Articles
Deterministic Chaos Theory: Basic Concepts
Teoria do Caos Determinístico: Conceitos Básicos
Mauro Cattani^{*
}^{1
}
Iberê Luiz Caldas^{1
}
Silvio Luiz de Souza^{2
}
Kelly Cristiane Iarosz^{1
}
^{1}Instituto de Física, Universidade de São Paulo, São Paulo, SP, Brasil
^{2}Departamento de Física e Matemática, Universidade Federal de São João delRei, Ouro Branco, MG, Brasil
ABSTRACT
This article was written to students of mathematics, physics and engineering. In general, the word chaos may refer to any state of confusion or disorder and it may also refer to mythology or philosophy. In science and mathematics it is understood as irregular behavior sensitive to initial conditions. In this article we analyze the deterministic chaos theory, a branch of mathematics and physics that deals with dynamical systems (nonlinear differential equations or mappings) with very peculiar properties. Fundamental concepts of the deterministic chaos theory are briefly analyzed and some illustrative examples of conservative and dissipative chaotic motions are introduced. Complementarily, we studied in details the chaotic motion of some dynamical systems described by differential equations and mappings. Relations between chaotic, stochastic and turbulent phenomena are also commented.
Keywords chaos theory; differential equations; Poincaré sections; mapping; Lyapunov exponent
RESUMO
Este artigo foi escrito para estudantes de matemática, física e engenharia. Em geral, a palavra caos pode se referir a qualquer estado de confusão ou a desordem, mas também se referir a mitologia ou filosofia. Em ciência e matemática é entendido como um comportamento irregular sensível às condições iniciais. Neste artigo vamos analisar a teoria do caos determinístico, um ramo da matemática e da física que lida com sistemas dinâmicos (equações diferenciais nãolineares ou mapeamentos), com propriedades muito peculiares. Conceitos fundamentais da teoria do caos determinístico são brevemente analisados e alguns exemplos ilustrativos de movimentos caóticos conservativos e dissipativos são introduzidos. Complementarmente, estudamos em detalhes o movimento caótico de alguns sistemas dinâmicos descritos por equações diferenciais e mapeamentos. As relações entre fenõmenos caóticos, estocásticos e turbulentos também são comentados.
Palavraschave teoria do caos; equações diferencias; seções de Poincaré; mapeamento; expoente de Lyapunov
1. Introduction
This paper was written for students of mathematics, physics and engineering. Are briefly analyzed essential aspects of the growing field of mathematics and physics that has been applied to study a large number of phenomena generically named chaotic. These are present in many areas in science and engineering [^{1}^{4}], including astronomy, plasma physics, statistical physics, hydrodynamics and biology. As in Greek the word chaos (
χα
oç) means confusion, random, stochastic, and turbulent processes may be misleading associated with chaos. However, rigorously they are different in the framework of physics and mathematics, as will be shown. This article analyzes only the basic points of chaos theory, as exactly as possible from the mathematical point of view, avoiding sometimes a rigorous approach. In Section 2 we define chaos, in the context of the deterministic chaos theory, as a consequence of peculiar properties of deterministic nonlinear ordinary differential equations (NLODE) ^{[5]}. These equations that describe dynamic systems have a time evolution strongly dependent on initial conditions. Chaotic motion occurs depending of initial conditions and parameters values of the nonlinear equations.
In Section 3 is seen the difference between chaotic and stochastic (or random) processes. In Section 4, to give a general idea about the chaos, we study in details the Duffing equation and the dissipative motion of a driven damped pendulum, introducing the Poincaré technique. In Section 5 we show that it is possible to get a good description of chaotic processes using an iterative algebraic model named mapping. As examples, we introduce the logistic and the Hénon maps. In Section 6 is presented the Lyapunov characteristic exponent, used to quantify the sensitive dependence on initial conditions for chaotic behavior. Finally, in Section 7 is briefly discussed an open problem: the relation between turbulence and chaos.
