Mathematics 6350     Dynamical Systems and Chaos      Spring, 1992
Evans M. Harrell II
Georgia Institute of Technology† 

version of 28 May 1992

I.  Introduction.

The subject of these notes is one of the most ancient of sciences, and yet it remains one of the freshest and most vital.  Throughout history the motion of the heavenly bodies through the sky has commanded attention, and in this context the study of dynamical systems has a claim on the title of the oldest mathematical science, next to geometry:  The Chaldeans predicted eclipses about 3000 years ago, and there is some evidence that earlier civilizations performed the same feat 1000 years before that.  Celestial mechanics has continued to fascinate scientists largely because planetary motion is at the same time regular enough to be predicted and complicated enough to be interesting.

After centuries of analysis of the regular motion of dynamical systems like the solar system, attention has more recently been focused on irregular aspects of behavior.  Indeed, most dynamical systems are highly irregular.  The weather, for instance, is notoriously difficult to predict, so that even the best programs on supercomputers have indifferent success at prediction more than a day or two into the future.  Since the work of E. Lorentz in the 60’s it has been believed that the limitations on our ability to predict the weather do not arise from inadequacies in the model so much as from the “butterfly effect”:  Small differences in initial conditions accumulate in a dramatically unstable way, so that after some time the futures of essentially identical initial conditions have nothing to do with each other.  Irregular dynamical systems may lie beyond our power to understand in the traditional way of making precise predictions.  While many of the traditional questions may be out of the question, however, we may still hope to answer a different selection of questions, of a geometric or statistical nature.  Thus it is sometimes easier to predict the climate than to predict the weather - climatic modeling is reasonably successful in predicting average temperatures and rainfall on a time scale of a season.  By the way, even celestial mechanics is now known to be have some chaotic features - notable examples of chaotic motion being the tumbling of Hyperion, a potato-shaped satellite of Saturn, and possibly even the evolution of Pluto [WISDOM, 1987].  

I haven’t yet defined a dynamical system.  The concept of an abstract dynamical system is almost so general as to elude definition.  It originates in the desire of mathematical scientists, in common with Wall Street financiers and admirers of horseflesh, to predict the future.  Let us begin by mentioning some particular systems, such as:

	The Solar System
	A pendulum confined to a plane
	The world economy
	The weather
	A hydrogen atom
	An electrical circuit, such as a programmed computer
	The water and plumbing in your kitchen
	The flora and fauna in your kitchen
	The flora and fauna in your bloodstream, 
	etc.

Some of these can be considered as isolated systems, and others are subject to external influences in a way we may or may not be able to control.  For our purposes, we will imagine that we have control over any such influences, but if we didn’t we could still build a theory that brings them in in a probabilistic way.

In order to make predictions, we need to quantify both the system and our notion of time.  Time may be either continuous, as is apparently the case in physical kinematics, or discrete.  Discrete time would be appropriate for studying a population of moths, which breed in one season in the year and lay a number of eggs proportional to the population of adults surviving until that season.  The number of moths at other times has no bearing on the propagation of the species, and it would be a needless confusion to try to incorporate it into a model for predicting the change in the population over several years.  The state of a system will typically be specified by a set of real numbers that can be thought of as abstract position coördinates.  Thus we have to have some set S = {x1, …, xn}, where the xj would typically range over

	the positive integers (populations)
	the real number line (position coördinates)
	the positive numbers (temperature)
	points on a circle (angular position variables)

etc.  I will usually denote any appropriate way to specify the state of a dynamical system with a single letter x or z, without bothering to equip it with vector arrows or the like.  Although the set S  could be completely abstract, in this course we will always assume that it is a metric space, i.e., that we have a way to gauge the distance between points of S .  In fact, in most cases, S  will either be a discrete set, such as the set of integers Z, or a manifold, i.e., a set that looks like the usual finite-dimensional Euclidean vector space when viewed locally, in the sense that we can make use of a local coördinate system.  You can think of a manifold as a surface like a torus or a sphere, except that it may be of a higher dimension and may not in any obvious way sit inside some larger space in the way that these surfaces are embedded in R3.  (R is the set of real numbers and Rn denotes n-dimensional real Euclidean space.)  S  will often be called the state space, or phase space of the system.  On occasion S  will be a more complicated set, like a fractal, and in that eventuality we shall have to discuss its structure more carefully.

Understanding the dynamics of the system would entail finding a family of functions ft,s(z) which, for any initial time s and elapsed time t, allow us to predict the future knowing the present.  This means reading off the state of the system at time s+t given the state at s.  In other words, for any s,t, ft,s is a function sending S  to itself.  The family of functions ft,s will be called a number of things depending on the context: the flow, the solution operator, or the time evolution.  The flow has the following two properties:

	fo,s(z) = z
								(1.1)
	ft',t+s ˚ ft,s (z)  =  ft+t',s(z).

