The period-doubling route to chaos has been observed for a large number of
experimental systems, and for numerical models associated with differential
equations (continuous flows) and difference equations (discrete maps). A classical
example of the latter kind is the logistic or quadratic map:
x k þ 1 ¼ lx k ð1 À x k Þ; k ¼ 0; 1; 2; . . .:
ð6:11Þ
This one-dimensional map originates from population biology. Nevertheless, it
turns out that many systems associated with differential equations can locally be
reduced to a quadratic map through the use of Poincaré sections.
Choosing an initial condition x 0 , the iteration of the map (6.11) first produces
some transient changes in x k before the solution settles down into some limit
behavior. Just as for continuous flows there are three possible limit sets: equilibrium
points (for maps they are called fixed points), periodic motion (including
quasiperiodic motion) and chaotic motion. The limit set depends on the system
parameter l, which is in this case considered a bifurcation parameter.
Fig. 6.9 shows a bifurcation diagram for the quadratic map. The map was
iterated for 300 different values of l 2 [2.5; 4.0] with x 0 = 0.1. For each value of l
the first 100 iterates were considered transients, whereupon the next 25 iterates were
plotted against l. It appears that for l 2 [2.5; l 1 ] the iterates settle down onto a
fixed point. We may consider this the ‘period-1’ motion. When l = l 1 the fixed
point becomes unstable and the period doubles, i.e. the iterates visit two different
values in turn. At a somewhat higher value of l the period-2 solution becomes
unstable in favor of a stable period-4 solution, i.e. the period has doubled once
again. This process of period-doubling continues with still smaller changes in l,
accumulating at a critical value beyond which the motion turns chaotic.
Feigenbaum, in his famous work (1978, 1980), discovered that the sequence of
bifurcation parameters l n at which the period doubles satisfy the relation:
lim
n!1
l n þ 1 À l n
l n À l nÀ1
¼
1
d
; d ¼ 4:6692 Á Á Á ;
ð6:12Þ
where the constant d is the Feigenbaum number or Feigenbaum’s constant.
Differentiating the right-hand side of (6.11) with respect to x k , or sketching the
function lx k (1 − x k ), it is found that the map has a zero tangent at x k ¼
1
2 . Hence,
the map is non-invertible, since given a value for x k+1 there are two possible values
for x k . The importance of Feigenbaum’s work is that he showed the phenomenon of
cascading period-doublings to be typical of maps that resemble (6.11), that is, maps
possessing a zero tangent. Hence the universal appearance of the period-doubling
route.
Examining Fig. 6.9 you might suggest that the crowding of dots in the chaotic
regime merely represents motion that has period-doubled a huge number of times.
This is not the case, as a calculation of the largest Lyapunov exponent would reveal
(actually, for maps they are called Lyapunov numbers, being larger than unity for
342
6 Chaotic Vibrations
experimental systems, and for numerical models associated with differential
equations (continuous flows) and difference equations (discrete maps). A classical
example of the latter kind is the logistic or quadratic map:
x k þ 1 ¼ lx k ð1 À x k Þ; k ¼ 0; 1; 2; . . .:
ð6:11Þ
This one-dimensional map originates from population biology. Nevertheless, it
turns out that many systems associated with differential equations can locally be
reduced to a quadratic map through the use of Poincaré sections.
Choosing an initial condition x 0 , the iteration of the map (6.11) first produces
some transient changes in x k before the solution settles down into some limit
behavior. Just as for continuous flows there are three possible limit sets: equilibrium
points (for maps they are called fixed points), periodic motion (including
quasiperiodic motion) and chaotic motion. The limit set depends on the system
parameter l, which is in this case considered a bifurcation parameter.
Fig. 6.9 shows a bifurcation diagram for the quadratic map. The map was
iterated for 300 different values of l 2 [2.5; 4.0] with x 0 = 0.1. For each value of l
the first 100 iterates were considered transients, whereupon the next 25 iterates were
plotted against l. It appears that for l 2 [2.5; l 1 ] the iterates settle down onto a
fixed point. We may consider this the ‘period-1’ motion. When l = l 1 the fixed
point becomes unstable and the period doubles, i.e. the iterates visit two different
values in turn. At a somewhat higher value of l the period-2 solution becomes
unstable in favor of a stable period-4 solution, i.e. the period has doubled once
again. This process of period-doubling continues with still smaller changes in l,
accumulating at a critical value beyond which the motion turns chaotic.
Feigenbaum, in his famous work (1978, 1980), discovered that the sequence of
bifurcation parameters l n at which the period doubles satisfy the relation:
lim
n!1
l n þ 1 À l n
l n À l nÀ1
¼
1
d
; d ¼ 4:6692 Á Á Á ;
ð6:12Þ
where the constant d is the Feigenbaum number or Feigenbaum’s constant.
Differentiating the right-hand side of (6.11) with respect to x k , or sketching the
function lx k (1 − x k ), it is found that the map has a zero tangent at x k ¼
1
2 . Hence,
the map is non-invertible, since given a value for x k+1 there are two possible values
for x k . The importance of Feigenbaum’s work is that he showed the phenomenon of
cascading period-doublings to be typical of maps that resemble (6.11), that is, maps
possessing a zero tangent. Hence the universal appearance of the period-doubling
route.
Examining Fig. 6.9 you might suggest that the crowding of dots in the chaotic
regime merely represents motion that has period-doubled a huge number of times.
This is not the case, as a calculation of the largest Lyapunov exponent would reveal
(actually, for maps they are called Lyapunov numbers, being larger than unity for
342
6 Chaotic Vibrations