2. Definition of Deterministic Systems and Chaos
Usually in physics basics courses [^{1},^{3},^{4},^{6},^{7}] we learn that all physical laws are described by differential equations. So, integrating, that is, solving analytically or numerically these equations, knowing the initial and boundary conditions (see Section 4), we would know the future of a physical system for all times. This is the deterministic view of nature. In other words, physics systems are deterministic because they obey deterministic differential equations. They can be conservative or dissipative. Remark that the deterministic development refers to the way as a system develops from one moment to the next, where the present system depends on the one just past in a welldetermined way through physical laws [^{1},^{3},^{4},^{6},^{7}]. If the initial states of deterministic systems were exactly known, future states would be precisely theoretically predicted.
This deterministic view survived to be questioned after the famous visionary works of Henri Poincaré on celestial mechanics ^{[8]} performed at the end of the
19th
. These works begin in 1880 when he found nonperiodic orbits in the threebody problem.
According to Poincaré [^{8},^{9}]: "If we knew exactly the law of nature and the situation of the universe at the initial moment, we could predict exactly the situation of that same universe at a succeeding moment. But even if it were the case that the natural laws had no longer any secret for us, we could still only know the initial situation approximately. If it enabled us to predict the succeeding situation with the same approximation, that is all we require, and we should say that the phenomenon had been predicted, that it is governed by laws. But it is not always so: it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible, and we have the fortuitous phenomenon".
In practice, as observed for many systems, knowledge about the future state is limited by the precision with which the initial state can be measured. That is, knowing the laws of nature is not enough to predict the future. There are deterministic systems whose time evolution has a very strong dependence on initial conditions. That is, the differential equations that govern the evolution of the system are very sensitive to initial conditions. Usually we say that even a tiny effect, such as a butterfly flying nearby, may be enough to vary the conditions such that the future is entirely different than what it might have been, not just a tiny bit different [^{1}^{3},^{10}]. In this way, measurements made on the state of a system at a given time may not allow us to predict the future situation even moderately far ahead, despite the fact that the governing equations are exactly known. By definition, these equations are named chaotic and that they predict a deterministic chaos.
Only in recent years, with advent of computers that was allowed chaos to be studied because now it is possible to perform numerical calculations of the time evolution of the properties of systems sensitive to initial conditions. We begin to understand the existence of chaos when computers were readily available to calculate the longtime histories required to explain the discussed behavior. It did not happen until the 1970s. After almost one century of investigations we learned that chaotic systems can only be solved numerically, and there are no simple, general ways to predict when a system will exhibit chaos [^{1}^{3},^{10}]. We have also learned that deterministic chaos is always associated with nonlinear systems; nonlinearity is a necessary condition for chaos but not a sufficient one.
3. Random or Stochastic Process
According to Section 2 the deterministic model will always produce the same output from a given starting condition or initial state. On the other hand, a random process, sometimes called stochastic process, is a collection of random variables, representing the evolution of some system of random values over time ^{[11]}. Instead of describing a process which can only evolve in one way (as, for example, the solutions of an ordinary differential equation), in a stochastic process there is some indeterminacy: even if the initial condition is known, there are several (often infinitely many) directions in which the process may evolve. There is a probabilistic evolution of the initial states.
As an example, let us consider the Langevin [^{11},^{12}] stochastic process. He proposed in 1908 the following stochastic differential equation to describe the Brownian (random) motion of a particle immersed in a fluid [^{11},^{12}]:
md2xdt2=λdxdt+η(t).
(1)
The degree of freedom of interest here is the position
x
of the particle,
m
denotes the particle's mass. The force acting on the particle is written as a sum of a viscous force proportional to the particle's velocity (Stokes' law), and a noise term
η
(t) (the name given in physical contexts to terms in stochastic differential equations which are stochastic processes) representing the effect of the collisions with the molecules of the fluid. The force
η
(t) has a Gaussian probability distribution with correlation function
<ηi(t)ηj(t′)>=2λκBTδi,jδ(tt′),
(2)
where
κB
is Boltzmann's constant and
T
is the temperature. The
δ
function form of the correlations in time means that the force at a time
t
is assumed to be completely uncorrelated with it at any other time. This is an approximation; the actual random force has a nonzero correlation time corresponding to the collision time of the molecules. However, Langevin's equation is used to describe the motion of a macroscopic particle at a much longer time scale, and in this limit the
δ
correlation and the Langevin equation become exact.
