Chaos
Chaos
in continuous systems:
About the expected
dynamics in continuous systems
Lorenz equations
Rössler
equations
Chaos in
discrete systems:
The
discrete world:
enlarging possible dynamics
The Grebogi equation
The logistic map
Chaos
in infinitedimensional systems:
(delay differential equations)
The
McKeyGlass model
Numerically
computed Lyapunov exponents
Introduction
Ordinarily, chaos is confusion or
disorder... deterministic chaos
implies,
scientifically talking, some upsets, but we can find outstanding
properties with important consequences. In
science, chaos describes an important conceptual paradox which has a
precise mathematical meaning. This paradox is that chaos is generated
by some deterministic systems which make difficult their future
prediction. Change and time are two fundamental vectorial form subjects
that form
altogether chaos
foundations. We normally think that a deterministic system is one whose
state at one time completely sets its state for all future times. Chaos
is not a mathematical contradiction, it is a conceptual contradiction
with practical consequences.
The main
question is how accurately can you predict over what length of time
given a certain amount of initial information. The reason a
deterministic
system can be difficult to predict is that what happens in the future
can be very sensitive to its current state, this clearly is given by
nonlinear relations between inputs and outputs of such systems. This
property, called
"sensitivity to initial conditions," has been described as the
Butterfly Effect and we could sum up this idea with Lorenz quotation: Does
the flap of a butterfly's wings in Brazil set off a tornado in Texas ?
A
technical way to describe sensitivity is through the divergence of
trajectories of the system. Over time, a system starting from one state
becomes less and less similar (farther and farther away in state space)
to a system which starts out in a similar, but not exactly the same,
state. It is worth emphasizing that this means that the more accurately
the initial state is known, the more accurate can be a prediction.The
problem with prediction is that the degree of accuracy needed in many
practical cases is likely to be impossible to obtain. We can not know
the air temperature until the infinite decimal. The
paradox of chaos strikes at the roots of traditional concepts of
science which suggest that increasing knowledge will lead to
predictability. Chaos is not the only
source of unpredictability of a system's behavior. Conceptually, there
are three sources for the lack of predictability. The first is the
influence of random noise, the second is the effect of the environment
on the system, and the third is lack of knowledge of the initial
conditions. The third one is related to the idea of chaos. Chaos and
fractal geometry are closely related, so attractors in phase space
responsible of nonperiodic flow have a fractal geometry. These kind of
attractors
are called strange
attractors and add another kind of dynamic behavior to the other
known attractors: as fixed points or
limit cycles. An attractor is a set of states
(points in the phase space), invariant under the dynamics, towards
which
neighboring states in a given basin of attraction asymptotically
approach in the course of dynamics evolution. An
attractor is defined as the smallest unit which cannot be itself
decomposed into two or more attractors with distinct
basins of attraction. This restriction is necessary since a dynamical
system may have
multiple attractors, each with its own basin of attraction.
For dissipative dynamical
systems volumes shrink exponentially so attractors have 0
volume in
ndimensional phase space. A stable fixed point
surrounded by a dissipative region is an attractor known as a map sink.
Regular attractors (corresponding to 0 Lyapunov characteristic
exponents) act as
limit cycles, in which trajectories circle around a limiting trajectory
which they asymptotically
approach, but never reach. Strange attractors are bounded regions of
phase space
(corresponding to positive
Lyapunov
characteristic exponents) having
zero
measure in the embedding phase space and a fractal dimension.
Trajectories within a strange
attractor appear to skip around randomly. A dynamical system is chaotic
if it a) has a dense (A set
A in a firstcountable space is dense in
B if
D_{n},where
L
is the limit of sequences of
elements of
A. For example, the rational numbers are dense in
the reals. In general, a subset
A of
X is
dense if its set closure n>=10)
collection of points with periodic orbits, b) is sensitive to the
initial condition of the system (so that initially nearby points can
evolve quickly into
very different states), and c) is tologically transitive (A function
f
is topologically transitive if, given any two intervals
U and
V,
there is some positive integer
k such that lim f
_{n} = f
.
