version of 3 June 1992
In much of the discussion above we have been concerned with the long-time behavior of dynamical systems, with what happens when we start up the system and let it run indefinitely. Periodicity has played a large rôle, and some other notions such as attraction and asymptotic or eventual periodicity have arisen on occasion. In this section we survey the possible types of asymptotic behavior in a more organized fashion, beginning with some precise definitions. If S is a subset of S , the notation [[phi]]t(S) means the image of S under [[phi]]t, i.e., {z [[propersubset]] S such that z=[[phi]]t(x) for some x[[propersubset]]S}. Similarly, [[phi]]t- 1(S) denotes the preimage of S, {z [[propersubset]] S such that [[phi]]t(z) [[propersubset]] S}.
The first definitions generalize the concepts of fixed points and closed orbits.
Definition. A trapping set S is any set with the property that [[phi]]t(S) Ì S for all t >= 0. The phrase positively invariant set may be used as a synonym for trapping set.
Some writers simply use the term invariant instead in this context, although it is not assumed that [[phi]]t(S) = S. In these notes an invariant set is defined as one for which [[phi]]t- 1(S) = S. This implies that [[phi]]t(S) = S, but also prevents other points from being mapped to S.
Definition. A special case of a trapping set is an attracting set, i.e., a closed set A for which there is a neighborhood U with the property that any trajectory that enters U is pulled into A, meaning that for z[[propersubset]]U, dist([[phi]]t(z),A)->0. Its basin of attraction, or stable set, denoted Ws(A) or B(A), is the set of all points z such that dist([[phi]]t(z),A)->0. A basin of attraction is essentially the same as a stable manifold, except that it may not have the smooth structure of a manifold. For iterated mappings, basins of attraction can be disconnected and frequently have fractal structure. An attractor A is usually defined as an attracting set that contains a trajectory that is dense in A. An attractive node or spiral is a simple type of attractor. Like basins of attraction, however, attractors frequently have fractal structure. A strange attractor is an attractor with fractal structure.
Definition. A recurrent point is a point z [[propersubset]] S such that if U is any neighborhood of z, then for any T > 0, there is a time t > T for which [[phi]]t(z) [[propersubset]] U.
A recurrent point returns infinitely often to any neighborhood of itself. A theorem of G.D. BIRKHOFF guarantees the existence of at least one recurrent point in many situations:
Theorem IV.1. If S is a compact metric space and the mapping [[phi]] = [[phi]]1 is continuous on S, then there is a recurrent point.
There might easily be only one recurrent point, for example a fixed point that is attracting for all of S. On the other hand, if there is a group structure analogous to the rotations of the circle (cf. Exercise III.2), then every point will be recurrent according to a theorem of KRONECKER.
Definition. Let K be a compact group, a[[propersubset]]K, and define T: K->K by Tx = ax. Then the dynamical system (K, T) is a KRONECKER system:
Theorem IV.2. Every point of a KRONECKER system is recurrent.
For more details about KRONECKER systems see [FURSTENBERG, 1981], from which this theorem is quoted.
Definition. A nonwandering point for a dynamical system is a point z [[propersubset]] S such that if U is a neighborhood of z, then for any T > 0, there is a time t > T for which [[phi]]t(U) [[intersection]] U != Ø. A nonwandering set is a collection of nonwandering points.
In other words, while the trajectory of z itself is not necessarily guaranteed to return to z, it will happen infinitely often that some nearby point returns arbitrarily closely to z.
Periodic points are recurrent, and recurrent points are nonwandering, but there are nonperiodic recurrent points and nonrecurrent nonwandering points. One reason for splitting hairs like this is that the theorems that one can prove often refer to the nonwandering property. For instance, a theorem of ANDRONOV (Androcentsnov) allows the nonwandering points of differentiable dynamical systems in two dimensions to be completely categorized. Theorems are, of course, not disconnected from reality, and in realistic dynamical systems all kinds of odd nonwandering sets frequently occur.
At the Infinite Institute of Technology, the academic terms never end, so students are never graduated or given grades. Instead, the students are categorized according to attendance. A fixed student attends every class. A periodic student attends regularly, let us say every Tuesday. A recurrent student comes to class infinitely often. Then there are the nonwandering students, who may not come back to class themselves, but occasionally send their friends along to take notes.
A situation that frequently arises, as will be seen in the next section, is that there is a trajectory that winds densely through S or through some distinguished subset of S. Such a trajectory is said to be dense, or, sometimes, ergodic. This sense of the word "ergodic" is somewhat different from its more modern sense in Section VII.
