iterations. Thus, the logistic map provides an illuminating example of
complexity and universality generated by interplay of nonlinearity
and discreteness.
7.3 CONTINUOUS SYSTEMS
While the discrete time series are the convenient framework for
financial data analysis, financial processes are often described using
continuous presentation [6]. Hence, we need understanding of the
chaos specifics in continuous systems. First, let us introduce several
important notions with a simple model of a damped oscillator (see,
e.g., [7]). Its equation of motion in terms of the angle of deviation
from equilibrium, u,is
d
2
u
dt
2
þ g
du
dt
þ v
2
u ¼ 0(7:3:1)
In (7.3.1), g is the damping coefficient and v is the angular frequency.
Dynamical systems are often described with flows, sets of coupled
differential equations of the first order. These equations in the vector
notations have the following form
dX
dt
¼ F(X(t)), X ¼ (X
1
,X
2
, ...X
N
)
0
(7:3:2)
We shall consider so-called autonomous systems for which the func-
tion F in the right-hand side of (7.3.2) does not depend explicitly on
time. A non-autonomous system can be transformed into an autono-
mous one by treating time in the function F(X, t) as an additional
variable, X
Nþ1
¼ t, and adding another equation to the flow
dX
Nþ1
dt
¼ 1(7:3:3)
As a result, the dimension of the phase space increases by one. The
notion of the fixed point in continuous systems differs from that of
discrete systems (7.2.4). Namely, the fixed points for the flow (7.3.2)
are the points X
at which all derivatives in its left-hand side equal
zero. For the obvious reason, these points are also named the equilib-
rium (or stationary) points: If the system reaches one of these points,
it stays there forever.
Nonlinear Dynamical Systems 75
Equations with derivatives of order greater than one can be also
transformed into flows by introducing additional variables. For
example, equation (7.3.1) can be transformed into the system
du
dt
¼ w,
dw
dt
¼gw v
2
u (7:3:4)
Hence, the damped oscillator may be described in the two-dimen-
sional phase space (w, u). The energy of the damped oscillator, E,
E ¼ 0:5(w
2
þ v
2
u
2
)(7:3:5)
evolves with time according to the equation
dE
dt
¼gw
2
(7:3:6)
It follows from (7.3.6) that the dumped oscillator dissipates energy
(i.e., is a dissipative system)atg > 0. Typical trajectories of the
dumped oscillator are shown in Figure 7.4. In the case g ¼ 0, the
trajectories are circles centered at the origin of the phase plane. If
g > 0, the trajectories have a form of a spiral approaching the origin
of plane.
2
In general, the dissipative systems have a point attractor in
the center of coordinates that corresponds to the zero energy.
Chaos is usually associated with dissipative systems. Systems with-
out energy dissipation are named conservative or Hamiltonian
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
−1.5 −0.5 0.5 1.5
FI
PSI
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
−1.5 −1 −0.5 0 0.5 1 1.5
FI
PSI
a)
b)
Figure 7.4 Trajectories of the damped oscillator with v ¼ 2: (a) g ¼ 2; (b)
g ¼ 0.
76 Nonlinear Dynamical Systems
systems. Some conservative systems may have the chaotic regimes,
too (so-called non-integrable systems) [5], but this case will not be
discussed here. One can easily identify the sources of dissipation in
real physical processes, such as friction, heat radiation, and so on. In
general, flow (7.3.2) is dissipative if the condition
div(F)
X
N
i ¼1
@F
@X
i
< 0(7:3:7)
is valid on average within the phase space.
Besides the point attractor, systems with two or more dimensions
may have an attractor named the limit cycle. An example of such an
attractor is the solution of the Van der Pol equation. This equation
describes an oscillator with a variable damping coefficient
d
2
u
dt
2
þ g[(u=u
0
)
2
1]
du
dt
þ v
2
u ¼ 0(7:3:8)
In (7.3.8), u
0
is a parameter. The damping coefficient is positive at
sufficiently high amplitudes u > u
0
, which leads to energy dissipation.
However, at low amplitudes (u < u
0
), the damping coefficient be-
comes negative. The negative term in (7.3.8) has a sense of an energy
source that prevents oscillations from complete decay. If one intro-
duces u
0
ffiffiffiffiffiffiffiffi
v=g
p
as the unit of amplitude and 1=v as the unit of time,
then equation (7.3.8) acquires the form
d
2
u
dt
2
þ (u
2
e
2
)
du
dt
þ u ¼ 0(7:3:9)
where e ¼ g=v is the only dimensionless parameter that defines the
system evolution. The flow describing the Van der Pol equation has
the following form
du
dt
¼ w,
dw
dt
¼ (e
2
u
2
) w u (7:3:10)
Figure 7.5 illustrates the solution to equation (7.3.1) for e ¼ 0:4.
Namely, the trajectories approach a closed curve from the initial
conditions located both outside and inside the limit cycle. It should
be noted that the flow trajectories never intersect, even though
their graphs may deceptively indicate otherwise. This property
follows from uniqueness of solutions to equation (7.3.8). Indeed, if the
Nonlinear Dynamical Systems 77
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