Asimov (Notes on the Topology of Vector Fields)

Notes on the Topology of Vector Fields and Flows

Daniel Asimov

Visualization ′93 San José, California October 1993

Vector fields are ubiquitous in science and Figure 1: Vector Field (unnormalized) mathematics, and perhaps nowhere as much as in computational fluid dynamics (CFD). In order to make sense of the vast amount of information that a vector field can carry, we present in these notes some important topological features and concepts that can be used to get a glimpse of “what a vector field is doing.”

1 Vector Fields

A vector ﬁeld arises in a situation where, for some reason, there is a direction and magnitude assigned to each point of space.

The classic example of a vector field in the real world is the velocity of a steady wind. We will draw a vector field as having its point of origin at the point x to which it is assigned. In this way a vector field resem- bles a hairdo, with all the hair being per- fectly straight A vector field is, mathematically, the choice of a vector for each point of a region of space. In general, let U denote an open set of Euclidean space Rn. (We will usu-

1 of 21 ally be interested in the cases n = 2 and 2 Differential Equations and Solu- n = 3.) Then a vector field on U is given by tion Curves a function f : U → Rn. We will assume that a vector field f is at least once continu- A differential equation (or more pre- ously differentiable, denoted C1. cisely, an autonomous, ordinary differential equation) defined on an open set U of More generally, one can also consider a Euclidean space Rn is an equation of the phenomenon such as real wind, which can form be represented by a vector field which dx = f() x changes from moment to moment. The td mathematical name for this is time- dependent vector field. Mathemati- where x : I → U denotes an unknown cally, a time-dependent vector field on an curve parametrized by some (as yet open set U in Rn is a function of the form unknown) interval I containing 0, and f : U × J → Rn, where J is some time inter- f : U → Rn is C1. val a ≤ t ≤ b. Given any point x0 ∈U, we may assume an Think of a time-dependent vector field as “initial condition” of the form x(0) = x0. a movie, each frame of which is an ordi- With this condition, the Existence Theo- nary vector field. rem for solutions of autonomous O.D.E.’s states the following: There exists a num- In these notes we will give more empha- ber c > 0 and a solution x : (−c, c) → U. In sis to steady (non-time-dependent) vector other words, x(0) = x0 and, for each t in fields. the interval (−c, c), we have x′(t) = f(x(t)).

Figure 2: Normalized Vector Field This solution is unique in that any curve y : (−c, c) → U which also satisﬁes the initial condition y(0) = x0 and the equation y′(t) = f(y(t)), must in fact be the curve x.

Think of f(x) as representing the velocity of a steady wind at the point x. Then the solution curve x(t) represents the hypo- thetical trajectory of a massless particle released at time = 0 at the point x0.

One may also consider non-autonomous differential equations. These are equations of the form dx = f() x, t td

2 of 21 Notes on the Topology of Vector Fields and Flows where x : I → U once again denotes an 2π. This cylinder, then is the phase unknown curve parametrized by some space for this problem. The differential interval I. However, in this case, equation governing the physics of the f : U × J → Rn is a C1 function taking not trashcan lid provides a vector ﬁeld on this one but two arguments. (Here J repre- phase space. sents some interval of time.) Trajectories of this vector ﬁeld decribe the To say that x is a solution of a non-autono- change in state that the lid would experi- mous O.D.E. like this means that x must ence form any initial position. Physically, = satisfy the initial condition, x(t0) x0. this is the classic “pendulum with fric- And there must exist some number c > 0 tion.” such that for each t lying in the interval (t0 − c, t0 + c), we have x′(t) = f(x(t), t). Figure 3: Pendulum with Friction (Note that unlike the interval (−c, c) in the autonomous case, which must contain 0, this interval may omit 0 but must contain the value t0.)

The classic example of such an equation is again wind: real wind, which varies with time. The function f(x, t) may be thought of as representing the velocity of the wind at location x and time t. And the solution curve x represents the trajectory of a particle that is released in the wind at time = t0 at the point x0.

3 Phase Space In this figure the horizontal coordinate In addition to concrete geometrical appli- represents the angular position (which cations, one of the most important uses of has been unrolled to lay the cylinder out differential equations is to understand the flat; hence the “phase portrait” above evolution of the state of an abstract physi- repeats with horizontal period 2π). The cal system as time progresses. For exam- vertical coordinate represents the angular ple, consider the hinged roof-shaped lid velocity, of the trashcan lid. For any initial atop some public trashcans. Its physical conditioni.e., a point of this phase state can be described by two coordinates: space the unique trajectory though that the angular position of the lid, and the point shows the time evolution of that ini- angular velocity with which it is rotating tial condition. It is apparent from the fig- about its hinge. The abstract representa- ure that, for almost all initial conditions, tion of all states is actually a cylinder, the trajectory eventually spirals into a because the angular position repeats after stationary point of the vector field. This

Notes on the Topology of Vector Fields and Flows 3 of 21 corresponds to going through some finite which takes the value p at time = 0 will be α number of full revolutions in angular posi- denoted by p. tion, and then swinging back and forth, α approaching the stationary equilibrium These trajectory functions p must, where state in the limit. Anyone who has scien- defined, satisfy the fundamental consis- tifically experimented with this kind of tency condition: Let p ∈U and let s and trashcan knows that this is exaactly what t denote any two lengths of time. Then the happens. result of following the trajectory of p for time s, and then following the trajectory of 4 Vector Fields ↔ O.D.E.’s that result for time t, is exactly the same as the result of following the trajectory of In a mathematical sense, vector fields and p for time s + t. differential equations may be considered to be the same thing. More precisely, time- In mathematical language: Let q denote α independent vector fields are the same the point p(s), the result of following the thing as autonomous ordinary differen- trajectory of p for time s. Then we must α = α tial equations. have q(t) p(s + t).