Definition. If a set [[Lambda]] Ì S contains a dense trajectory, it is said to be dynamically transitive. If the set [[Lambda]] has the property that for any pair of (relatively) open sets U and V, there is a time t for which [[phi]]t(U) [[intersection]] V != Ø, it is topologically transitive.
The phrase "relatively open" refers to the topology of subsets of S. If, for example, S is a Cantor set in the interval [0,1], its open subsets are intersections of open intervals in [0,1] with S. It should be clear that dynamically transitive implies topologically transitive, which implies nonwandering. It is easy for a set to be nonwandering but not topologically transitive. Topological and dynamical transitivity are equivalent in many cases.
Definition. Let [[phi]]t be the flow of a differentiable dynamical system on a state space S, with at least two equilibria p1 and p2. If Ws(p1) intersects Wu(p2), then any trajectory in the intersection is called heteroclinic. (It should be clear that if there is a single point in the intersection, then the trajectory through that point lies in the intersection, and that the intersection comprises all such trajectories.) If Ws(p1) intersects Wu(p1), for the same point p1, then the trajectories composing Ws(p1) [[intersection]] Wu(p1) are called homoclinic.
In order to be able to define an unstable set, it is necessary for [[phi]]1 to be invertible. In this case the unstable set Wu(p) of a fixed point p is defined as the stable set under the action of the flow
[[phi]]-n = ([[phi]]n)-1 = ([[phi]]1-1)[[ring]]n.
When this is possible, points in the intersection of Ws(p) and Wu(p) are called homoclinic points.
Definition. Now suppose S = R2. A set of fixed points p1, ... , pn together with heteroclinic trajectories connecting each pi to pi+1 and connecting pn to p1 is called a homoclinic cycle.
Theorem IV.3 (ANDRONOV). Let S = R2 (or a planar region), and assume that [[phi]]t is the flow of a differentiable dynamical system. Then nonwandering points comprise: fixed points, closed orbits, fixed points together with homoclinic trajectories, and homoclinic cycles.
Warning. There are other possibilities in two-dimensional, nonplanar regions, such as the torus.
Exercise IV.1. Sketch the possibilities mentioned in ANDRONOV's theorem to get geometric intuition about the theorem.
Exercise IV.2. Find examples for differentiable dynamical systems of nonperiodic recurrent points and of nonrecurrent nonwandering points.
What sorts of behavior can a trajectory exhibit as time marches on to infinity? If S is an unbounded set, the trajectory can head off to infinity, but any trajectory that does not go to infinity will have at least one accumulation point:
Definition. The [[omega]]-limit set of a trajectory [[phi]]t(p) is the set of all points z [[propersubset]] S such that there exist times tj->[[infinity]] with
limj->[[infinity]] [[phi]]sDO2(tsDO2(j))(p) = z.
The [[alpha]]-limit set is defined similarly, with tj->-[[infinity]].
Exercise IV.3. Find the trapping, attracting, and nonwandering sets for the examples we have discussed above (DUFFING's equation, quadratic map, tent map, etc), and also for:
a) d[[theta]]/dt = u - sin([[theta]]), [[theta]] an angular variable.
b) d2[[theta]]/dt2 + sin([[theta]]) = .5, [[theta]] an angular variable.
Exercise IV.4. Show that the [[omega]]-limit set of any trajectory is nonwandering and trapping. Show that it need not be recurrent.
Another theorem characterizing certain two-dimensional [[omega]]-limit sets is due to POINCARé and BENDIXSON:
Theorem IV.4. Let R be a closed, bounded region of the phase plane S Ì R2 for a differentiable dynamical system. Suppose that R contains no equilibria. If z(t) is a trajectory lying in R for all t >= to, then either
a) z(t) is a closed orbit, or
b) z(t) spirals toward a closed orbit as t -> [[infinity]].
Such a closed orbit is called a limit cycle. Clearly, this theorem means that if some trajectory stays in a closed, bounded region forever, then there must be a closed orbit or a fixed point.
Exercise IV.5. Prove, as a corollary of the POINCARé-BENDIXSON Theorem, that if a differentiable dynamical system on R2 has a closed orbit, then the closed orbit encloses an equilibrium. (Hint: you may wish to run time backwards.)
Exercise IV.6. An equation arising in the study of an ancient and rather primitive device called the vacuum tube is called VAN DER POL's equation:
f(d2x(t),dt2) + u(x2-1) f(dx(t),dt) + x = 0. (4.1)
Convert this equation into a system, and show that there is an attracting closed orbit. (This is a difficult problem. The analysis due to LIéNARD is reproduced in [SIMMONS, 1972].)