It can be difficult to tell from data whether a physical or other observed process is random or chaotic [^{11},^{13}]. In reference ^{[3]} one can see some procedures proposed to distinguish between deterministic chaos and stochastic behavior.
Finally, in quantum mechanics, the Schr
ö
dinger equation, which describes the continuous time evolution of a system's wave function, is deterministic ^{[14]}, besides the well known relationship between the wave function and the observable properties of the system.
4. Deterministic Chaos
According to Section 2, after 130 years of investigations, it is known that chaotic phenomenon may be observed when dynamic systems obey nonlinear ordinary differential equations (NLODE)
1
or partial differential equations [PDE]
2
[^{22},^{23}]. In this article to avoid complex mathematical analysis we only consider chaos generated by NLODE.
In this way, let us recall the definitions of NLODE. An ordinary differential equation is an equation containing a function of one independent variable and its derivatives [^{15},^{16},^{19}]. The term ordinary is used in contrast with the term PDE which may be with respect to more than one independent variable. Let
x
be an independent variable,
y=(x)
a function of
x
, and
y(n)=dny∕dxn
the derivative of order n of the function
y(x)
. An ODE of order
n
can be generally written as
F(x,y,y′,...,y(n))=0
. If
x
,
y(x)
and
y(n)
are linear functions and
F
is a linear function of these functions we say that
F
is an ODE without any chaotic solutions (see Footnote ^{1}). When nonlinear terms are present,
F
is a NLODE. In the Ndimensional case it is assumed that the time evolution of the dynamic of a system is described by continuous and continuous flux created by ordinary nonlinear differential equation
dxdt=fα[x(t)],
(6)
with,
x(0)=xo,x,fα
(flow equation) are Nvectors
ϵ
Rm
,
m
is the number of degrees of freedom of the system,
fα
is explicitly independent of time and
α
is a control parameter. Usually it is assumed that any NLODE can be integrated in the sense that they are resolved analytically or numerically and that the solutions obtained are unique. Note that rigorously in Mathematics, differential equations can be integrated [^{28},^{29}] when are manifested the following features: (a) existence of enough number of conserved quantities; (b) existence of an algebraic geometry and (c) ability to give explicit solutions.
To give a general idea about the chaos theory we study in details two examples of dissipative chaotic systems. Thus, in Section (5.1) a dissipative non linear oscillator and in (4.2) the dissipative motion of a damped and driven pendulum.
There are, however other illustrative examples of conservative chaotic systems. We suggest the lecture of two conservative processes [^{30}], described by the Hamiltonian formalism, with chaotic solutions. One is the motion of a particle of mass
m
in a double quartic nonharmonic potential (Duffing potential) governed by the Duffing Hamiltonian:
H(p,x,t)=p22mkx2+x4+Fcos(ωt),
(7)
where the oscillating term
Fcos(ωt)
is a perturbative potential. A didactic approach of this case was done, for instance, in ^{[30]}. The second case is the conservative motion of a double pendulum seen, for instance, in reference ^{[31]} where are found animation pictures of the chaotic motion.
Another classical example is the chaos in the solar system (see, for instance, reference [^{3}]).
4.1. Duffing Equation
A dissipative illustrative case is the motion of a particle with mass
m
submitted to a Duffing potential and to a dissipative force
β(dx∕dt)
. That is, the motion is governed by the NLODE (Duffing equation) [^{2}^{4}]:
ẍ+βẋx+γx3=Fcos(ωt),
(8)
The NLODE (8) can only be solved for
x
using numerical methods, given the parameters
β
,
k
, and
ω
.