This is simply because each one is deter- It is useful to unify the trajectory func- α mined by a function f : U → Rn. Given a tions p for all p into one single function of ∈ ∈ vector field defined by a function two variables. For any p U and t Ip φ( α φ f : U → Rn, one may write the differential define p, t) to be p(t). Now (p, t) is just equation given by dx/dt = f(x). And con- the result of following the trajectory of p versely, given a differential equation for time t. dx/dt = f(x), one may extract the vector field f(x). Thus both vector fields and Then the consistency condition becomes just autonomous ordinary differential equa- φ( φ (), , )= φ (), + tions carry the same information. p s t p s t wherever it is defined. (Note that this And similarly, time-dependent vector holds for negative as well as positive val- fields may be thought of as the same thing ues of s and t.) We also have as non-autonomous differential equations: φ ()p, 0 = p for all p in U. each one is determined by a function By the way, the existence and uniqueness f :U × J → Rn. theorem for O.D.E.’s also tells us that the φ 5 Flows flow function of the two variables p and t must be continuously differentiable, or C1.

Vector fields and differential equations Now we can see that for each fixed value give rise to families of transformations of of t, there is a mapping of U to itself which space called flows. takes p to φ(p, t). This mapping is often φ Let us introduce new notation for the tra- denoted by t. Thus, by definition, we jectory of a point p: the solution curve

4 of 21 Notes on the Topology of Vector Fields and Flows φ = φ have t(p) (p, t) wherever this is Let a be any number. Then the curve defined. given by t → (t, (t − a)3) is a trajectory of this vector field. But since V(x, 0) = (1, 0) Finally, we can define what is meant by for all x, it is clear that the x-axis itself is the flow associated with a differential also a trajectory, via the curve given by equation: This is the family of all the t → (t, 0). These two trajectories cross φ → transformations t : U U. The consis- each other. tency condition for this flow can now be φ φ ≡ φ stated as s o t s+t for all s and t for At the point where they cross, two distinct which this is defined. Due to the equiva- trajectories start from the same initial lence of differential equations and vector conditions.This violates the uniqueness of fields, the flow of the differential equation solutions which must hold for C1 vector dp/dt= f(p) is also called the flow of the fields. vector field f(p). Figure 4: Vector Field V(x, y) = ( 1, 3y2/3) No convergent trajectories of a C1 vector field

The uniqueness of solutions to O.D.E.’s tells us important information about the conﬁguration of trajectories of a C1 vector ﬁeld: Each point p of the domain U lies on one and only one trajectory.

In particular, this tells us that trajectories can never truly converge, cross each other, or branch. (Two trajectories may appear to converge, for example, if as time → ∞ both trajectories approach the same point. But this can only occur when that point is in fact its own trajectory: a stationary point.)

It is enlightening to see what can happen when we drop the condition that the vector field be continuously differentiable. An interesting example (Fig. 4) can be con- structed if a vector field is defined on all of R2 via V(x, y) = ( 1, 3y2/3). (The derivative − ∂(3y2/3)/∂y is 2y 1/3, which does not exist when y = 0, so this vector field is not continuously differentiable. )

Notes on the Topology of Vector Fields and Flows 5 of 21 tory is called a closed orbit. (More about The ﬂow of a linear vector ﬁeld these later.)

In the case where a vector field on Rn is 3) The trajectory stays put. The entire tra- defined by a matrix, then there is a simple jectory is a single point: x(t) = p for all t. explicit formula for the flow of V. This happens at locations where the vector field is 0, and nowhere else. This kind Suppose V(p)= Lp at each point p, where L of trajectory is called a stationary point. is some n × n matrix. Then the flow of V is (More about these later.) given by a very simple and beautiful formula: φ(p, t) ≡ etLp. Here the exponential Terminology: Other terms for a station- eM of an n × n matrix is the n × n matrix ary point include equilibrium, singular- defined as ity, fixed point, and zero. (It is also ∞ sometimes called a “critical point,” but Mk eM = this term is perhaps best reserved for its ∑ k! original definition: where the derivative of k = 0 some mapping to R1 vanishes.) A vector field without stationary points is called 6 Classification of Trajectories non-singular.

We are now in a position to make an ele- mentary classiﬁcation of the kinds of tra- 7 Some Essential Concepts of jectories that can occur. We may think of a Topology trajectory as a mapping x: R → Rn from the real numbers to Euclidean space1. Some fundamental concepts of topology in Then one of three conditions must n hold: Euclidean space R that will be used or assumed here include the following:

1) The mapping is one-to one. In this case An ε-neighborhood of a point p for a the trajectory is a curve that never number, denoted Nε(p), is the set of all returns to where it has been. This is called points at a distance < ε from p. a regular trajectory. (More about these later.) A point p is a limit point (or accumula- tion point) of a set X if for every ε > 0, 2) The mapping has a least period > 0, say Nε(p) − {p} contains at least one point of t , after which it repeats exactly where it 0 = X. has been before: x(t + t0) x(t) for all t. The trajectory must form a simple closed An open set is any set that is an arbi- curve in Euclidean space. Such a trajec- trary union of sets of the form Nε(p). Note that the empty set is open, as is the entire 1. In these notes, we will often not distinguish trajectories which n arise from different initial conditions but constitute the same set space R . of points.

6 of 21 Notes on the Topology of Vector Fields and Flows A closed set is any set X whose comple- Two vector ﬁelds or ﬂows are called topo- ment Rn − X is open. Equivalently, a logically equivalent (or homeomorphic) closed set is any set which contains all its if there is a homeomorphism between limit points. their domains which carries trajectories to trajectories, and preserves the direction of A neighborhood of a point p is any open increasing time. set containing p.