As remarked above, dynamics on a torus can be more complicated than dynamics on a flat region of the same dimension. The two-dimensional torus T 2 can be represented as the square {0<=x<=1, 0<=y<=1}, except that the values 0 and 1 are considered equivalent. (See figure.) The one-dimensional version of a torus is a circle, so the phenomena of Exercise III.2 happen, with additional complications.
Figure IV.2. Making a rectangle into a torus.
Example. Free motion on the torus. Let the equations of motion be
f(d,dt) b(a(x,y)) = b(a(v1,v2)),
i.e., x(t) = x(0) + v1t, y(t) = y(0) + v2t, but both quantities calculated mod 1. If v1/v2 is rational, then every trajectory closes, and the motion is periodic. If, on the other hand, v1/v2 is irrational, then the trajectories never close. In this case, T 2 is the [[omega]]-limit set for every trajectory. Although S is two-dimensional, it is not a planar region, so ANDRONOV's theorem does not apply.
Figure IV.3. Free motion on the torus.
Exercise IV.7. Analyze the discrete-time version of free motion on the torus, i.e., take t = n to be an integer. Distinguish the cases where v1 and v2 are separately rational and irrational. Find the periodic points and their periods, [[omega]]-limit sets, etc.
Example. A remarkable example of a two-dimensional iterated mapping, studied by ADLER, WEISS, ANOSOV (Anocentssov) and ARNOL'D (Arnol`d), is often called the cat map. On the torus T 2 it acts like the linear transformation defined by the matrix
A = b(a(1 1,1 2)), i.e.,
Ab(a(x,y)) = b(a(x+y mod 1,x+2y mod 1)). (4.2)
Exercise IV.8. Find the periodic points and the homoclinic points for the cat map.
Other mappings of this type, with a matrix A with integer coefficients and determinant 1 are known as toral automorphisms.
Recurrence can be caused by a property enjoyed by many dynamical systems, of being area-preserving, volume-preserving, or, more generally, measure-preserving. For example, the toral automorphisms are area-preserving because the Jacobian of the transformation is the determinant of the matrix A. Recall that a measure u is a function assigning positive numbers or zero to reasonable subsets of a set S and having the intuitive properties of an area. The measure of the empty set is always 0, and the measure of any countable union of reasonable subsets that do not intersect is the sum of their measures, possibly [[infinity]]. The formal notion of "reasonable" that is used goes by the name measurable. A collection of measurable subsets, known as a [[sigma]]-ring, has to allow complementation and countable unions and intersections. For a particular S there are many different [[sigma]]-rings that one might use, but for our purposes we will always work with the BOREL measurable sets, which are the sets that one naturally considers if one starts with all the open and closed sets. Likewise, a measure can be very abstract, but for our purposes it can be thought of as an integral:
u(S) = i(S,,w(z)dz) =: i(S,,du), (4.3)
where w(z) is a nonnegative weight function, and dz is the usual volume element (often dnx or dnxdnp). If w(z) = 1, then u(S) is the usual area or volume of S. The theory of measure is treated for example, in [KOLMOGOROV-FOMIN, 1970].
In measure theory there is a way of not quite admitting that one is telling small lies, with the words "almost everywhere" or "almost every," denoted by almost everyone a.e. A statement is true a.e. if it is possibly false on a null set, a set S of measure zero: u(S)=0.
One extra assumption about measures is helpful, namely that the measure of any open set is positive. I shall refer to such a measure as a volume (more formally we would say that u is mutually absolutely continuous with respect to Lebesgue measure). This implies that the weight in (4.3) is an honestly defined function and is positive a.e. It excludes point masses (d functions) in Euclidean regions, for example.
Definition. Assume that [[phi]]t is a discrete or continuous invertible flow on a measure space S, with measure u. The dynamical system is measure-preserving if for all measurable sets U,
u(U) = u([[phi]]t(U)). (4.4)
When [[phi]]t is not invertible, the definition usually made works with inverse images [[phi]]t-1(U), and states that if the total set of points mapped to another set is considered, then the measures are the same:
u([[phi]]t-1(U)) = u(U). (4.5)
The inverse image [[phi]]t-1(U) is defined as {z [[propersubset]] S: z = [[phi]]t-1(w) for some w [[propersubset]] S}. The same definition with t = n applies to discrete dynamical systems. If the preserved measure is a volume, the dynamical system is said to be volume-preserving.