The motion in the phase space associated with Eq.(8) can be efficiently studied using the technique invented by Poincaré, named Poincaré sections, illustrated in Fig. 4 and 5. First is constructed a 3dim phase space with orthogonal axis (
x
,
y
,
z
), where
y=dx∕dt′
and
z=ωt′
and second, are taken parallel planes (
y
,
x
) orthogonal to the axis
z
distant one of the other by a given interval
Δz
(see Fig.4(a)). These planes, or Poincaré sections, are used to drawn a stroboscopic map of the flow. This name is given because such map consists in observe the system in discrete times
tk=n∕ω
(
n=1,2,…,n
). Taking for
t=0
the initial values
x(0)=xo
and
y(0)=yo
we integrate numerically Eq.(8) up to the instant
t1
determining the point
A1=
[
x(t1)
,
y(t1)
] of the path. These values are now taken as new initial values to calculate the next point
A2=
[
x(t2)
,
y(t2)
] for
t2
and so on. Note that the calculated path is a continuous curve. The calculated path in the phase space (
x
,
y
,
z
) pierces the planes (stroboscopic sections) as a function of speed (
y=dx∕dt
), time (
z=n∕ω
) and the coordinate
x
, according to Fig. 4 (a). The points on the intersections are labelled as
A1
,
A2
and
A3
, etc. This set of points
Ai
forms a pattern (stroboscopic map) when projected on the plane (
y
,
x
) (see Figs. 4(b) and (5). Poincaré realized that the simple curves represent motion with possibly analytic solutions, but the many complicated, apparently irregular, curves represent chaos.
Now let us analyze results seen in Fig.1 (a). The displays
x
versus time
t
when transient effects have died out. The value
ε
=4.0 shows a simple periodic motion (only one vibrational frequency), but the results for
ε
=6.0 is not periodic Fig.2 (a). In Figs.1 (b) and 2 (b), we observe the phasespace plot, namely,
y=dx∕dt
versus
x
^{[6]}. These results indicate the beautiful and surprising behavior obtained from nonlinear dynamics: the motion is periodic for
ε
=4.0, but chaotic for
ε
=6.0.
In Fig.6 is displayed the stroboscopic map, for
F
=6.0. This Poincaré section represents a chaotic motion: the system never comes back to the same position (
x
,
y
) after
z
goes through multiples of
z=n∕ω
. The illustrated motion presents a complicated variation of points expected for the chaotic motion (with a period T
→∞
). In these cases we have aperiodic motions which is a characteristic of the deterministic chaos ^{[32]}.
Finally, we remark that only for dissipative systems there are set of points (attractors) or a point on which the motion converges. In chaotic motion, nearby trajectories in phase space are continually diverging from one another following the attractor. This effect is shown in Fig. 3, for two motions obtained for the same parameters but with two different neighbor initial conditions. Due to these attractors, named strange or chaotic attractors, the motions in the phase space are necessarily bounded.
The attractors create intricate patterns, folding and stretching the trajectories must occur because no trajectory intersects in the phase space, which is ruled out by deterministic dynamical motion ^{[6]}. The figures reveal a complex folded, layered structure of the attractors. Amplifying figure we would note that the lines are really composed of a set of sub lines. Amplifying a sub line we would see another set of sub lines and so on
…
verifying that the strange attractors usually are fractals [^{3},^{31},^{33}].
4.2. Chaos in damped and driven pendulum
Another example of onedimensional nonlinear motion is the one described by the damped and driven pendulum around its pivot point shown in Fig.7
^{[1]}.
The torque
τ
around the pivot point can be written as
τ=Id2θdt2=bdθdtmglsin(θ)+Ndcos(ωdt),
(9)
where
I
is the moment of inertia,
b
the damping coefficient and
Nd
is the driving force of angular frequency
ωd
. Dividing Eq. 9 by
I=ml2
results the nonlinear equation
d2θdt2=bml2dθdtglsin(θ)+Ndml2cos(ωdt).
(10)
If we want to deal with Eq.(10) with a computer it is more convenient to use dimensionless parameters. So, let us divide Eq.(10) by
ωo2
=g/l and define the dimensions less parameters: time
t′=t∕to
with
to=1∕ωo
and driving frequency
ω′=ωd∕ωo
. The new dimensionless variables and parameters are presented in Table 1:
Table 1 Damped and driven pendulum dimensionless variables and parameters.
Variables and parameters 
Damping coefficient
c=bml2ωo

Driving force strength
F=Ndml2ωo=Ndmgl

Dimensionless time
t′=tto=tgl

Driving angular frequency
ω=ωdωo=ωdlg

Using the variables and parameters defined in Table1, we verify that Eq.11 becomes,
d2θdt′2=cdθdt′sin(θ)+Fcos(ωt).