These assumptions just say that if no time elapses, the system stays the same, and if time t elapses and then time t' elapses, time t+t' has elapsed - Our observations do not affect the system.

For theoretical purposes we will generally assume that ft,s is independent of s, and will denote it simply ft.  This is like assuming that the force is independent in time, and the system is then said to be autonomous.  We can always accomplish this simplification by using a new state space S' having a new coördinate s in addition to the coördinates z.  Then a new flow ft is defined by

	ft(z,s) := (ft,s(z), s+t).

This is an absolutely worthless trick for analyzing actual problems, but it simplifies the theoretical framework of our analysis.  Since the coördinate appended to the state space evolves in such a simple way, it is often said that the effect of explicit time-dependence is equivalent to half an extra dimension.

Observe that autonomous dynamical systems have the semigroup properties, 

	fo(z) = z
	ft ˚ ft'(z) = ft+t'(z),					(1.2)

for all z Œ S  and all times t,t'.

In the abstract that is all there is to the definition of a dynamical system - a family of functions on a set with some very general properties.  

Exercise I.1.  Specify suitable sets S for the systems listed above, or others of your choice.  For instance, if we consider only the planetary motions, the solar system could be specified by S = R54.  In order to solve NEWTON’s equations in principle, we need  to know 6 coördinates for each planet, viz., both a position and a momentum vector.  Explain why the phase space for a pendulum could be considered as a circular cylinder.

I wish to keep this course fairly specific, so as an antidote to the abstract definition given above, let us look at several particular examples.

In a physics class one starts with an analysis of one-dimensional Hamiltonian systems, such as the harmonic oscillator.  A harmonic oscillator is a system where there is a restoring force proportional to the displacement z Œ R. (As HOOKE expressed it, “ceiiinosssttvv” in an anagram for his law:  “vt tensio sic vis”.  In those days people engaged in scientific priority battles by intentionally publishing articles that nobody could understand.)  In the Skiles Classroom Building at Georgia Tech, masses and spring constants have a striking tendency to equal 1, so the equations of motion are:

		dx/dt = p     (definition of the momentum)
										(1.3)  
		dp/dt = -x    (NEWTON’s law for a force  -x).

The phase space in this case is a phase plane, (x,p) Œ S  = R2.

The force in this case is equal to - —V(x) for a function V(x) = x2/2, known as the potential energy.  A system with this property is said to be Hamiltonian, or conservative, and satisfies a conservation of energy principle.  (Actually, Hamiltonian is a somewhat more general adjective.)

If the phase space is two-dimensional, as for (1.3), then the geometry of a phase-plane analysis of the motion is relatively simple.  In this specific example, we discover that ft is simply a uniform rotation of the plane, as follows:

If z = (x,p) denotes a position in the phase plane, then (1.3) is of the form

	f(dz,dt) = v(z),

where v(z) is a vector field defined on S .  We can understand its effect by drawing velocity vectors for z at some representative points of the plane.  The time-evolution of any point z Œ S  will take it along a trajectory that is always parallel to the velocity vector at z.  We can thus visually connect the arrows representing the velocity vectors and sketch out the trajectories, or flow lines, of this dynamical system:

To see analytically that the flow-lines are circles, we need only note that in the case of the harmonic oscillator (1.3), the velocity is always perpendicular to the position vector (slope -x/p compared with p/x).  Geometrically, this can only happen for a circle centered at the origin.  A second way to see that the trajectories are circles is to note that the total energy, p2/2 + x2/2 = E is independent of time, so the trajectories are circles with radii r(2E).

This sort of analysis is called sketching the phase portrait of the dynamical system.

The phase portrait does not indicate the rate at which the system follows a trajectory, but it is easy to solve (1.3) for x and p as functions of time:  Differentiating the first formula and substituting from the second, we get d2x/dt2 = - x.  Thus we find 

	x(t) = A cos(t - a),       p(t) = - A sin(t - a),

with the amplitude A and the phase a arbitrary.  

Exercise I.2. Sketch phase portraits for the dynamical systems

	f(dx,dt)  =  p
					(the free particle)
	f(dp,dt)  =  0

	f(dx,dt)  =  p
					(the pendulum)
	f(dp,dt)  =  - sin x

An elementary treatment of phase-plane analysis is to be found in [SIMMONS, 1972].  More advanced treatments can be found in [ARNOLD, 73; Hale-Koçak, 1989; JORDAN-SMITH, 1987; LEFSCHETZ, 63].  In addition, many examples of phase portraits can be studied on computers with such software as Mathematica  or H. Koçak’s Phaser   (MS-DOS based machines) or B. WALL’s Chaos  (Macintosh).

