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4

Figure 7.2 The bifurcation diagram of the logistic map in the parameter

region 3 A < 4.

74 Nonlinear Dynamical Systems

Another manifestation of universality that may be present in cha-

otic processes is the Feigenbaumâ€™s observation of the limiting rate at

which the period-doubling bifurcations occur. Namely, if An is the

value of A at which the period-2n occurs, then the ratio

dn Â¼ (An Ã€ AnÃ€1 )=(AnÃ¾1 Ã€ An ) (7:2:6)

has the limit

lim dn Â¼ 4:669 . . . : (7:2:7)

n!1

It turns out that the limit (7.2.7) is valid for the entire family of maps

with the parabolic iteration functions [5].

A very important feature of the chaotic regime is extreme sensitiv-

ity of trajectories to the initial conditions. This is illustrated with

Figure 7.3 for A Â¼ 3:8. Namely, two trajectories with the initial

conditions X0 Â¼ 0:400 and X0 Â¼ 0:405 diverge completely after 10

1

Xk

0.8

0.6

0.4

0.2

X0 = 0.4

X0 = 0.405

k

0

1 11 21

Figure 7.3 Solution to the logistic map for A Â¼ 3.8 and two initial condi-

tions: X0 Â¼ 0:400 and X0 Â¼ 0:405.

75

Nonlinear Dynamical Systems

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

d2 u du

Ã¾ g Ã¾ v2 u Â¼ 0 (7:3:1)

dt2 dt

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

Â¼ F(X(t)), X Â¼ (X1 , X2 , . . . XN )0 (7:3:2)

dt

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, XNÃ¾1 Â¼ t, and adding another equation to the flow

dXNÃ¾1

Â¼1 (7:3:3)

dt

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 XÃƒ 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.

76 Nonlinear Dynamical Systems

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 dw

Â¼ Ã€gw Ã€ v2 u

Â¼ w, (7:3:4)

dt dt

Hence, the damped oscillator may be described in the two-dimen-

sional phase space (w, u). The energy of the damped oscillator, E,

E Â¼ 0:5(w2 Ã¾ v2 u2 ) (7:3:5)

evolves with time according to the equation

dE

Â¼ Ã€gw2 (7:3:6)

dt

It follows from (7.3.6) that the dumped oscillator dissipates energy

(i.e., is a dissipative system) at g > 0. Typical trajectories of the

dumped oscillator are shown in Figure 7.4. In the case g Â¼ 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 2.5

PSI

PSI b)

a)

2

2

1.5

1.5

1

1

0.5

FI

0.5 0

FI âˆ’1.5 âˆ’1 âˆ’0.5 0 0.5 1 1.5

âˆ’0.5

0

âˆ’1.5 âˆ’0.5 0.5 1.5 âˆ’1

âˆ’0.5

âˆ’1.5

âˆ’1

âˆ’2

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