2.7 The Definition of Chaos
43
and stretching in the p direction, very much like the flow in the neighborhood of a
hyperbolic fixed point.
The connection between the Lyapounov exponents and the KS metric entropy
was established by Piesin (1976). The KS metric entropy may be related to the
Lyapounov exponents in the following way. Let
ρ(X
N ) =
N −1
i=1
λ i (X
N ),
(2.89)
where λ i (X N ) denotes the Lyapounov exponent in a region of phase space in the
interval X N → X N + dX N on the energy surface. (Remember that the Lyapounov
exponents are constant and nonzero throughout a stochastic region and are zero in
regular regions.) The KS entropy is then Benettin et al. (1979)
h(E) =
E
ρ(X
N )dμ E ,
(2.90)
where dμ E denotes an invariant volume element of the energy surface. Thus the KS
entropy is directly related to the Lyapounov exponents. Benettin et al. (1976) have
made an estimate of the KS metric entropy as a function of energy for the HenonHeiles system. Their result, the dotted line, is shown in Fig. 2.12. The KS metric
entropy has an energy dependence and qualitative behavior similar to that of the
largest Lyapounov exponent for this system.
The Henon-Heiles system is one whose phase space contains a mixture of regular
and chaotic trajectories. The fraction of the phase space occupied by each can be
varied by varying parameters of the system. This is the most common type of
behavior found in Hamiltonian systems and is characteristic of systems with smooth
differentiable Hamiltonians.
One of the few systems that is known to be a K-flow for all values of its
parameters is the hard sphere gas. This was proven by Sinai (1963b) for the Sinai
billiard, which consists of a particle confined in a box that has periodic boundary
conditions and a hard circular barrier placed inside the box (see Fig. 2.15). The
Fig. 2.15 Sinai (1963b)
proved that the phase space
flow of a moving particle
confined to a box containing a
hard circular barrier is a
K-flow. The box is assumed
to have periodic boundary
conditions
43
and stretching in the p direction, very much like the flow in the neighborhood of a
hyperbolic fixed point.
The connection between the Lyapounov exponents and the KS metric entropy
was established by Piesin (1976). The KS metric entropy may be related to the
Lyapounov exponents in the following way. Let
ρ(X
N ) =
N −1
i=1
λ i (X
N ),
(2.89)
where λ i (X N ) denotes the Lyapounov exponent in a region of phase space in the
interval X N → X N + dX N on the energy surface. (Remember that the Lyapounov
exponents are constant and nonzero throughout a stochastic region and are zero in
regular regions.) The KS entropy is then Benettin et al. (1979)
h(E) =
E
ρ(X
N )dμ E ,
(2.90)
where dμ E denotes an invariant volume element of the energy surface. Thus the KS
entropy is directly related to the Lyapounov exponents. Benettin et al. (1976) have
made an estimate of the KS metric entropy as a function of energy for the HenonHeiles system. Their result, the dotted line, is shown in Fig. 2.12. The KS metric
entropy has an energy dependence and qualitative behavior similar to that of the
largest Lyapounov exponent for this system.
The Henon-Heiles system is one whose phase space contains a mixture of regular
and chaotic trajectories. The fraction of the phase space occupied by each can be
varied by varying parameters of the system. This is the most common type of
behavior found in Hamiltonian systems and is characteristic of systems with smooth
differentiable Hamiltonians.
One of the few systems that is known to be a K-flow for all values of its
parameters is the hard sphere gas. This was proven by Sinai (1963b) for the Sinai
billiard, which consists of a particle confined in a box that has periodic boundary
conditions and a hard circular barrier placed inside the box (see Fig. 2.15). The
Fig. 2.15 Sinai (1963b)
proved that the phase space
flow of a moving particle
confined to a box containing a
hard circular barrier is a
K-flow. The box is assumed
to have periodic boundary
conditions