A set in Rn is compact if it is closed and bounded. Fundamental theorem of O.D.E. s

A set X is disconnected if there exist two In a sense, closed orbits and stationary disjoint open sets, each of whose intersec- points are much rarer than regular trajec- tion with X is non-empty. A set is con- tories, at least for most vector ﬁelds. If p is nected if it is not disconnected. a point on a regular trajectory, we can Intuitively, X is connected if it is all in one state exactly what nearby trajectories “piece.” look like in some neighborhood of p.

A sequence p1, p2,...,pi,... approaches Theorem: If p is a point on a regular the limit p if for every ε > 0, there exists trajectory, then there is a neighborhood U an integer M = M(ε) > 0, such that i ≥ M of p and a homeomorphism h : U → Rn implies that pi lies in Nε(p). This situation which carries each piece of a trajectory = is denoted by lim pi p. (If you are not lying in U onto a straight line in Rn paral- familiar with this definition, it is worth lel to the x-axis. taking the time to understand it.) Note that a sequence can approach at most one In plain English, this theorem says that { } limit, although a set of points pi may near a point of a regular trajectory, have many limit points. trajectories fill up space in the same way that parallel lines do, topologically A mapping f : U → V between two open speaking. sets U and V is continuous if whenever there are points {p } and p in U for which 8 Alpha- and Omega-limit Sets = i = lim pi p, then lim f(pi) f(p). (In brief: a continuous function is one which pre- We wish to define how a regular trajectory serves limits.) x(t) behaves as t approaches infinity. So we define the omega-limit set of the A homeomorphism or topological trajectory x to be all those points of the equivalence between two sets X and Y is → domain which are limits of a sequence of a mapping h : X Y that is continuous, the form x(t) for values of t approaching one-to-one, onto, and has continuous infinity. We denote this set by ω(x). inverse h-1. Topology is concerned with properties of shapes that remain Example: Consider a vector field in which unchanged under homeomorphism. one trajectory x approaches a closed orbit

Notes on the Topology of Vector Fields and Flows 7 of 21 by spiraling around it closer and closer. Deﬁnition: A limit cycle of a vector ﬁeld Then the entire closed orbit is the omega- V is a closed orbit C of V that is contained limit set of x. in α(x) or ω(x) for of some trajectory x ≠ C.

Example: Consider the vector field V(x,y) Here is an important result about limit = (−x, y) in the plane. The trajectory x(t) cycles in the plane: satisfying the initial condition x(0) = = (x0, y0) is given by the curve x(t) Theorem (Poincaré-Bendixson): Suppose −t t 2 (x0e , y0e ). Thus each trajectory lies on a V is a vector field in R , and let x be a tra- hyperbola of the form xy = k for some jectory of V. Suppose that ω(x) is non- value of k (and this includes the degener- empty, compact, and contains no station- ate hyperbola xy = 0 as well). ary point. Then ω(x) must be a closed orbit. Clearly, the only trajectories having a non-empty omega-limit set are those lying What makes the limit cycle particularly on the x-axis. important is that it will persist even after a small perturbation of the vector field. Similarly, the alpha limit set, of a trajectory x(t), denoted by α(x), is defined as Theorem: Let V be a vector field having a the omega-limit set of the reversed trajec- limit cycle C. Let V1 be a sufficiently small 0 tory x(−t). In the above example, only tra- (in the C sense) perturbation of V. Then jectories lying on the y-axis have a non- V1 will also have a limit cycle. empty alpha-limit set. 10 Classification of Isolated Station- Suppose there is a number t0 for which ary points the trajectory x(t) remains inside a com- ≥ pact subset of the domain for all t t0. A stationary point is called isolated if is Then the following must be true concern- not a limit point of other stationary ing the omega-limit set of x: points. If the point p is an isolated station- 1) ω(x) is non-empty ary point of a vector field 2) ω(x) is closed V(x , x ) = (V (x ,x ),V (x ,x )), or 1 2 = 1 1 2 2 1 2 3) ω(x) is invariant by the flow (i.e., it is V(x1, x2, x3) (V1(x1,x2,x3),V2(x1,x2,x3),V3(x1,x2,x3)), a union of trajectories), and then we can further investigate the struc- 4) ω(x) is connected. ture of the trajectories nearby by examin- ing the so-called Jacobian matrix of = ∂ ∂ partial derivatives, JV (Vi/ xj). This matrix will of course be 2×2 in the 2- 9 Limit Cycles dimensional case and 3×3 in the 3-dimensional case. The limit cycle is an important topological feature of some vector fields. When we evaluate the Jacobian matrix at the stationary point p, we obtain a matrix whose values are numbers. As with any

8 of 21 Notes on the Topology of Vector Fields and Flows such matrix, we may calculate its eigen- Definition: A source of a vector field V is values and eigenvectors. an isolated stationary point p which is a sink of the reversed field, −V. Definition: A stationary point is called hyperbolic if the real parts of the eigen- 11 Geometric Classification of values of its Jacobian are all non-zero . Hyperbolic Stationary points