Vaguely,
this means that neighborhoods of points eventually get
flung out to "big" sets so that they don't necessarily stick together
in one localized clump).
The boundary between linear
and
chaotic behavior is often characterized by period doubling, followed by
quadrupling, etc., although other routes to chaos are also possible
(Abarbanel
et al. 1993; Hilborn 1994; Strogatz 1994,
pp. 363365).
About
attractors and strange
sets:
The term attractor is difficult to rigorously define. There is
still not a complete agrement about the definition of attractor.
Roughly speacking, an attractor is a set of the phase space to which
all or some trajectories converge. Extensively, an attractor is a closed set A
with
the following properties (Strogatz, 2000) :
1.
A
is an invariant set: any
trajectory
x(t) that starts in
A stays in
A for all time.
2.
A attracts an open set of
initial conditions: there is an open set
U containing
A such
that if x(0) belongs
to
U, then the distance from
x(t) to
A tends to 0 as t > infinite.
This means that
A attracts all trajectories
that start sufficiently close to it. The largest such
U is the
socalled the basin of attraction of
A.
3.
A is minimal:
there is no proper subset of
A
that satisfies conditions 1 and 2.
A chaotic attractor exhibits
sensitive
dependence on initial conditions. Such attractors are often fractal
sets, which
gave the name of strange set. Although chaos motion and its associated
aperidocity is highly complex, it can be found in "simple"
deterministic equations. For example, the discrete iterative function
x_{
t+1} = 1.9 
x^{2}_{t},
generates chaotic series
(Grebogi et al. 1983; Williams, G.P.P 1999). Another classic example is
given by the quadratic recurrence
equation (logistic map)
x_{t+1} = r
x_{t}(1

x_{t}), widely used in the context of
economics or ecology, which also has a
chaotic domain when the growth rate,
r,
is about 3.79.
It is surprising to find
in such onedimensional
discrete equation such complex behavior. The question is... is chaos a
real phenomenon or it is only a mathematical curiosity ? In this sense,
nonlinear
analysis tools and mathematical theorems related to turbulent fluid
dynamics have
improved a lot since first chaos papers were published. Even, many
experimental data in chemical kinetics or electronics have shown that
chaos can be found in such systems. So more than a
mathematical artifact chaos seems to be the essence of
nonlinear
and longrange unpredictable systems. The concept of lowdimensional
chaos arises in a huge amount of scientific literature, and has proven
to be useful to understand many complex phenomena. However very few
natural systems have actually been found to be lowdimensional
deterministic in the sense of the theory.
Chaos in continuous systems:
About the
expected dynamics in continuous systems:
The PoincaréBendixson theorem is one of the central results of
nonlinear dynamics. This theorem states: Let
be an
open subset of
,
and
.
Consider the planar differential equation
.
Consider a fixed
. Suppose that the omega
limit set
is compact, connected, and contains only finitely many equilibria. Then
one of the following holds:
 is a fixed orbit (a
periodic point with period zero, i.e., an equilibrium).
 is a regular periodic
orbit.
 consists of (finitely
many) equilibria and nonclosed orbits such that
and
(where is the alpha limit set
of ).
The same result holds when replacing
omega limit sets by alpha limit
sets. Since
was chosen such that existence and
unicity hold, and that the system is planar, the Jordan curve theorem
implies that it is not possible for orbits of the system satisfying the
hypotheses to have complicated behaviors. Typical use of this theorem
is to prove that an equilibrium is globally asymptotically stable
(after using a Dulac type result to rule out periodic orbits).
Roughly speacking, this theorem says that the dynamical possibilities
in the two dimensions (i.e. phase plane) are very
limited. If a trajectory is confined to a closed, bounded region that
contains no fixed points, then the trajectory must eventually approach
a closed orbit (i.e. a limit cycle attractor). Nothing more complicated
is possible. In systems of higher dimensionality (n >= 3), the
PoincaréBendixson theorem no longer applies, and something
radically
new can happen: trajectories may wander around forever in a bounded
region
without settling down to a fixed point or a limit cycle. In some
cases, the trajectories are attracted to a complex geometric object
called a strange attractor, a fractal set on which the motion is
aperiodic and sensitive to initial conditions. This sensitivity makes
the motion unpredictable in the long run.