Now let’s get a little more formal:

Definition.  Let S  be either a Euclidean set (a subset of Rn) or a manifold.  Let v be a differentiable vector field, i.e., a function assigning to each point of S  a tangent vector, and doing so in a differentiable way.  (If we use coördinates defined at least in a neighborhood of z Œ S , then a tangent vector is differentiable provided that each of its coördinates, as expressed in terms of some basis, depends differentiably on each coördinate of z.)  Then a differentiable dynamical system is the dynamical system specified by the equations of motion

		f(dz,dt)  =  v(z).						(1.4)

The basic existence-uniqueness theorem for ordinary differential equations states:

Given any initial point  z(0), there exists a time-interval  -a ≤ t ≤ b, a,b > 0, on which the differentiable dynamical system (1.4) has a unique solution z(t) = ft(z).  This solution depends differentiably on  t and  z.

(This theorem actually requires less than differentiability, viz., a Lipschitz condition.)

Definition.  The general solution of (1.4) is said to define a complete flow if solutions exist for all times for all initial points.  When speaking of flows below, I will tacitly assume they are complete, unless they aren’t.

Warning.  Even if V is very nice, the flow may not be complete.  For example, if dx/dt = p, and dp/dt = 4x3, then most trajectories reach infinity in a finite amount of time, as shown by the following calculation:  The potential energy here is -x4, so the total energy E = p2/2-x4 is conserved (i.e., independent of time.  For those innocent of physics, just calculate 
dE/dt = pÇÇ◊(4x3) -4x3◊p = 0).

Therefore the time of transit between two points x1 and x2 is:

	T(x1, x2) = i(x1,x2, f(dt,dx) dx)  =  i(x1,x2, f(1,p) dx) 

		   = i(x1,x2, f(dx,r(2E+2x4))) . 
This integral is finite even if x2 = ∞.

Exercise I.3.  Formulate a condition on the force dp/dt, so that the system cannot reach infinity in a finite time.

The basic existence-uniqueness theorem has some immediate consequences for continuous dynamical systems (assumed complete):

a)   Time can run backwards:  Knowing the future, we can predict the past.  This means that ft has the algebraic structure of a group.  (In addition to (1.2), a group has the property that every element has an inverse, which, by the composition law (1.2), satisfies ft-1 = f-t.)  Not every interesting dynamical system will have this property.

b)  Flow lines (= trajectories) cannot intersect.  (In the case of equilibria, as we shall see later, infinite-time trajectories can have the single-point representing the equilibrium as an end point, but such a trajectory is like an open interval, and does not contain its end point.)

c)    The state space is precisely the union of the trajectories.  In fancier language, the state space is fibered  by the trajectories.

Suppose that the vector field V Œ Cr, which by definition means that all its derivatives of r‑th order or less exist and are continuous, where r is a positive integer.  Then for each t, the function ft is a Cr‑diffeomorphism of S , i.e., ft and its inverse f-t are defined on all of S  and Œ Cr.  Some abstract works define a flow as a continuous group of diffeomorphisms parametrized by time.

Now consider some examples from biology.  If p(t) denotes the population, as measured in grams, of an initially small amount of pond slime (algae) introduced into a pond full of good phosphate pollutants, then the slime reproduces at a steady rate, proportional to p(t), i.e., p'(t) = ap(t).  In this case S  = R+, and, as everyone knows, ft(p) = exp(at) p.  Of course this model cannot be realistic when the limits of the resources are neared (when the pond fills up with slime).  A plausible improvement of the model would be to assume an equation of motion

	p'(t) = ap - b p2,						(1.5)

since then the population p cannot grow indefinitely:  p' would be negative if p > a/b.  In fact it is easy to see that a/b is the asymptotic value of the population, regardless of its starting value.  A straightforward calculation allows us to calculate the solution operator, and

	ft(p) = f(ap,bp+(a-bp)exp(-at)) .

This is a decent model for microbes, and not at all bad for larger creatures:  In 1845 VERHULST used curve-fitting to determine a and b for the American population from census data, and extrapolated to the future.  His prediction for 1930 was 123.9 million, compared with an actual value of 123.2 million.

If there are more species, we could use continuous dynamical systems to model such things as competition and predation (assuming that the populations are so large that we may treat them as continuous variables).  For example, if there are two species x and y, and y preys on x, which is herbivorous, a plausible model, developed by VOLTERRA and LOTKA in the nineteenth century, for the development of the two species is:

		S  = R+ ¥ R+

		f(dx,dt) =  a x - b xy
									(1.6)
		f(dy,dt) =  - c y + d xy.

