 dimensional continuous chaotic system, is described (Section 7.4).
Finally, the main pathways to chaos and the chaos measures are
outlined in Section 7.5 and Section 7.6, respectively.
7.2 DISCRETE SYSTEMS: THE LOGISTIC MAP
The logistic map is a simple discrete model that was originally used
to describe the dynamics of biological populations (see, e.g.,  and
references therein). Let us consider a variable number of individuals
in a population, N. Its value at the k-th time interval is described with
the following equation
N
k
¼ AN
k1
BN
k1
2
(7:2:1)
Parameter A characterizes the population growth that is determined
by such factors as food supply, climate, etc. Obviously, the popula-
tion grows only if A > 1. If there are no restrictive factors (i.e., when
B ¼ 0), the growth is exponential, which never happens in nature for
long. Finite food supply, predators, and other causes of mortality
restrict the population growth, which is reflected in factor B. The
maximum value of N
k
equals N
max
¼ A=B. It is convenient to intro-
duce the dimensionless variable X
k
¼N
k
=N
max
. Then 0 X
k
1,
and equation (7.2.1) has the form
X
k
¼ AX
k1
(1 X
k1
)(7:2:2)
A generic discrete equation in the form
X
k
¼ f(X
k1
)(7:2:3)
is called an (iterated) map, and the function f(X
k1
) is called the
iteration function. The map (7.2.2) is named the logistic map. The
sequence of values X
k
that are generated by the iteration procedure
is called a trajectory. Trajectories depend not only on the iteration
function but also on the initial value X
0
. Some initial points turn out
to be the map solution at all iterations. The value X
that satisfies the
equation
X
¼ f(X
)(7:2:4)
is named the fixed point of the map. There are two fixed points for the
logistic map (7.2.2):
Nonlinear Dynamical Systems 71 X
1
¼ 0, and X
2
¼ (A 1)=A(7:2:5)
If A 1, the logistic map trajectory approaches the fixed point X
1
from any initial value 0 X
0
1. The set of points that the trajec-
tories tend to approach is called the attractor. Generally, nonlinear
dynamical systems can have several attractors. The set of initial values
from which the trajectories approach a particular attractor are called
the basin of attraction. For the logistic map with A < 1, the attractor
is X
1
¼ 0, and its basin is the interval 0 X
0
1.
If 1 < A < 3, the logistic map trajectories have the attractor
X
2
¼ (A 1)=A and its basin is also 0 X
0
1. In the mean time,
the point X
1
¼ 0 is the repellent fixed point, which implies that any
trajectory that starts near X
1
tends to move away from it.
A new type of solutions to the logistic map appears at A > 3.
Consider the case with A ¼ 3:1: the trajectory does not have a single
attractor but rather oscillates between two values, X 0:558 and
X 0:764. In the biological context, this implies that the growing
population overexerts its survival capacity at X 0:764. Then the
population shrinks ‘‘too much’’ (i.e., to X 0:558), which yields
capacity for further growth, and so on. This regime is called period-
2. The parameter value at which solution changes qualitatively is
named the bifurcation point. Hence, it is said that the period-doubling
bifurcation occurs at A ¼ 3. With a further increase of A, the oscilla-
tion amplitude grows until A approaches the value of about 3.45. At
higher values of A, another period-doubling bifurcation occurs
(period-4). This implies that the population oscillates among four
states with different capacities for further growth. Period doubling
continues with rising A until its value approaches 3.57. Typical tra-
jectories for period-2 and period-8 are given in Figure 7.1. With
further growth of A, the number of periods becomes infinite, and
the system becomes chaotic. Note that the solution to the logistic map
at A > 4 is unbounded.
Specifics of the solutions for the logistic map are often illustrated
with the bifurcation diagram in which all possible values of X are
plotted against A (see Figure 7.2). Interestingly, it seems that there is
some order in this diagram even in the chaotic region at A > 3:6. This
order points to the fractal nature of the chaotic attractor, which will
be discussed later on.
72 Nonlinear Dynamical Systems 0.25
0.35
0.45
0.55
0.65
0.75
0.85
0.95
1 11213141
k
X
k
A = 2.0
A = 3.1
A = 3.6
Figure 7.1 Solution to the logistic map at different values of the
parameter A.
0X
3
A
4
1
Figure 7.2 The bifurcation diagram of the logistic map in the parameter
region 3 A < 4.
Nonlinear Dynamical Systems 73

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