(11)
Defining
y=dx∕dt
and
z=ωt
, the secondorder nonlinear differential equation (11) is substituted by a system of two first orderdifferential equations:
y=dxdt′
(12)
dydt′=cysin(x)+Fcos(z)
(13)
Integrating numerically Eq. (12) and (13), we find periodic and chaotic attractors which depend on the chosen parameters and initial conditions. As an example, in Fig. 8 we present the only three periodic oscillations (represented by blue, red, and green lines) that are obtained by a specific choice of parameters, for all possible initial conditions. These solutions correspond to three different periodic attractors.
Furthermore, to show the attractor dependence on initial conditions, we present in Fig.9 the parameter space obtained by a grid of initial conditions. For each initial condition we obtain the numerical solution and identify the corresponding atractor, associated with one of the three lines shown in Fig.8, and represent it in Fig.9 as a point with the same color used in Fig.8. Figure 9 (a) is denominated basin of attraction of teh solutions of Eq. (7)
^{[32]}. The successive amplifications of the basin of attraction, shown in Fig.9 (b), (c) and (d), indicate the basin of attraction fractality.
5. Mapping
In some cases it is very difficult to study the evolution of a nonlinear system integrating their differential equations. Sometimes it is also difficult to construct an exact nonlinear mathematical model to study physical system. In these cases it is possible to get a good description of the chaotic process using an iterative algebraic model named mapping. To understand the origin of this model let us assume that the motion of a system is described by nonlinear firstorder differential equations of the form ^{[8]}
dxdt=f(x),
(14)
where
x
and
f(x)
are explicitly independent of time and that the motion is represented in Poincaré section
∑R
in Fig. 5.
The Poincaré map is found by choosing a point
xn
on
∑R
and integrating Eq. (14) to find the next intersection
xn+1
of the orbit with
∑R
. In this way we construct the map
xn+1=f(xn)
.
In a few words, denoting by
n
the time sequence of a system and by
x
the physical observable of this system we can describe the progression of a nonlinear system at a particular moment by investigating how the
(n+1)th
state depends on the
nth
state. The evolution
n→n+1
can be written as a difference equation using a function
f(α,xn)
as follows
xn+1=fα(xn),
(15)
where
α
is a model dependent control parameter,
α
and
x
are real numbers. The function
falpha(xn)
generates the value
xn+1
from
xn
and the collection of points generated is said to be a map of the function itself. The difference equation (14), which is an evolution equation in the Poincaré section is considered a milestone in the field of nonlinear phenomena. Note that
n
must be iterated from
n=1
up to
N>>1
.
5.1. Logistic Equation and Logistic Map
There are innumerous chaotic systems studied with the mapping approach. Famous examples are the map models for ecological and economic interactions: symbiosis, predator prey and competition [^{34},^{35}]. Malthus, for instance, claimed that the human population p grows obeying the law ^{[34]}
dpdt=kp.
(16)
Verhulst ^{[35]} argued that the population grow has inhibitory term
ap2
so that Eq. (16) is actually given by a nonlinear equation, called logistic function
dpdt=kpap2,
(17)
which shows that the population tends asymptotically to the constant
k∕a
.
One century later, indicating the population by
x
the differential equation (17) was substituted by the logistic equation [^{34},^{35}]
xn+1=αxn(1xn),
(18)
where
0<α<4
in order to assure that
0<xn<1
.
Note that the Eq. (18) must be calculated (iterated) from
n=1
up to the cycle
n>>1
. An
n
cycle is an orbit that returns to its original position after
n
iterations. In reference ^{[1]} are presented logistic maps of
xn+1
as a function of
xn
showing that
x
assume one stable value and only two discrete values for
α
values in the interval
2.8

3.1
, characterizing a periodic motion.
A more general view of the evolution can be obtained plotting a bifurcation diagram [^{1},^{34},^{35}] (see Fig. 11) where the
xn
is calculated numerically after many 1nteractions to avoid initial effects is plotted as a function of the parameter
α
^{[1]}.