A hyperbolic stationary point must neces- Let V denote a vector field with a hyper- sarily be isolated. bolic stationary point p, at which the Jaco- bian matrix is JV = (∂V /∂x ). Theorem: If p is a hyperbolic station- i j ary point, then the trajectories near p are The 2-dimensional case: determined up to topological equivalence by the number of positive and the number The eigenvalues of JV are the roots of the of negative real parts that the various quadratic polynomial P(λ) defined by P(λ) eigenvalues have. = det(J − λI). Since this is a quadratic polynomial with real coefficients, the roots Just how often does this “non-zero real are either both real or else they are com- part” condition hold? Fortunately for us, it plex conjugates of each other (say K + Li is the rule and not the exception. In a and K − Li, where K and L are real). We sense that can be made precise, this condi- consider the cases: tion holds with probability = 1. 1. Both roots real: This theorem is a remarkable application a. Both positive: of topology to the local classification of source stationary points. b. Both negative: sink According to this theorem, we can catego- c. Opposite signs: rize “all but a set of measure 0” of the iso- saddle lated stationary points in 2 dimensions in 2. Complex conjugate roots: terms of just 3 kinds: both eigenvalues a. K positive: with positive real part; one positive and spiral source one negative, and both negative. Simi- b. K negative: larly, in 3 dimensions there are 4 distinct spiral sink types (except for that set of measure 0). Note: The sense of the spiral in case 2. can be determined from the sign of curl(V) at Definition: A sink is an isolated station- the point p. If curl(V) (thought of as the ary point p each of whose nearby trajecto- scalar quantity ∂V /∂x − ∂V /∂x ) is pos- ries x satisfies ω(x) = {p}. (Alternatively, 1 2 2 1 itive, then the swirl will be counterclock- for each nearby trajectory x, the limit of wise, and vice versa. x(t) as t → ∞ is p.) The 3-dimensional case:

Notes on the Topology of Vector Fields and Flows 9 of 21 The eigenvalues of JV are the roots of the values of V near p.) Conclusion: the vector λ λ = polynomial P( ) defined by P( ) field V1 must also have its own stationary − λ det(J I). Since this is a cubic polyno- point at some point p1 near p, and further- mial with real coefficients, the theory of more the trajectories of V1 near p1 equations tells us that there are either 3 must be topologically equivalent to real roots, or 1 real root and a pair of com- the trajectories of V near p. In brief: a plex conjugate roots. Case by case: local perturbation of a hyperbolic stationary point does not change the topology. 1. All 3 roots real: a. All positive: This kind of stability is called local source structural stability. b. 2 positive, 1 negative: saddle (2 dims. out, 1 in) This is a very important property of c. 1 positive, 2 negative: hyperbolic stationary points. It means saddle (1 dim. out, 2 in) that they will show up independent of d. All negative: small errors of measurement as invari- sink ably occur in the real world. 2. 1 real, 1 complex conjugate pair (again, say K + Li and K − Li, where K Conversely, any isolated stationary and L are real): point possessing local structural sta- I) Real root positive: bility must be hyperbolic. a. K positive: spiral source b. K negative: spiral saddle (1 out, 2 dims. in ) 13 Area- or Volume-Preserving Vec- II) Real root negative: tor Fields a. K positive: In many applicationsfor example, spiral saddle (2 dims. out, 1 in)  b. K negative: hydrodynamics the vector fields encoun- spiral sink tered will often be volume-preserving. (In this context we use the term “volume” to mean both ordinary volume in 3 dimen- 12 What is So Important About sions and area when we are referring to 2 Hyperbolic Stationary Points? dimensions.) By definition, a vector field is volume- If V is a vector field with a hyperbolic sta- φ preserving when its flow t carries any tionary point at the point p, then this is φ ( ) open set S in its domain to a set t S of stable in the following sense: Suppose V1 , is another vector field which is sufficiently the same volume for all times t. 1 “C close” to V in a neighborhood of p. The condition for a vector field V to pre- (This means that V1 and its first serve volume is that div(V) = 0. (In 2D derivatives are close to the corresponding ∂ ∂ ∂ ∂ ≡ this means V1/ x1+ V2/ x2 0; in 3D it

10 of 21 Notes on the Topology of Vector Fields and Flows ∂ ∂ ∂ ∂ ∂ ∂ ≡ means V1/ x1+ V2/ x2 + V3/ x3 0.). excluded because they cannot preserve For this reason, the term divergence- area). free is used as a synonym for volume-preserving. A center is important because it possesses constrained local structural stabil- A fundamental topological consequence of ity: Any area-preserving vector field that being volume-preserving is that no open is sufficiently close (in the C1 sense) to one set of finite volume may ever flow to a having a center, must also have a center. proper subset of itself by the flow, whether in positive or negative time. Thus if the only vector fields that can arise in some 2-dimensional situation This strong condition immediately rules must be area-preserving, then the only out both sources and sinks from occurring isolated stationary points that can happen in volume-preserving vector fields. But it are 1) saddles and 2) centers. is easy to see that any type of saddle is possible in a volume-preserving field. 14 The Poincaré Map Centers In 2 dimensions, there is another impor- By the use of the Poincaré map, we may tant kind of isolated stationary point, a reduce the topology of surrounding trajec- center. A center is a stationary point in 2 tories to a question of understanding a dimensions for which all nearby trajec- map of a smaller disk to a larger disk. tories are closed orbits. The simplest example of a center is the point (0, 0) in We will consider a vector field in 3 dimen- the vector field given by V(x, y) = (−y, x) sions. The 2-dimensional case, being sim- (uniform counterclockwise rotation). pler, will follow from this.

The eigenvalues of the Jacobian matrix at Choose any point p belonging to C, and a center must be pure imaginary. consider any open disk D whose intersection with the circle C is the point p, and The converse is not true in general: If an which is not tangent to the circle C at p. arbitrary 2-dimensional vector field has (In the 2-dimensional case we would use an isolated stationary point with pure an arc instead of a disk.) We may assume imaginary eigenvalues of the Jacobian, it that D is chosen so that the vector field is is not necessarily a center. It could also be nowhere tangent to D. Such a disk (or arc, a source or a sink (due to non-linear terms if in 2 dimensions) is called a local sec- of its Taylor series). tion of the flow.

