40
2 Fundamental Concepts
Fig. 2.12 A plot of
k(E) = lim n→∞ k n as a
function of energy for
trajectories in the chaotic
regime (black squares) and
trajectories in the regular
regime (black circles) of the
Henon-Heiles system. The
dotted line is an estimate of
the KS metric entropy as a
function of energy (Benettin
et al. 1976)
exponent, k n , is computed for six different initial conditions, three taken from
the chaotic region and three taken from the regular region (it is useful to locate
these initial conditions in the surfaces of section for the Henon-Heiles system in
Fig. 2.3). For initial conditions in the chaotic regime, all three exponents approach
the same final value as n → ∞, even though the initial conditions are taken
from quite different regions of the phase space. For initial conditions in the regular
region, the three exponents steadily decrease toward zero. In Fig. 2.12, the exponent
k(E) = lim n→∞ k n is plotted as a function of energy in both the chaotic and regular
regimes for the Henon-Heiles system. The rate of divergence of trajectories appears
to increase with increasing energy.
Regions of phase space for which neighboring trajectories have positive Lyapounov exponents are said to exhibit sensitive dependence on initial conditions,
which is the definition of classical chaos. Any small change in the initial trajectories
can lead to quite different final states.
2.7.2 KS Metric Entropy and K-Flows
There is a relation between the Lyapounov exponents and the KS metric entropy.
In order to build some intuition about the KS metric entropy, let us consider the
baker’s map (Arnol’d and Avez 1968; Penrose 1970), which is the simplest case of
a Bernoulli shift (Moser 1973). The baker’s map consists of an alphabet with two
“letters,” 0 and 1, and the set, {S}, of all possible doubly infinite sequences
S = (. . . , s −2 , s −1 , s 0 ; s 1 , s 2 , . . .)
(2.81)
2 Fundamental Concepts
Fig. 2.12 A plot of
k(E) = lim n→∞ k n as a
function of energy for
trajectories in the chaotic
regime (black squares) and
trajectories in the regular
regime (black circles) of the
Henon-Heiles system. The
dotted line is an estimate of
the KS metric entropy as a
function of energy (Benettin
et al. 1976)
exponent, k n , is computed for six different initial conditions, three taken from
the chaotic region and three taken from the regular region (it is useful to locate
these initial conditions in the surfaces of section for the Henon-Heiles system in
Fig. 2.3). For initial conditions in the chaotic regime, all three exponents approach
the same final value as n → ∞, even though the initial conditions are taken
from quite different regions of the phase space. For initial conditions in the regular
region, the three exponents steadily decrease toward zero. In Fig. 2.12, the exponent
k(E) = lim n→∞ k n is plotted as a function of energy in both the chaotic and regular
regimes for the Henon-Heiles system. The rate of divergence of trajectories appears
to increase with increasing energy.
Regions of phase space for which neighboring trajectories have positive Lyapounov exponents are said to exhibit sensitive dependence on initial conditions,
which is the definition of classical chaos. Any small change in the initial trajectories
can lead to quite different final states.
2.7.2 KS Metric Entropy and K-Flows
There is a relation between the Lyapounov exponents and the KS metric entropy.
In order to build some intuition about the KS metric entropy, let us consider the
baker’s map (Arnol’d and Avez 1968; Penrose 1970), which is the simplest case of
a Bernoulli shift (Moser 1973). The baker’s map consists of an alphabet with two
“letters,” 0 and 1, and the set, {S}, of all possible doubly infinite sequences
S = (. . . , s −2 , s −1 , s 0 ; s 1 , s 2 , . . .)
(2.81)