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s2 Вј 2(l ГЂ 1)= ln (2) (6:2:7)

The multifractal spectrum of the lognormal cascade that satisfies

(6.2.7) equals

(a ГЂ l)2

f(a) Вј 1 ГЂ (6:2:8)

4(l ГЂ 1)

Note that in contrast to the binomial cascade, the lognormal

cascade may yield negative values of f(a), which requires interpret-

ation of f(a) other than the fractal dimension.

Innovation of multifractal process, DX Вј X(t Гѕ Dt) ГЂ X(t), is de-

scribed with the scaling rule

E[j(DX)jq ] Вј c(q)(Dt)t(q)Гѕ1 (6:2:9)

where c(q) and t(q) (so-called scaling function) are deterministic func-

tions of q. It can be shown that the scaling function t(q) is always

concave. Obviously, t(0) Вј ГЂ1. A self-affine process (6.1.4) can be

treated as a multifractal process with t(q) Вј Hq ГЂ 1. In particular, for

the Wiener processes, H Вј 1вЃ„2 and tw (q) Вј q=2 ГЂ 1. The scaling func-

tion of the binomial cascade can be expressed in terms of its multi-

pliers

t(q) Вј log2 (m0 q Гѕ m1 q ) (6:2:10)

67

Fractals

The scaling function t(q) is related to the multifractal spectrum f(a)

via the Legendre transformation

t(q) Вј min[qa ГЂ f(a)] (6:2:11)

a

which is equivalent to

f(a) Вј arg min[qa ГЂ t(q)] (6:2:12)

q

Note that f(a) Вј q(a ГЂ H) Гѕ 1 for the self-affine processes.

In practice, the scaling function of a multifractal process X(t) can

be calculated using so-called partition function

X

NГЂ1

jX(t Гѕ Dt) ГЂ X(t)jq

Sq (T, Dt) Вј (6:2:13)

iВј0

where the sample X(t) has N points within the interval [0, T] with the

mesh size Dt. It follows from (6.2.9) that

log {E[Sq (T, Dt)]} Вј t(q) log (Dt) Гѕ c(q) log T (6:2:14)

Thus, plotting log {E[Sq (T, Dt)]} against log (Dt) for different values

of q reveals the character of the scaling function t(q). Multifractal

models have become very popular in analysis of the financial time

series. We shall return to this topic in Section 8.2

6.3 REFERENCES FOR FURTHER READING

The MandelbrotвЂ™s work on scaling in the financial time series is

compiled in the collection [1]. Among many excellent books on frac-

tals, we choose [2] for its comprehensive material that includes a

description of relations between chaos and fractals and an important

chapter on multifractals [5].

6.4 EXERCISES

*1. Implement an algorithm that draws the Sierpinski triangle with

r Вј 0:5 (see Figure 6.2).

Hint: Choose the following fixed points: (0, 0), (0, 100), (100,

0). Use the following method for the randomized choice of the

68 Fractals

fixed point: i Вј [10 rand()] %3 where rand() is the uniform

distribution within [0, 1] and % is modulus (explain the ration-

ale behind this method). Note that at least 10000 iterations are

required for a good-quality picture.

*2. Reproduce the first five steps of the binomial cascade with

m0 Вј 0:6 (see Figure 6.3). How will this cascade change if

m0 Вј 0:8?

Chapter 7

Nonlinear Dynamical Systems

7.1 MOTIVATION

It is well known that many nonlinear dynamical systems, including

seemingly simple cases, can exhibit chaotic behavior. In short, the

presence of chaos implies that very small changes in the initial condi-

tions or parameters of a system can lead to drastic changes in its

behavior. In the chaotic regime, the system solutions stay within the

phase space region named strange attractor. These solutions never

repeat themselves; they are not periodic and they never intersect.

Thus, in the chaotic regime, the system becomes unpredictable. The

chaos theory is an exciting and complex topic. Many excellent books

are devoted to the chaos theory and its applications (see, e.g., refer-

ences in Section 7.7). Here, I only outline the main concepts that may

be relevant to quantitative finance.

The first reason to turn to chaotic dynamics is a better understand-

ing of possible causes of price randomness. Obviously, new infor-

mation coming to the market moves prices. Whether it is a

companyвЂ™s performance report, a financial analystвЂ™s comments, or a

macroeconomic event, the companyвЂ™s stock and option prices may

change, thus reflecting the news. Since news usually comes unexpect-

edly, prices change in unpredictable ways.1 But is new information the

only source reason for price randomness? One may doubt this while

observing the price fluctuations at times when no relevant news is

69

70 Nonlinear Dynamical Systems

released. A tempting proposition is that the price dynamics can be

attributed in part to the complexity of financial markets. The possi-

bility that the deterministic processes modulate the price variations

has a very important practical implication: even though these pro-

cesses can have the chaotic regimes, their deterministic nature means

that prices may be partly forecastable. Therefore, research of chaos in

finance and economics is accompanied with discussion of limited

predictability of the processes under investigation [1].