However, if an area-preserving vector By the continuity of solutions to C1 differ- ﬁeld has an isolated stationary point with ential equations, there must exist a pure imaginary eigenvalues of the Jaco- smaller disk D0, contained in D and con- bian, then it must be a center (since the taining p, with the following property: only alternativessource or sinkare

Notes on the Topology of Vector Fields and Flows 11 of 21 For any point q in D0, there is a equivalent (in fact, differentiably equiva- smallest value of t = t(q) > 0 such lent). that following the trajectory of p for time t(q) results in a point q′ that is The Poincaré map is important because it once again in the larger disk D (but contains all the information about the not necessarily in D0). topology of trajectories close to the closed orbit C. (In principle, given the Poincaré The Poincaré map (or first-return map of a closed orbit, one could recon- map) of the closed orbit C is the mapping struct the topological structure of trajecto- → f : D0 D that takes any point q of D0 to riesnear C.) But it is simpler to think the point q′ described above. If V is a C1 about a mapping between two disks than vector field (as we have usually been all that spaghetti. assuming), then the Poincaré map f is also continuously differentiable. Since f has a fixed point at p, we can again look at the Jacobian matrix, not of a vec- This Poincaré map f must carry the origi- tor field, but of the mapping f. The Jaco- nal point p to itself. This is because f(p) bian of a mapping f is again defined as the must lie on D, and it also lies on the closed square matrix of partial derivatives, in = ∂ ∂ orbit C through p; the intersection of C this case Jf (fi/ xj). (In order for this to and D is {p} by assumption. make sense, we need to pretend that we have given 2-dimensional coordinates to Figure 5: The Poincaré Map the disk D.)

Important fact about eigenvalues: Regardless of the choice of coordinates used for the disk D, (which will affect the Jacobian matrix Jf), the eigenvalues of the matrix Jf will always be the same for a given closed orbit.

And so once again we can play the eigenvalue game. Analogous to the terminology for stationary points, we call a closed orbit hyperbolic if the eigenvalues of Jf lie off the unit circle in the complex plane.

As in the case of hyperbolic stationary Important fact about Poincaré maps: points, hyperbolic closed orbits are Regardless of the choices involved in the locally structurally stable: a small per- definition (the point p, the disks D and turbation of the vector field near a hyper- D0), any two resulting Poincaré maps of bolic closed orbit will result in a the same closed orbit will be topologically topologically equivalent vector field.

12 of 21 Notes on the Topology of Vector Fields and Flows And conversely: if any closed orbit of an ii) both roots negative: arbitrary vector field is locally structur- twisted saddle closed orbit1 ally stable, then it must be hyperbolic. 2. Complex conjugate roots: a. Both outside the unit circle: (Note how this differs from hyperbolicity spiral source closed orbit for stationary points, where the eigenval- b. Both inside the unit circle: ues of the appropriate Jacobian matrix JV spiral sink closed orbit need to lie off the imaginary axis. The And for completeness, we also mention unit circle is the image of the imaginary the rather simple classification of hyper- axis under the complex exponential map- bolic closed orbits in 2 dimensions, where ping, and this reflects the exponential the Jacobian matrix is only 1 × 1, and that relationship between vector fields and single element is the lone, necessarily mappings.) real, eigenvalue of Jf:

Once again, the eigenvalues are the roots Hyperbolic Closed orbits in 2D of a polynomial: in this case the polynomial is det(Jf - λI). Since Jf is a 2 × 2 1. Root > 1: matrix, this polynomial is just a quadratic source closed orbit polynomial, with real coefﬁcients. 2. Root < 1 : sink closed orbit Hence the roots are either both real, or else complex conjugates. In addition, the Intrinsic vs. Extrinsic Topology: product of the eigenvalues must be positive, since Euclidean space is orientable. Topology is often called upon to describe a (This is the 3-dimensional version of the situation where one space X is sitting fact that a Möbius band cannot be a sub- inside (“embedded in”) another space Y. In set of the plane.) So if both eigenvalues this case, a distinction is made between- are real, they must have the same sign. properties that depend only on X, and We consider the possibilities : those that come about from the way in which X is positioned inside Y.

Hyperbolic Closed orbits in 3D Those properties depending only on X are (Root1 and Root2 refer to the two eigenval- called intrinsic, and those depending on ues; the numbering is of no signiﬁcance)

1. Both roots real:   >   > 1. An additional complexity is possible in the case of a. Root1 1, Root2 1: saddle closed orbits which cannot occur for saddle sta- source closed orbit tionary points: twisting . This is detectable by the sign of     the eigenvalues (which must be the same sign by a pre- b. Root1 < 1, Root2 < 1: vious comment). This tells us that the eigendirections sink closed orbit are taken by the Poincaré map to their opposite direc-   >   tions. The only way this can be accomplished by trajecto- c. Root1 1, Root2 < 1: ries near a closed orbit is by twisting. i) both roots positive: saddle closed orbit

Notes on the Topology of Vector Fields and Flows 13 of 21 the manner in which X sits inside Y are the unit circle in the xy-plane. Any simple called extrinsic. closed curve in the knot type of the unknot unknotted. Extrinsic Topology of Closed Orbits: The problem of classifying knots is a very A. Twisting due to the embedding: difficult one; much progress has been made, but much remains to be under- In the case of a closed orbit, we may think stood. As an example of the difficulty of of X as representing a small neighborhood the subject, we cite one result that was N(C) of the closed orbit C, and Y as repre- conjectured for a long time, but proved senting Euclidean space R3. only a couple of years ago: Theorem: Two simple closed curves in Whether or not C is a twisted closed orbit, R3 are of the same knot type if and only if the neighborhood N(C) may or may not be their complementary regions in R3 are 3 embedded in R with twisting due solely toplogically equivalent spaces. to the choice of embedding. This extrinsic twisting can occur in any integer mul- Anything that a simple closed curve can tiple of 360° twists. do in R3, a closed orbit of a vector field can potentially do as well. So any knot type can occur as the knot type of a closed orbit in some vector field.