Analyzing this figure we verify that for
2.80<α<3.00
there is a stable population (the period is one cycle;
xn+1=xn
). At
α=3.1
we see a bifurcation (because of obvious shape of the diagram) where there is a period doubling effect (
xn+2=xn
):
x
begins to oscillate periodically between 0.558 and 0.765. At
α=3.45
there are two different points of bifurcation: now there appear four possible periodic oscillations. The bifurcation and period doubling continues up to an infinite number of cycles near 3.57. Chaos (black regions) occurs for many of
α
values between 3.57 and 4.0, but there are still windows of periodic motions (white region). Detailed description of these regions can be seen, for instance, in references [^{34},^{36}], where is also shown a cobweb diagram of the logistic map showing chaotic behavior for most values of
α>3.57
. The special case of
r=4
can in fact be solved exactly ^{[10]}, as can the case with
α=2
; however the general case can only be calculated numerically. For
α=4
is
xn=sin2(2nθϕ
) where the initial condition parameter
θ
is given by
θ=(1∕ϕ)arcsin(xo1∕2
). For rational
θ
after a finite number of iterations
xn
maps into a periodic sequence. But almost all
θ
are irrational, and, for irrational
θ
,
xn
never repeats itselfit is nonperiodic. This solution equation clearly demonstrates the two key features of chaos stretching and folding: the factor
2n
shows the exponential growth of stretching, which results in sensitive dependence on initial conditions, while the squared sine function
xn
keeps folded within the range
{0,1}
.
5.2. Hénon Map
Another example is the bidimensional dissipative Hénon map given by the equations
xn+1=a+bynxn2
(19)
yn+1=xn,
(20)
where the parameters
a
and
b
are the control parameters ^{[26]}.
Examples of numerical solutions of Eq. (19) and (20) are in Fig. 12, which shows a periodic and a chaotic attractors, obtained, respectively, for (a)
a=1.45
,
b=0.2
and (b)
a=0.2
,
b=1.48
.
To show how the numerical solutions depend on the control parameters, we present in Fig. 13 (a) the bifurcation diagram of Eq. (19) solutions for a fixed a and
1.42<b<1.48
. An interval with a period 5 attractor can be observed in Fig. 13. In the parameter space of Fig. 14 (a) we indicate the period 5 window in the parameter space. The amplification in Fig. 13 (b) shows better the same periodic window. Such windows are also called shrimps ^{[37]} and have been observed in several dynamical systems [^{38},^{39}].
6. Lyapunov Exponents
The nonlinear terms of the differential equations amplify exponentially small differences in the initial conditions. In this way the deterministic evolution laws can create chaotic behavior, even in the absence of noise or external fluctuations. In the chaotic regime it is not possible to predict exactly the evolution of the system state during a time arbitrarily long. This is the unpredictability characteristic of the chaos. The temporal evolution is governed by a continuous spectrum of frequencies responsible for an aperiodic behavior (see, for instance, 4). The motions present stationary patterns, that is, patterns that are repeated only nonperiodically [^{2},^{3}]
Lyapunov created a method [^{1}^{3},^{34}] known as Lyapunov characteristic exponent to quantify the sensitive dependence on initial conditions for chaotic behavior. It gives valuable information about the stability of dynamic systems. With this method it is possible to determine the minimum requirements of differential equations that are necessary to create chaos (see footnote ^{2}). To each variable of the system is a Lyapunov exponent. Let us study the case of systems with only one variable ^{[1]} that assume two initial states
xo
and
xo+ε
, differing by a small amount
ε
. We want to investigate the possible values of
xn
after
n
iterations from the two initial values. The difference
dn
between the two
xn
values after
n
iterations (omitting for simplicity the subscript
α
) is given approximately by
dn=f(xn+ε)f(xn)=εexp(nλ),
(21)
where
λ
is the Lyapunov exponent that represents the coefficient of the average exponential growth per unit of time between the two states. From Eq.(21) we see that if
λ
is negative, the two orbits will eventually converge, but if positive, the nearby trajectories diverge resulting chaos. The difference
d1
between the two initial states is written as
d1=f(xo+ε)f(xo)≈εdfdxx0.