There have been several attempts to find possible strange attractors

in the financial and economic time series (see, e.g., [1вЂ“3] and refer-

ences therein). Discerning the deterministic chaotic dynamics from a

вЂ˜вЂ˜pureвЂ™вЂ™ stochastic process is always a non-trivial task. This problem is

even more complicated for financial markets whose parameters may

have non-stationary components [4]. So far, there has been little (if

any) evidence found of low-dimensional chaos in financial and eco-

nomic time series. Still, the search of chaotic regimes remains an

interesting aspect of empirical research.

There is also another reason for paying attention to the chaotic

dynamics. One may introduce chaos inadvertently while modeling

financial or economic processes with some nonlinear system. This

problem is particularly relevant in agent-based modeling of financial

markets where variables generally are not observable (see Chapter

12). Nonlinear continuous systems exhibit possible chaos if their

dimension exceeds two. However, nonlinear discrete systems (maps)

can become chaotic even in the one-dimensional case. Note that the

autoregressive models being widely used in analysis of financial time

series (see Section 5.1) are maps in terms of the dynamical systems

theory. Thus, a simple nonlinear expansion of a univariate autore-

gressive map may lead to chaos, while the continuous analog of this

model is perfectly predictable. Hence, understanding of nonlinear

dynamical effects is important not only for examining empirical

time series but also for analyzing possible artifacts of the theoretical

modeling.

This chapter continues with a widely popular one-dimensional

discrete model, the logistic map, which illustrates the major concepts

in the chaos theory (Section 7.2). Furthermore, the framework for the

continuous systems is introduced in Section 7.3. Then the three-

dimensional Lorenz model, being the classical example of the low-

71

Nonlinear Dynamical Systems

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

Nk Вј ANkГЂ1 ГЂ BNkГЂ1 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 Вј 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 Nk equals Nmax Вј A=B. It is convenient to intro-

duce the dimensionless variable Xk Вј Nk =Nmax . Then 0 Xk 1,

and equation (7.2.1) has the form

Xk Вј AXkГЂ1 (1 ГЂ XkГЂ1 ) (7:2:2)

A generic discrete equation in the form

Xk Вј f(XkГЂ1 ) (7:2:3)

is called an (iterated) map, and the function f(XkГЂ1 ) is called the

iteration function. The map (7.2.2) is named the logistic map. The

sequence of values Xk 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 X0 . Some initial points turn out

to be the map solution at all iterations. The value XГѓ that satisfies the

equation

XГѓ Вј f(XГѓ ) (7:2:4)

is named the fixed point of the map. There are two fixed points for the

logistic map (7.2.2):

72 Nonlinear Dynamical Systems

XГѓ Вј 0, and XГѓ Вј (A ГЂ 1)=A (7:2:5)

1 2

If A 1, the logistic map trajectory approaches the fixed point XГѓ 1

from any initial value 0 X0 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 XГѓ Вј 0, and its basin is the interval 0 X0 1.

1

If 1 < A < 3, the logistic map trajectories have the attractor

Гѓ

X2 Вј (A ГЂ 1)=A and its basin is also 0 X0 1. In the mean time,

the point XГѓ Вј 0 is the repellent fixed point, which implies that any

1

trajectory that starts near XГѓ tends to move away from it.

1

A new type of solutions to the logistic map appears at A > 3.

Consider the case with A Вј 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 вЂ˜вЂ˜too muchвЂ™вЂ™ (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 Вј 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.

73

Nonlinear Dynamical Systems

0.95

Xk

0.85

0.75

0.65

0.55

0.45

0.35

A = 2.0

A = 3.1 k

A = 3.6

0.25

1 11 21 31 41

Figure 7.1 Solution to the logistic map at different values of the

parameter A.

0 X 1

3

A

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 FeigenbaumвЂ™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 Вј (An ГЂ AnГЂ1 )=(AnГѕ1 ГЂ An ) (7:2:6)

has the limit

lim dn Вј 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 Вј 3:8. Namely, two trajectories with the initial

conditions X0 Вј 0:400 and X0 Вј 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 Вј 3.8 and two initial condi-

tions: X0 Вј 0:400 and X0 Вј 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

Гѕ g Гѕ v2 u Вј 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

Вј F(X(t)), X Вј (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, XNГѕ1 Вј t, and adding another equation to the flow

dXNГѕ1

Вј1 (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 XГѓ 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

Вј ГЂgw ГЂ v2 u

Вј 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 Вј 0:5(w2 Гѕ v2 u2 ) (7:3:5)

evolves with time according to the equation

dE

Вј ГЂgw2 (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 Вј 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

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