B. Knotting: C. Linking:

Any simple closed curve in R3 (which any Any collection of two or more disjoint sim- closed orbit must be!) is intrinsically topo- ple closed curves in R3 may exhibit link- logically equivalent to any other simple ing, another extrinsic property. Linking is closed curve, purely as a topological space. really just the generalization of knotting appropriate to a ﬁnite union of simple From the extrinsic point of view, however, closed curves. there are many ways in which a simple closed curve can be embedded in R3. Two Analogous to knotting, we deﬁne the col- 3 { } simple closed curves C1 and C2 in R are lection C1,...,Ck of disjoint simple closed considered to have topologically equiva- curves in R3 to be link equivalent to { } lent embeddings if there is a homeomor- another such collection D1,...,Dk if there 3 3 phism from R to itself that carries C1 is a homeomorphism of R to itself that onto C2. (Technically, the situation is carries the union C ∪...∪ C onto the ∪ ∪ 1 k slightly more complicated than this, but union D1 ... Dk. (This implies that, in not enough to be worth mentioning.) fact, each Ci is carried onto one of the Dj.) As with knotting, an equivalence class of The resulting equivalence classes are links is called a link type. called knot types. The knot type known as the unknot is the equivalence class of

14 of 21 Notes on the Topology of Vector Fields and Flows Any collection of closed orbits of a vector 15 Stable Manifolds field is a collection of disjoint simple closed curves in R3, and as such repre- If H represents a hyperbolic stationary sents a link type. Much is known, but point or closed orbit of a vector field, there much remains unknown, about what link are important topological features lurking types can constitute the set of source nearby: the so-called stable and unstable closed orbits, saddle closed orbits, and manifolds of H. sink closed orbits of a vector field defined on R3 or a subset of R3. For convenience we make the

As an example of the intriguing state of Definition: A critical element of a vec- ignorance of this situation, we mention a tor field V is any stationary point or closed famous unsolved problem known as the orbit of V. Seifert conjecture, posed over 40 years ago: The stable manifold is the set of all points flowing into H in positive time; the unsta- The Seifert conjecture: ble manifold is the set of all points flowing into H in negative time. Suppose V is a non-singular smooth (i.e., ∞ C ) vector field defined on a solid torus Of particular interest is how the stable (anchor ring) shaped region S in R3. Fur- manifold of one critical element intersects ther suppose that V is pointing inward the unstable manifold of another, or possi- normal on the torus boundary of S. bly of itself. (It will be left to the reader to Question: Must V necessarily have any understand why two different stable man- closed orbits at all? ifolds cannot intersect each other.)

The first examples of such a vector field that one comes up with usually have a closed orbit that goes around the hole in Stable manifolds of hyperbolic sta- the solid torus S. Surprisingly enough, tionary points: ∞ there are examples of C vector fields satisfying the premises of Seifert’s Conjec- We now consider another important topo- ture, but whose closed orbits do not go logical features of any hyperbolic critical around the hole in S. In addition, the element: the stable and unstable mani- answer is known to be false if the differen- folds. Assume there is a vector field V 2 3 tiability condition is relaxed to only C1 or defined on some open set U in R or R . C2. Definition: Let p be a hyperbolic sta- ∞ Addendum: In late 1993 the C Seifert tionary point of the vector field V. The sta- conjecture was found to be false when ble manifold of the point p, denoted s Krystyna Kuperberg of Auburn Univer- by W (p), is the set of points whose trajec- sity announced a counterexample, which tories approach p as t → ∞. The unstable as of March 1994 appears correct. manifold of the point p, denoted by

Notes on the Topology of Vector Fields and Flows 15 of 21 Wu(p), is deﬁned as the stable manifold of A point is a 0-manifold. A curve is a 1- p for the reversed ﬂow. (In other words, it manifold. A surface is a 2-manifold. And a is the set of points x whose trajectories 3-manifold, if it is a subset of R3, is any approach p as t → −∞.) open subset of R3.

In all cases the stable manifold of a hyper- As the terminology implies, any stable bolic stationary point is topologically manifold is, in fact, an n-manifold for equivalent to a Euclidean space. This copy some n. The n in question, as the above of Rk is mapped into the domain of the examples suggest, is the number of eigen- vector field C1 (as long as V is) and one-to- values having negative real part. one. It follows from the definition that a sta- In 2 dimensions, a source p has a 0- ble or unstable manifold is invariant dimensional stable manifold: the point p under the action of the flow. For this itself. If p is a saddle, then p has a reason they are sometimes collectively 1-dimensional stable manifold, consisting called “invariant manifolds.” However, of p and the two orbits along the negative since there are many other manifolds that eigendirection. If p is a sink, the stable are invariant under the flow, we prefer not manifold is 2-dimensional, consisting at to use this terminology. least of all the points sufficiently close to p, and all the points that eventually flow into them. Stable manifolds of hyperbolic closed In 3 dimensions, a source p has a orbits: 0-dimensional stable manifold. A saddle of type (2 out, 1 in) has a 1-dimensional Definitions: Let C be a hyperbolic closed orbit of the vector field V. The sta- stable manifold: again it will be the point s p and the precisely two orbits along the ble manifold of C, denoted by W (C), is the set of points x whose trajectories negative eigendirection. A saddle of type → ∞ (1 out, 2 in) has a 2-dimensional stable approach C as t . (Here the word “approach” indicates simply that the dis- manifold (i.e., it is a surface), consisting of φ the point p and the trajectories which get tance of t(x) to C gets arbitrarily small as t gets large.) The unstable manifold of arbitrarily close to the negative real part u eigenplane as t → ∞. Finally, a sink p has C, denoted by W (C), is defined as the sta- a 3-dimensional stable manifold. ble manifold of C for the reversed flow. (In other words, it is the set of points x whose The term n-manifold in topology means trajectories approach C as t → −∞.) a space each point of which has a neighborhood that is topologically equivalent to The stable manifold of a hyperbolic closed an open neighborhood in Euclidean space orbit is topologically equivalent to either Rn. In brief, a manifold is a space that is the ordinary or twisted product of the cir- “locally Euclidean” of a fixed dimension n. cle and a Euclidean space. If the stable manifold is 2-dimensional, for example,