(22)
Now, in order to avoid confusion that sometimes is found in the chaotic literature, we remember that
xn+1=f(xn)=f(f(xn1))=f(f(f(xn2)))=...=f(f(f(...(f(xo))))),
(23)
that also is written as
xn+1=f(xn)=fn(xo),
(24)
where the superscript
n
indicates the
nth
iterate of the map.
After a large number
n
of iterations the difference between the nearby states, using Eq. (21) and Eq. (23), will be given by
dn=f(xn+ε)f(xn)=fn(xo+ε)fn(xo)=εexp(nλ).
(25)
Dividing Eq. (23) by
ε
and taking the logarithm of both sides, results
lnfn(xo+ε)fnxoε=ln[exp(nλ)]=nλ.
(26)
Taking into account that
ε
is small we obtain from Eq. (24),
λ(xo)=1nlndfnxodxo.
(27)
Since
fn(xo)
is obtained iterating
f(xo)
n
times we have
fn(xo)=f(f((f(xo)))
, that is,
fn(xo)=f(fn1(xo))=f(fn1(fn2(xo)))=
, where
xi=fi(xo)
is the result of the
ith
iteration of the map
f(x)
from the initial condition
xo
. So, using the derivative chain rule we get
dfnxodxo=dfn(xn1)dxodfn(xn2)dxo…dfn(xo)dxo.
(28)
Thus, for
ε→∞
we get, using Eq. (25) and Eq. (26),
λ(xo)=lim→∞1nln∏dfxidxo=lim→∞1nlndf(xi)dxo,
(29)
where
xi=fi(xo)
. In the lim n
→∞
the Lyapunov exponent becomes independent of the initial condition
xo
. This occurs because when is done an infinite numbers of iterations. the attractor is entirely covered by
x(t)
, and it does not matter the initial point
xo
. As in practice
n
are large, but finite numbers, we calculate
λ
for different initial conditions and take an average of these values.
From Eq. (21) we verify that if
λ
is negative, the two orbits will eventually converge; but if
λ
is positive, the nearby trajectories diverge resulting chaos. From Eq. (23) we see that at the bifurcation
λ=0
because
df∕dx=1
(the solution becomes unstable). When
df∕dx=0
we have
λ=∞
(the solution becomes super stable).
The
λ
estimation using simply the flow equations (6), (8) and (3), that is, without maps, are in general difficult because one has to deal with solutions of NLDE and analytic calculations. This kind of calculation for the damped and driven pendulum is seen, for instance, in reference ^{[1]}. Using maps these calculations become easier. This is shown in what follows for logistic map and triangular map.
6.1. Lyapunov exponents for logistic map
According to Eq. (25) or Eq. (19) to obtain
λ
are used the iterated functions
fn(xo)
. For the logistic map we have the logistic equation (18).
As an example, the second order iterated function
f2(x)
is given by
f2(x)=f(f(x))=f(αx(1x))=α(f(x)(1f(x)))=α2x(1x[1αx(1x)]
.
So, to get
λ(xo)
we can continue to iterate
f(x)
up to
n>>1
and use Eq. (25) or use Eq. (26) taking into account
f(xi)
, with
i=1,2,…,n
, remembering that
f(xi)=fi(x)
.
In reference [^{36}, ^{40}] are seen cobweb plots (web diagrams) or Verhulst diagrams that are graphs that can be used to visualize successive iterations of the function
f(x)
. In particular , the segments of the diagram connect the points (
x,f(x)),(f(x),f(f(x))),(f(f(x)),f(f(f(x))))
. The diagram is sonamed because its straight lines segments anchored to the functions
x
and
f(x)
resemble a spider web. The cobweb plot is a visual tool used to investigate the qualitative behavior of onedimensional iterated functions such as the logistic map. With this plot it is possible to infer the long term status of an initial condition under repeated application of a map.
In Fig. 11 are shown the Lyapunov exponents
λ
calculated numerically as a function of the parameter
α
for the logistic map
x
seen in Fig. 6.
