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., [5] 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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