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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