16 of 21 Notes on the Topology of Vector Fields and Flows these are the cylinder and the Möbius pose that every vector in Rn is the band, respectively. As in the case of a sta- sum of a vector in T1 and a vector in tionary point, these stable manifolds are T2. Then M1 and M2 are said to intersect also mapped C1 and one-to-one into the transversely at the point p. domain of the vector ﬁeld. ∩ If for every point p in M1 M2, it is the In 3 dimensions, a source closed orbit C case that M1 and M2 intersect trans- has a 1-dimensional stable manifold: just versely at p, then M1 and M2 are said sim- itself. A saddle closed orbit has a 2-dimen- ply to intersect transversely. Note that ∩ sional stable manifold. A sink closed orbit if M1 M2 is empty, then M1 and M2 nec- has a 3-dimensional stable manifold. essarily intersect transversely.

Stable manifolds that are 1- or 2-dimen- When two manifolds of dimensions d1 and sional look like ordinary curves or sur- d2 intersect transversely in Rn, their faces, respectively, near their associated intersection N is again a manifold. If closed orbit or stationary point. However, d1 + d2 < n, then in fact N must be empty. as they get farther away they get wound Otherwise, N will be a manifold of dimen- up in the dynamics of other parts of the sion equal to d1 + d2 - n. vector field, such as other closed orbits or stationary points. This can cause them to When two manifolds intersect trans- be curly beyond belief. versely, then a sufficiently small perturbation of each of them will not change this fact. Like hyperbolicity, transversality is a property which will not 16 Transversality be washed away by small errors of measurement. Suppose that a line lies in a plane in R3. An arbitrarily small perturbation of the And as in the case of hyperbolicity, if two line or the plane or both can (and usually manifolds do not intersect transversely, will) change this situation: after the per- then they can be made to do so after an turbation, the line will almost certainly arbitrarily small perturbation. intersect the plane in one point. 17 The Non-Wandering Set Now suppose instead that a line intersects a plane in just one point. In this case a For an arbitrary vector field V, we can sufficiently small perturbation of the line define the set of points that “recirculate,” and/or plane cannot alter this state of in a sense that can be made precise. affairs. Definition: We say that a point is wan- Suppose two manifolds M1 and M2 in dering if every neighborhood of the point, Euclidean space Rn intersect at a point p. after flowing for a sufficiently large Let T denote the set of all vectors amount of time, never intersects itself i = tangent to Mi at p, for i 1,2. Now sup- again.

Notes on the Topology of Vector Fields and Flows 17 of 21 Precisely: p is wandering point for V if with s2 ≥ s1 + T, such that φ ∩ φ ≠ ∅ there exists a neighborhood N of p and s1(N) s2(N) , then by applying a T > 0 such that, for all t with |t| ≥ T, we φ− to both sides of this expression we φ ∩ = ∅. s1 ∩ φ ≠ ∅ have t(N) N would obtain N s2−s1(N) contradiction. Definition: The non-wandering set Ω(V) of a vector field V consists of all Thus whenever t increases by at least T, φ points that are not wandering. t(N) must occupy a new region of U with equal positive volume, which eventually The non-wandering set must be a will exhaust the necessarily finite volume closed set invariant under the action of Ucontradiction. ■ of the flow. 18 Structural Stability Any critical element of a vector field is necessarily contained in its non- If we are ever going to be able to accu- wandering set. rately describe the topology of a real- world vector field, then the necessarily Example: For any r > 0 consider the con- inaccurate numerical definition we have stant vector field V on the unit square r of the vector field must not affect the 0 ≤ x ≤ 2π, 0 ≤y ≤ 2π, given by V (x, y) ≡ r topology. We are thus interested in vector (1, r2). If we identify opposite sides of this fields which keep their global topology square via (x,0) ~ (x,2π) and (0,y) ~ (2π,y), even after a sufficiently small perturba- we get a torus with a vector field on it. tion. Finally we can fill up a solid anchor ring A 3 in R by nesting an infinite family of such This is exactly the consideration which led tori of revolutionone for each r satisfy- /  to our giving prominence to hyperbolicity, ing 0 < r < 1 2 about a core circle. We and transversality, except this is not just inscribe the vector field Vr on the torus of on a local scale: it is on a global scale. small radius r. Then the non-wandering set of this vector field on A is all of A. Definition: A vector field V on U is called structurally stable if for any new vector Theorem: Suppose the vector field V is field V1 on U that is sufficiently near to V volume-preserving on a bounded (in the C1 sense), there exists a homeo- domain U and has no trajectories exit- morphism H : U → U which carries the ing its domain. Then Ω(V) must be all of trajectories of V to those of V1 and pre- U. serves their sense.

Sketch of proof: Suppose, to the con- In brief, a small enough perturbation of V trary, that there is some point p of U and results in a new vector ﬁeld that is topo- some neighborhood N of p such that for all | | ≥ > ∩ φ = ∅ logically equivalent to the old one. t T 0, we have N t(N) . Note: It might be nice to be able to say Now, if there were two numbers s1 and s2 that the old and new vector ﬁelds were

18 of 21 Notes on the Topology of Vector Fields and Flows smoothly equivalent (i.e., topologically Theorem: In 2 dimensions, every struc- equivalent via a homeomorphism that is turally stable vector field is Morse-Smale. smooth and has smooth inverse). How- ever, smooth equivalence preserves eigen- It would be convenient if this were also values at stationary points, but the case in higher dimensions, but alas, eigenvalues can be easily changed via a this is not the case. There exist structur- C1 perturbation. ally stable vector fields in dimensions ≥ 3 whose non-wandering set is the entire Example: Consider the constant vector domain (the so-called Anosov flows). field V(x, y) = (a, b) on R2. This is structurally stable, as you might guess.