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Lorenz equations
Lorenz
paper was very important in the science of chaos because the first
strange attractor of a
deterministic system was found, described and ploted. This is
the Lorenz
attractor and was discovered by Edward N.
Lorenz in 1962. By solving numerically Lorenz equations
(taking sigma = 10, r = 28 and b = 8 / 3) we can obtain time series and
the state variables that can be represented in phase space. In
Equations set 1 we show the Lorenz system and in figure 1 the time
series for its variables x, y and z.
Equations in Eq. set 1 are the
convection equations
which were obtained by projecting the infinitedimensional space of
solutions on a threedimensional subspace. Especifically, x is
proportional to the circulatory fluid velocity, y characterizes the
temperature difference between ascending and descending fluid elements,
and z is proportional to the distortionof the vertical temperature
profile from its equilibrium (which is linear with height). Sigma, r
and b are the physical parameters of the system given by positive
values. Roughly, the Lorenz system is a model of thermal convection
which includes a description of the motion of some viscous fluid or
atmosphere and the information about heat distributionwhich actually
represents the driving force of thermal convection. The simplicity of
this model hiddens a wide range of dynamical behaviors for various
values of one control parameter. The Lorenz system has either stable or
unstable fixed points, a globally attracting periodic or nonperiodic
solutions, a homoclinic orbit embedded in a twodimensional stable
manifold, bistability and hysteresis, an a variety of cascading
bifurcations (see Fig. 4) as well as the phenomenon crises (see Fig.
5).
Equations set 1.
Lorenz's dynamical equations
Fig. 1.
Time series for
Lorenz's
system variables. Initial conditions
are (0.3, 0.3, 0.3) for black lines.
In x(t) series initial conditions are changed (0.300000005, 0.3,
0.3) (red line)
Fig. 2. Another view of Lorenz
attractor in a 3D phase space
Trajectories flow
in phase space show two spiral unstable foci
suspended in an attracting surface, and mutually connected in such a
way that the outer portion of either spiral is "glued" toward the side
of the other spiral, whereby the outermost parts of the first spiral
map onto the more inner parts of the second, and viceversa
(Rössler,
1976).
Fig. 3.
Projections of
Lorenz
attractor in 2D phase spaces. Left: xy, center: xz and right: yz views
The Lorenz system is simple but the array of dynamical behaviors is
very huge.
r
Fig. 4.
Bifurcation diagram
for the Lorenz system by using r
as the order parameter
Fig. 5a. (x,z) phase
portrait in which the Lorenz attractor
collapses, via
chaotic transient, to a fixed
point placed inside the right wing
Fig. 5b. (x,z) phase
portrait for several asymptotic dynamics. (a) strange attractor;
(b), (c), (d) and (e) asymptotic fixed point dynamics via
chaotic
transient; (f) fixed point
Lorenz's
conclusions about weather forescasting stated:
" [...] When our results
concerning the instability of nonperiodic flow are applied to the
atmosphere, which is ostensibly nonperiodic, they indicate that
prediction of the sufficiently distant future is impossible by any
method, unless the present conditions are known exactly. In view of the
inevitable inaccuracy and incompleteness of weather observations,
precise verylongrange forecasting would seem to be nonexistent.
[...] ".
return up
Another famous attractor appears from
a simple set of three
differential equations in Rössler's system (see Equations set 2).