Here a, b, c, and d  are positive constants, known as the vital parameters of the species.  In the absence of prey, the species y will decrease exponentially - some last longer than others, perhaps because of cannibalism.  One might try to improve the validity of the model somewhat by adding a self-limiting term proportional to -x2 to the formula for dx/dt.

Exercise I.4.  Sketch a phase portrait for this system.  Use your computer to study it.

For a discrete-time problem, we shall often denote the time variable by n.  Discrete-time problems are the same as iterated maps, or iterated functions, that is, functions on S  which are composed with themselves many times.  Although some authors denote this procedure,

	          f˚f˚…f(x)    =    f(f(....f(x))),
		   (n times)          (n times)

by fn(x), this can easily be confused with taking powers, so I prefer the notation

	f˚n(x)  = f˚f˚…f(x),							(1.7)
		   (n times)

which is also somewhat widespread.

Discrete-time dynamical systems arise in practice in a number of ways.  Some systems, like the population of butterflies mentioned above, are naturally set up from the beginning with a discrete dependence on time.  Moreover, even if a dynamical system depends continuously on time, as soon as we model it on a computer, we may discretize it by using equal minimal time intervals.  This does not necessarily involve making approximations; we could analytically solve a system of differential equations, such as (1.3), but only choose to examine the solution at integer times:

		x(n) = fn(x) = f1°n(x).						(1.8)

Another way in which discrete-time dynamical systems arise is via a Poincaré section.  Instead of evaluating a differentiable dynamical system at equal time intervals, we may evaluate it whenever the trajectory passes through a distinguished submanifold (surface) in S.  For example, a planar pendulum has a state space S with coördinates (q,w) corresponding to angular displacement and angular velocity.  Suppose that the pendulum has a small amount of friction.  We might only be able to measure the maximum displacement of a pendulum from equilibrium, which would correspond to having the trajectory pass through the surface w=0, and would like to predict q(n), the n-th maximum displacement, from the value of q(n-1).  There is a definite functional relationship of some sort q(n) = F(q(n-1)), which can be considered a discrete dynamical system, and is known as the Poincaré map, or first-return map.  Unlike the harmonic oscillator, the oscillations of the pendulum do not have precisely the same period regardless of amplitude, so F(q) is certainly different from evaluating ft at fixed time intervals.  

Not every submanifold is suitable for Poincaré sections; for example if the submanifold M actually contained a trajectory, then there would not be a well-defined map taking points of M to the next points where their trajectories lie in M.  For this reason M is generally chosen so that all trajectories cut through it transversely.  In additional to arising naturally in experiments, since measurements are often taken at convenient values of spatial or other physical variables rather than at regular time intervals, the notion of a Poincaré section is useful in computation, because it is much more efficient to iterate a discrete map than to integrate a differential equation.  One can solve for a Poincaré map by integrating some trajectories over some important piece of the state space, and then iterate the Poincaré map instead of continuing to repeat the integrations of the trajectories.  It is also useful conceptually, as often one can use a Poincaré section to get some dimensional intuition about the behavior of a higher-dimensional dynamical system.  

Exercise I.5.  Consider the pendulum again, but take the Poincaré section with q=0.  To what experimental situation does this correspond?  Can you write an expression for the Poincaré map?  Can you give a physical interpretation for it in terms of energy loss?
												
In analogy with (1.5), we might expect to model the growth of a single population with finite resources by

	x(n) = Fµ(x(n-1)), 							(1.9)

with Fµ(x) being a quadratic expression such as µx(1-bx).  The subscript µ here is a parameter, and not a time-step; for iterated maps, the subscript 1 in (1.8) is not very helpful, and the solution operator will usually be denoted in this context by something like F(x) rather than ft(x).  Let us set b = 1, so that

	Fµ(x) = µx(1-x),							(1.10)

allowing us to focus on the effect of a single positive parameter µ.  The flow of this dynamical system will be given by the functions Fµ˚n.  It is not necessary here to solve equations of motion before beginning to ask what the solution means.

There is a surprising amount of structure in the theory of this simple, one-dimensional dynamical system.  The current vogue for the theory of iterated mappings as dynamical systems began with the analysis of this structure independently by R.M. MAY [May, 1976], a mathematical biologist, and M. FEIGENBAUM [1978], a theoretical physicist.  Perhaps the most astonishing fact is that this intricate structure is pretty representative of what goes on in a wide variety of dynamical systems with many more degrees of freedom.