For a complete differentiable dynamical system we know that for each t, the flow [[phi]]t is a diffeomorphism S -> S. If the Jacobian of [[phi]]t is 1, then the flow is volume-preserving, since
V([[phi]]t(U)) = i([[phi]]t(U),, dz). = i(U,, J([[phi]]t(z'))dz'), (4.6)
where we think of z -> z' such that [[phi]]t(z') = z as a change of variables, and write J for the Jacobian determinant for the transformation,
J([[phi]]t(z)) = bBC|(det b(f([[partialdiff]][[[phi]]t(z)]i,dzj))).
Many important dynamical systems are volume-preserving, notably HAMILTONIAN dynamical systems, such as those arising from a conservative force law: For x [[propersubset]] Rn, let
dx/dt = p/m, for some constant mass m
and
dp/dt = - grad V(x), for some smooth potential energy V.
This is a dynamical system on S = R2n, and LIOUVILLE's Theorem states that the flow of any Hamiltonian dynamical system is volume-preserving (for the measure dnx dnp). It is important to remember that this is the phase-space volume and not the configuration-space volume dnx. A more general form of LIOUVILLE's theorem states:
Theorem IV.5. Consider a differentiable dynamical system on an n-dimensional state space S, governed by equations of motion
dz/dt = v(z),
where
tr[Dv] = 0 for all x. (4.7)
Then [[phi]]t preserves the volume measure dnz.
Recall that for any matrix M, tr[M] := [[Sigma]] Mjj, so (4.7) means that div(v) = 0 (when the divergence is calculated with respect to all the phase space variables). This theorem follows from differentiating (4.6) and using the fact, known since the time of LIE, that
f(d,dt)det(exp(At))|t=0 = tr[A],
for any nxn matrix A. Hamiltonian vector fields all have zero trace (remember to replace n by 2n and z by <x,p>, where p represents the momentum coordinates).
POINCARé was the first to realize the connection between volume-preservation and recurrence. The weak form of his recurrence theorem reads:
Theorem IV.6 (POINCARé): Let S be a measure space with u(S) < [[infinity]], and suppose that [[phi]] = [[phi]]1 is measure-preserving. Then for any set U with u(U) > 0, there exists k arbitrarily large, such that U [[intersection]] [[phi]]k(U) != Ø.
If u is a volume, this says that every point of S is nonwandering.
Proof. Consider the collection of sets [[phi]]-j(U) := [[phi]]j-1(U), j = 1, 2, ....Since the total measure of S is finite, and each [[phi]]-j(U) has positive measure equal to u(U), some of these sets must intersect,
[[phi]]-m(U) [[intersection]] [[phi]]-n(U) != Ø (4.8)
for some (actually infinitely many) m,n. Suppose that n>m, and observe that n-m can be arbitrarily large, i.e, given any prespecified N, we can find n,m such that this difference is larger than N. If these sets intersect then so do their images under F or F[[ring]]n. If we apply F[[ring]]n to (4.8), we see that
[[phi]]n-m(U) [[intersection]] U != Ø,
and we take k = n-m. x( )
Whether time is continuous or discrete, for this theorem we need only consider integer times. The strong form of POINCARé's theorem specifically guarantees recurrence:
Theorem IV.7 (POINCARé): If a dynamical system is volume-preserving on a set S, with u(S) < [[infinity]], then almost every point of S is recurrent.
Historical remark. According to DIEUDONNé [1975, p. 58], this theorem was probably the first significant instance where the concept of a null set was used in analysis.
Proof.
The proof will actually show that given any open set B, almost all of its points return to B infinitely often. But if the theorem were false, then by a covering argument there would have to be a set of positive measure inside some ball, for which the trajectories return only finitely often.
Let B be an open subset of S, and let Bk := [[union]]j>=k[[phi]][[ring]]-j(B) for k >= 0, where [[phi]][[ring]]-j = ([[phi]]1[[ring]]j)-1. Let
Kk := B[[intersection]]Bk. This is the set of points of B that return to B at some time >= k (and possibly earlier as well). The recurrent part of B, denoted R := [[intersection]]k>=0 Kk consists of the points that will return to B arbitrarily far in the future. Observe that [[phi]]-1(Bk-1) = Bk, so u(Bk) = u(Bl) for all k and l. Since BkÌBk-1, it follows that u(Bk-1\Bk) = 0. Now estimate the measure of R:
u(R) = u(K0) - isu(k=1,[[infinity]],u(Kk-1\Kk)) = u(B) - isu(k=1,[[infinity]],u(B[[intersection]](Bk-1\Bk)))
= u(B) - 0 = u(B)
In other words, almost all points of B belong to R x( )
Exercise IV.9. What rôle is played in this proof by the assumption of finiteness? By the assumption that du is a volume?