Rössler analysed
such equations in order to achieve a qualitative understanding of the
chaotic flow also shown for Lorenz model. Lorenz system has two
nonlinearities (xz and xy in equations set 1)
and Rössler has only one (zx
in equations set 2). In Rössler flow we find a
(diskembedded)
single spiral. The outer portion returns, after an appropiate twist and
the formation of a Möbius band is involved. Flow is nonperiodic
and structurally stable, although all trajectories are unstable. A
closer inspection of Rössler attractor shows that flow is not
confined to a (folded) twodimensional surface, but rather to a
(folded) disk of finite width (Rössler, 1976). Rösslerlike
equations, as this author claims, can be used as guideline for
the identification of systems (i.e. natural or artificial) showing the
same behavior, for instance in astrophysics, chemistry, biology as also
in economics.
Equations
set 2
Rössler dynamical
equations
If we want to see chaotic phenomenon
we
can solve Rössler equations numerically and get time series for
the
three variables involved in such system (see figure 6), we can also
plot phase
space to represent the vector field i.e. the flow. Figure 7 shows the
tridimensional phase
space where the Rössler attractor lives. Figure 8 shows
three
projections of this attractor in twodimensional phase space. Time
series
show nonperiodic and irregular oscillations or pulses for all of the
three
variables. Phase space in figures 7 and 8 show the flow, and if
we try to mentally follow such flow it is "easy" to see a
Möbius
band structure.
Fig. 6. x(t), y(t) and z(t) for the dynamical equations of
the Rössler system starting
from the initial condition (0.1, 0.2, 0.3) with a = 0.2, b = 0.2
and c = 6.7.
Fig.
7.
Rössler
attractor in a 3
dimensional euclidean space
The main property of chaotic systems
is
the high sensitive dependence on
initial conditions. Such systems cannot be broken down or descomposed
into two subsystems (two invariant open subsets) which do not interact
under f because of topological transitivity. And, in the midst of
this random behavior, we nevertheless have an element of regularity,
namely the priodic points which are dense. In figure 9 we can see this
effect. If we change
initial conditions in a far decimal place (the change is extremely
little) we can observe that series diverge in time.
Fig. 8. Trajectories
in two dimensional phase space for
Rössler System.
Top view (left) and sides views (centre and right)
Fig. 9.
Sensitivity to initial
conditions. In red, x(t) under initial
conditions (0.1, 0.2, 0.3). In black, x(t) under
(0.100005, 0.2,
0.3)
Rössler concludes his paper
quoting: "[...] continuous chaos is "strangely attractive" as a
physical phenomenon."
return up
Chaos
in discrete systems:
The discrete
world: enlarging possible dynamics
In
discrete dynamical systems (also called: difference equations,
recursion relations, iterated maps or simply maps) the presence of
chaos is possible from the one dimension. This is because the points
can hop along their orbits rather than continuously flow.
The Grebogi equation
One of the examples of a simple discrete
equation generating chaos is given by the equation of Grebogi. Such
iterative function is given by
x_{
t+1} = 1.9  x^{2}_{t}
(1)
Here x_{
t+1} (spoken as "x of t") is the
value of x at time t, and x_{
t+1} is the value of x at some interval (day, year,
...) later. That shows one of the requirements for chaos: the value at
any time depends, at least, in part on the previous model (this is the
characteristic of determinism).
Fig. 10. Irregular discrete time
series
obtained from Eq. (1)
Equation (1) also shows sensitive dependence on initial conditions. For
instance, in Fig. 10 the iteration starts with x(0) = 0.5. If we now
represent the same time series starting with x(0) = 0.5000000000001. We
obtain the time series in red, put on top of the time series of figure
10 and shown in Fig. 11. Here, approximately at the
iteration number 50 the system behaves completely different.
Fig. 11. Sensitive dependence on initial conditions,
one of the dynamical properties of strange attractors.
We have seen the dynamical behavior of Eq. (1) with the constant 1.9.
