5) degrees
of freedom. Meyer (1986) has been able to show that for sufficiently smooth
Hamiltonians there are at least 2N vanishing Lyapounov exponents if there are
N independent isolating integrals of the motion. In Fig. 2.11 and Eq. (2.79), we
show some of the results of Benettin et al., who computed the Lyapounov exponent
and KS metric entropy for the Henon-Heiles system. In Fig. 2.11, the Lyapounov
Fig. 2.11 Plot of Lyapounov exponent k n for the Henon-Heiles system for initial conditions
E = 0.125, q 1 = 0, p 1 > 0 and six different initial values for q 2 , p 2 , three chosen from the
chaotic regime (black circle, diamond, and square) and three chosen from the regular regime (open
circle, diamond, and square). For all initial conditions, typically d o = 3 × 10 −4 and τ = 0.2 (see
Fig. 2.10). As n → ∞, the exponent k n approaches a positive constant value for trajectories in the
chaotic regime, and approaches zero for trajectories in the regular regime (Benettin et al. 1976)