Chaos in discrete systems:
The discrete world: enlarging possible dynamics
The Grebogi equation
The logistic map

Chaos in infinite-dimensional systems:  (delay differential equations)

The McKey-Glass model

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 n-dimensional 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 first-countable space is dense in B if Dn,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 fn = 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. 363-365).

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 so-called 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 - x2t, generates chaotic series (Grebogi et al. 1983; Williams, G.P.P 1999). Another classic example is given by the quadratic recurrence equation (logistic map)  xt+1 = r xt(1 - xt), 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 one-dimensional 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 long-range unpredictable systems. The concept of low-dimensional 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 low-dimensional deterministic in the sense of the theory.

Chaos in continuous systems:

About the expected dynamics in continuous systems:
The Poincaré-Bendixson 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:
1. is a fixed orbit (a periodic point with period zero, i.e., an equilibrium).
2. is a regular periodic orbit.
3. consists of (finitely many) equilibria and non-closed 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 Poincaré-Bendixson 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 infinite-dimensional space of solutions on a three-dimensional 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 two-dimensional 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 very-long-range forecasting would seem to be non-existent. [...] ".

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Rössler Equations
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 (disk-embedded) 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) two-dimensional surface, but rather to a (folded) disk of finite width (Rössler, 1976). Rössler-like 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 two-dimensional 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."

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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 - x2t             (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 so-called 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 period-doubling, and is one of the routes to the chaotic behavior. A successive number of period-doublings 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)

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.

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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 well-known route to chaos: the period doubling. This one-dimensional model can be represented as

x t+1 = r xt (1 -  xt)           (1)
where r is the growth rate and (1 - xt) an intraspecific growth function. These kind of functions find applications in a wide range of fields, from biology to economics.

Fig. 14. The logistic bifurcation diagram

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. 31-50; Gulik 1992, pp. 112-126; Holmgren 1996, pp. 69-85). 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 host-parasite population dynamics. Such chaotic flow, characterized by near-zero, positive Lyapunov exponents has been labeled homeochaos.

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Chaos in infinite-dimensional 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 McKey-Glass 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.

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In view of the inevitable inaccuracy and incompleteness of weather observations,
precise very-long-range forecasting would seem to be non-existent
.

Edward N. Lorenz  1962
Journal of the Atmospheric Sciences, vol. 20 pp. 130-141

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