Examples. The circle rotation, the cat map, and the harmonic oscillator when restricted to a region x2+p2 < 2E all preserve the usual measures, respectively d[[theta]], dxdy, and dxdp, on their state spaces. POINCARé says that almost all points are recurrent. In fact, in these cases all points are recurrent.
Exercise IV.10. Does the dyadic map preserve the usual measure dx?
Example. The baker's transformation roughly models the procedure whereby a mixes dough with a rolling pin. Take the unit square, and stretch it by a factor of 2 in the x-direction but, to conserve area, squeeze it by a factor 1/2 in the y-direction. Then snip off the right half and place it above the left half to reconstruct a unit square. More exactly, let the state space be S = {0 <= x < 1, 0 <= y < 1}, and set
Bb(a(x,y)) = blc{(a(b(a(2x,y/2)) 0 <= x < 1/2, ,b(a(2x-1,(y+1)/2)) 1/2 <= x < 1)). (4.9)
While this looks discontinuous, it can be regarded an area-preserving map on a structure similar to the torus, like the cat map. The equivalence between the square and the surface differs from that for the cat map by having extra half twists: Set (x,y) [[congruent]] (x+1,y+1/2), and (x,y) [[congruent]] (x+1/2,y+1). See Figure IV.5.
Figure IV.5. Two iterations of the baker's map.
Exercise IV.11. Consider the transformation on [0,1) defined by T(x) = 1/x mod 1. (On the computer, this would be frac(1/x), where frac(z) is the fractional part of z, i.e., z minus the greatest integer <= z). Show that the measure
du = f(1,ln2) f(dx,1+x)
is an invariant under the action of T. (The constant just serves to make u((0,1)) = 1.) Do this by considering first intervals of the form I = (0,[[alpha]]), noting that T-1(I) consists of the infinite number of intervals of the form Ik := ((k+[[alpha]])-1,k-1), and calculating that
= ln(1+[[alpha]]) = ln2 u(I).
A final theorem, known as SCHWARZSCHILD's capture theorem, shows that volume-preservation also constrains the motion of a system so as to relate past and future limits:
Theorem IV.8: If a dynamical system is volume-preserving on a set S, with u(S) < [[infinity]], and A is a measurable set with u(A) < [[infinity]], then, except possibly for a subset of A of measure 0, if the trajectory through p[[propersubset]]A remains in A for all future times, it must have been in A for all times in the past.
Exercise IV.12. Proof this theorem by defining the set of points that remain in A for all future times, the set of points that were in A for all past times, and show that they have the same measure.
Notice that the system might be unstable, and the set remaining in A might have measure 0 to begin with. A reference for this is [THIRRING, 1978].
Another class of two-dimensional systems on which we can study recurrence and other dynamical properties is that of twist maps. These are constructed by lifting the circle rotations and similar circle maps to a two-dimensional setting by adding a coördinate. For example, we could add a coördinate J to the circle making it into a cylinder, and study the very simple mapping
J -> J
[[theta]] -> [[theta]] + [[alpha]].
Here nothing new will occur, but we can find interesting phenomena if we couple the two coördinates to make the CHIRIKOV-TAYLOR standard map.
J -> J + K sin [[theta]]
[[theta]] -> [[theta]] + J + K sin [[theta]]
The linearization of this mapping is the mapping corresponding to the matrix
which has determinant 1. The standard map is thus area-preserving on the cylinder.
Most of the well-understood conservative systems are integrable, which roughly means that there is a maximal set of constants of the motion. When this happens, there is a choice of variables in terms of which the motion is analogous to the free motion on a torus, and the state space can be regarded as the union of these invariant tori. A famous theorem of KOLMOGOROV, ARNOL'D, and MOSER states that for sufficiently small conservative perturbations, the invariant tori are distorted but remain in existence. The standard map is interesting for seeing how the tori break up as one moves away from integrability. This does not happen uniformly; as K increases, certain regions of the state space remain stable while the rest of the state space begins to exhibit complicated motion. This is fairly easy to see computationally.
There are several other popular two-dimensional mappings which are similarly constructed by "lifting" popular one-dimensional systems into two dimensions:
The Hénon map is a family of mappings on the x-y plane related to the quadratic map Fu:
The Lozi map : Hénon map as the tent map: quadratic map, i.e.
The last two examples are area-preserving only when b=-1, but are interesting for many other values of the parameters as well.