Example: Consider the vector field Decomposing Morse-Smale vector given by fields into handles V(x,y) = (−y + x(x2+y2−1), x + y(x2+y2−1)). This vector field has one sink stationary It would be rather complicated to describe point, and one source closed orbit. It, too, exactly how all the stationary points and is structurally stable. closed orbits of a Morse-Smale vector field in n dimensions (think of n = 2 or 3) fit 19 Morse-Smale Vector Fields together. But it is a remarkable fact that Morse-Smale vector fields can be decom- Morse-Smale vector fields are important posed into simple pieces, each of which because they are the simplest large class corresponds to a critical element. of structurally stable vector fields. Let Dj denote the unit ball in Rj: all points Definition: A vector field V is called at a distance of ≤ 1 from the origin. Let S1 Morse-Smale if the following hold: denote the unit circle in R2: all points at a 1) V has a finite number of stationary distance of exactly 1 from the origin. points and closed orbits, all of which are hyperbolic. For each stationary point with unstable 2) The non-wandering set Ω(V) is the manifold of dimension k, we will use one k n−k union of the closed orbits and the station- building block of the form D × D (a ary points. “handle”). For each closed orbit with 3) If H1 and H2 each represent a closed unstable manifold of dimension k, we will use one building block of the form orbit or stationary point, then the stable − − s 1 × k 1 × n k manifold W (H1) of H1 has transverse S D D ( a “round handle”). By intersection with the unstable manifold taking the union of these building blocks, u W (H2) of H2. which are allowed to intersect only along specified parts of their boundaries, we can Theorem: Morse-Smale vector fields are reconstructin a topological sensethe structurally stable. original Morse-Smale vector field.

Notes on the Topology of Vector Fields and Flows 19 of 21 The partial ordering (Note, however, that there is still additional topological informationhow the Consider a Morse-Smale vector field V. stable and unstable manifolds intersect each otherthat is not covered in the Observation 1: Let H1, H2, and H3 be above description.) critical elements of V. Suppose that u ∩ s ≠ ∅ W (H1) W (H2) , and also 20 Beyond Morse-Smale Flows u ∩ s ≠ ∅ W (H2) W (H3) . Then it must also be true that Morse-Smale flows are important to u ∩ s ≠ ∅ W (H1) W (H3) . So there is a kind understand, but they represent one of transitivity operating here. extreme of a spectrum. The non-wandering set of a Morse-Smale vector field is Observation 2: Suppose instead that we concentrated in a finite number of 0- and have critical elements H1 and H2 of V 1-dimensional sets: the stationary points u ∩ s ≠ ∅ with W (H1) W (H2) , and also and closed orbits. Wu(H ) ∩ Ws(H ) ≠ ∅. Then it must fol- 2 = 1 low that H1 H2. At the opposite extreme are the Anosov flows mentioned in Section 19. The non- Now let H1 and H2 be any two critical ele- wandering set of an Anosov flow is the ments of a Morse-Smale vector field V. entire domain of the flow. And there is a ≤ Then we shall use the notation H1 H2 to kind of hyperbolicity which applies to this u ∩ s ≠ ∅ mean that W (H1) W (H2) . entire non-wandering set: each point has a set of stable directions emanating from From the two observations above, we see ≤ it, on which the flow is “contracting,” and that the condition that is a bona fide a set of unstable directions on which the partial ordering on the set of critical ele- flow is “expanding.” ments of V. And the concept of hyperbolicity can be Describing a Morse-Smale vector field by extended to include a much wider variety A. labeling the critical elements of sets than manifolds. When this is done according to properly, the resulting flows are, like 1) stationary point or closed orbit Morse-Smale flows, structurally stable. 2) dimension of the stable Much progress has been made, but as of manifold late 1992 such so-called basic sets are far 3) if closed orbit, whether or not from being completely classified. stable manifold is twisted, and B. specifiying the pairs H1 and H2 21 Further Areas of Study for which H1 ≤ H2 goes a long way towards conveying a very One fiction we have assumed to avoid good description of “what the vector field complications in these notes, is that the V is doing topologically.” domain of a vector field must be an open set in Euclidean space. But in fact, the

20 of 21 Notes on the Topology of Vector Fields and Flows natural domain for a vector ﬁeld is an arbitrary n-dimensional manifold.

On the one hand, the mathematics is usually most elegant if the domain of a vector ﬁeld is a compact manifold without boundary, and if the vector ﬁeld is time-independent (like the ones we have been considering for most of these notes).

But in reality, domains for real-world vector ﬁelds are most commonly manifolds with boundary, and vector ﬁelds found in the real world are frequently time-dependent.

Much in the same way that the Poincaré map can be used to understand the orbit structure around a closed orbit, the dynamics of mappings in general sheds a great deal of light on the dynamics of ﬂows, and is well worthy of further study.

Very little is known about what kind of dynamics real-world vector fields tend to actually have, for example in aerodynam- ics. Valuable progress could be made by creating software to detect topological features of real-world vector fields, and reporting on the kinds of fields encoun- tered.

Much progress could also be made by devising new, rough descriptions of how a vector ﬁeld ﬂows around, computable with a modern computer, and which are robust enough to be accurate despite numerical imprecision.

We shall see.

Notes on the Topology of Vector Fields and Flows 21 of 21