6.2. Lyapunov exponents for triangular map
In the particular case of a triangular map [^{8},^{31}]
λ
can be calculated analytically. This map, represented in Fig. 15, obey the following equations:
xn+1=2βXn,0<x≥0.5xn+1=α(1xn),0.5<x<1,0<β≥1.
(30)
Equations (30) can be rewritten as
xn+1=f(xn)
, where the function
f(x)
is given by
f(x)=β[120.5x].
(31)
The
nth
application on
2βx
of the first region
0<x<0.5
give
fn(x)=(2β)nxn
.
The maximum value of
fn(x)
is
βn
at the point
x=2n
. By symmetry the next point of minimum must be
2.2n
and of maximum at
3.2n
and so on.
This implies that
fn(x)∕dx=(2β)n
for the two regions. Taking into account Eq. (25) we get
λ(xo)=1ndf(xo)dxo=ln(2β).
(32)
Consequently, there is chaos only for
β>0.5
, since
λ>0
.
7. Turbulent Processes
As usually seen in basic physic courses [^{4},^{41}], turbulence is originated from studies of fluid motion in classical mechanics. The general equation of motion for a viscous fluid is given by the NavierStokes nonlinear partial differential equation (NLPDE)
∂v∂t+(v.grad)v=grad(P)ρgrad(ϕ)+ηρlapl(v),
(33)
where
v(r,t)
is the velocity of the fluid at point
r
,
P
is the pressure,
ρ
the density of fluid,
ϕ(r)
the gravitational potential and
η
the viscosity. This equation is a miracle of brevity, relating a fluid's velocity, pressure, density and viscosity ^{[20]}. Since Eq. (33) is a NLPDE, it is not submitted to any general method of solution (see Footnote ^{2}).
Laminar flow occurs for very small Reynolds number
Re=νLρ∕η<<1
[^{17},^{20}], where
ν
is a typical fluid velocity and
L
is some characteristic length in the flow. In these conditions Eq. (33) can be approximated by a linear partial differential equation (LPDE) and all elements of volume of the fluid describe well defined trajectories
r=r(t)
. Since there are an infinite number of elements of volume
δV
the resulting LPDE has an infinite number of degrees of freedom which is a characteristic of the PDE (see Footnote ^{2}). For
Re>>1
the nonlinear effects become dominant being responsible for the phenomenon called turbulence. In these conditions the flux becomes disordered: the trajectories of the fluid elements
δV
are irregular and develop eddies, ripples and whorls. In spite of this yet there is some sort of order found within the disorder or turbulence which could be described as selfsimilar or fractal ^{[25]}. An open problem is to find a mathematical formalism able to describe this disordered state [^{25}^{27}].
Turbulence in fluid dynamics is being understood in infinite dimensional phase space under the flow defined by the NavierStokes equation. We have seen that in the finite dimensional phase space physical systems can be described with very good precision by LODE and NLODE that can solved exactly or numerically. They can in principle reveal all detailed structures of the dynamical systems. Turbulence in fluid mechanics is generated by a NLPDE anchored in an infinite dimensional phase space. Is turbulence a chaotic process? Up to nowadays it is wellknown that the theory of chaos in finitedimensional dynamical systems has been welldeveloped. Such theory has produced important mathematical theorems and led to important applications in physics, chemistry, biology, engineering, etc ^{[17]}.
Note that, in the contrary, theory of chaos in PDE has not been welldeveloped. In terms of applications, most of important natural phenomena are described by linear and nonlinear partial differential equations (wave equations, Yang

Mills equations, Navier

Stokes ,General Relativity, Schr
ȯ
dinger equations, etc) (see Footnote ^{2}). In spite of extensive investigations it was not possible to prove, in the general case, the existence of chaos in infinitedimensional systems [^{10},^{17},^{18},^{20}].
Among the NLPDE there is a class of equations called soliton equations that are integrable Hamiltonian PDE and natural counterparts of finitedimensional integrable Hamiltonian systems ^{[10]}. Many works have also been developed investigating the existence of chaos in perturbed soliton equations [^{20},^{27}].
Acknowledgements
This study was possible by partial financial support from the following Brazilian government agencies: CNPq, CAPES and FAPESP (2015/073117 and 2011/192961)
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