Lets now play a little with such constant defining it as an order
parameter. To do this we define k
as a variable taking the values of a constant in Eq. (1). If we plot a
diagram to analyze the dynamics of Eq. (1) as a function of k we obtain the socalled
bifurcation diagram. In Fig. 12 we can see all the universe of dynamic
behaviors in a window of the parameter k (here 0 < k < 2). The line
indicates that the attractor involved in the asymptotic dynamics is a
fixed point, two lines indicate a periodic dynamics with oscillations
among two constant values. The transition from the line to the double
line is called perioddoubling, and is one of the routes to the chaotic
behavior. A successive number of perioddoublings brings to the chaotic
domain, which is represented by a cloud of disperse points. We can also
see that in between the chaos can appear periodic windows indicating
and ordered and predictable dynamics. Notice that the value of k initially used i.e. 1.9, falls in
the chaotic region.
Fig. 12. Bifurcation diagram for Eq. (1)
click here to see the
algorithm in C
In Fig. 12 we have enlarged a little region of the bifurcation diagram
(in red), we can see that this little part is exactly the same as the
whole bifurcation diagram. This means that such diagram is fractal.
Hence, it has an infinte repetitive structure. In fact, its structure
is said to be invariant to the change of scale. The dynamics shown by
Eq. (1) is very similar to the famous logistic equation, explained in
the next section.
Fig. 13. One dimensional map of dynamics of Eq. (1)
The one dimensional map is a discrete equation or function that gives
the value of a variable as a function of its value at the previous time.
return up
The logistic map
The logistic map is an
iterative function able to give chaotic dynamics in some of its
parameter space. The parameter r
is the responsible to cause the bifurcation scenario characterized by
one of the most wellknown route to chaos: the period doubling.
This onedimensional model can be represented as
x_{
t+1} = r x_{t}
(1  x_{t})
(1)
where
r is the growth rate and (1  x_{t}) an intraspecific
growth function. These kind of functions find
applications in a wide range
of fields, from biology to economics.
Actually, the chaotic domain leaves a
cloud of points in parameter space with a fractional dimensionality.
Such structure is a
Cantor set. It
is relatively easy to show that the logistic map is
chaotic on an invariant
Cantor set for
(Devaney 1989, pp. 3150; Gulik 1992,
pp. 112126; Holmgren 1996, pp. 6985)
.
The logistic map actually made
scientifics to think that chaos was not possible to find in population
dynamics because its intrinsic unstability (Berryman and Millstein (
TREE, vol.4, nº1, 1989). The
initial debate because of May's papers were if the apparent random
fluctuations and the unpredictability in natural ecosystems may
actually be due to deterministic chaos. Berryman and Millstein argued
that in the bifurcation map of the logisitc equation
(see Fig. 14), the population spends more time at extremely low
densities, where there is a higher probability for deterministic
extinction given for example because of external noise. In
populations with small size the probability of extinction once the
chaotic domain is reached is extremely high. They also said that living
systems are antichaotic and that populations could enter in the chaotic
domain because of the human action i.e. perturbations. This paper, with
sentences like "deterministic chaos ... is more illusory than
scientific ..." , actually provoked an intensive debate. Since then,
lots of papers have shown that chaos do not involve a higher
probability of extinction. The role of chaos as a stabilizing
factor has been pointed out by Kaneko and Ikegami, who standed out a
high dimensional, weak chaotic flow responsible to give stability in
hostparasite population dynamics. Such chaotic flow, characterized by
nearzero, positive
Lyapunov
exponents has been labeled homeochaos.
return up
Chaos
in infinitedimensional systems:
Typically
autonomous ordinary differential equations are used to model the rate
of change of quantity which depends on its present value. However, for
some particular systems, its reasonable to assume that the rate of
change of a variable depends not only on its value at the present, but
also on its value at some time in the past.
The McKeyGlass model
dx /
dt = P(x(t  tau))  e x(t)
where P is the function that controls
the production of x and e is a decay constant.
Lyapunov exponents
One feature of
chaos is sensitive dependence on initial conditions.
return up
In view of the inevitable inaccuracy and
incompleteness of weather observations,
precise verylongrange forecasting would seem to be nonexistent.
Edward N. Lorenz 1962
Journal of the Atmospheric Sciences,
vol. 20 pp. 130141