Chapter 5: The Global Attractor

Chapter 4 produced the two structural ingredients that the abstract attractor theory asks for: a continuous semiflow on the enlarged phase space , and a compact absorbing set

That was enough to prove that every bounded trajectory is eventually trapped in a fixed compact region. It was not yet enough to say what the long-time behavior actually is.

The absorbing set is deliberately generous. It contains every state the system can reach after the transient, but many points of are not persistent in any asymptotic sense. In the Chapter 0 example, the box certainly contains the three equilibria, but it also contains many points that no late-time trajectory should visit. The right asymptotic object is therefore smaller than : it is the set of all states that can still be approached arbitrarily late.

This chapter constructs that set and proves that it is the global attractor. More precisely, we define , the omega-limit set of the absorbing set, and we show that is nonempty, compact, invariant, and attracting for every bounded subset of . We then prove uniqueness. This is the prescribed-closure attractor theorem promised in Chapter 1 (Theorem 1.6).

Two scope boundaries matter.

First, no chain-side hypotheses are used here. The generator regularity and uniform mixing assumptions of Chapter 6 do not enter. Chapter 5 belongs entirely to the autonomous ODE theory of Chapters 2—4.

Second, the attractor theorem is an existence theorem, not a classification theorem. It tells us that all long-time behavior is captured by one compact invariant set, but it does not yet say whether that set is a single equilibrium, several equilibria joined by connecting trajectories, or something more complicated.

The chapter proceeds in seven steps.

1. Omega-limit construction (Section 5.1). We specialize the general definition from Chapter 1 to the compact absorber .
2. Nonemptiness and compactness (Sections 5.2—5.3). Compactness of supplies convergent subsequences and keeps all late-time limit points inside a compact set.
3. Invariance (Section 5.4). Continuity of the semiflow and the semigroup property show that is mapped onto itself, not merely into itself.
4. Attraction (Section 5.5). First we prove that attracts . Then the absorbing property of upgrades that to attraction of every bounded subset of .
5. Uniqueness (Section 5.6). Any other compact invariant attracting set must contain , and because itself attracts bounded sets, the reverse inclusion also holds.
6. Synthesis (Section 5.7). We assemble the previous steps into Theorem 5.7, the prescribed-closure instance of the existence theorem announced as Theorem 1.6 in Chapter 1, and then record two useful geometric consequences proved along the way.
7. Discussion (Section 5.8). We describe what the attractor theorem determines and what it leaves open, and bridge forward to Chapter 6.

Chapter 1 introduced the omega-limit set of a subset as the set of all late-time accumulation points of trajectories starting in . For the present chapter, the relevant set is the absorber from Proposition 4.5.

We therefore define

By Definition 1.4, this means that a point belongs to if and only if there exist sequences and such that

The equivalent tail-set formula is

For later use, write

Then .

Because is forward invariant, for every . Therefore each tail set is contained in :

The family is nested downward. If , then

so after taking closures,

This is the right setup for the attractor construction: we start with the entire absorbing set, discard earlier and earlier portions of the dynamics, and keep only the states that survive every late-time cutoff.

For the reinforcement-learning reader. The absorbing set is a safe long-run envelope: after some transient, every deterministic mean trajectory is inside it. The omega-limit set is sharper. It keeps only those states that can still be seen arbitrarily late under some bounded initialization. In algorithmic language, is the complete menu of asymptotically possible deterministic mean behaviors.

For the dynamical-systems reader. Nothing infinite-dimensional is hiding here. Once Chapter 4 gave a compact absorber in a finite-dimensional phase space, asymptotic compactness became automatic. The rest of the work is the explicit omega-limit construction, not a separate compactness theory.

Verification in the Chapter 0 example. In the worked example, . The set is the part of that box that remains dynamically relevant at arbitrarily large times. Chapter 0 identified three equilibria inside , but it deliberately left open whether the attractor is just those equilibria or a larger invariant set. Chapter 5 answers that question at the theorem level.

The first task is to show that the omega-limit construction does not collapse to the empty set.

Lemma 5.1 (Nonemptiness of the omega-limit set). The set is nonempty.

Proof. Choose any point . Because is forward invariant,

The set is compact by Proposition 4.5, so the sequence has a convergent subsequence. Write

with . Since and the times tend to infinity, the definition of gives . Thus is nonempty.

The mechanism is exactly the finite-dimensional one we expected. We do not need an abstract asymptotic compactness theorem because the orbit is already trapped in a compact set.

Verification in the Chapter 0 example. In Chapter 0 we explicitly found three equilibria in the absorbing box. So nonemptiness is not surprising there. The point of Lemma 5.1 is that we no longer rely on explicit equilibrium computations: every model satisfying the Chapter 4 hypotheses automatically has some asymptotic content.

Nonemptiness alone is not enough. The attractor must be compact.

Lemma 5.2 (Compactness of the omega-limit set). The set is compact.

Proof. By Section 5.1,

Each is closed by construction. Since and is compact, each is a compact subset of . Therefore , as an intersection of closed subsets of the compact set , is itself closed and contained in . A closed subset of a compact set is compact. Hence is compact.

Compactness is the point where the absorber stops being merely a bound and starts acting as a container for all asymptotic behavior.

One extra geometric fact, previewed in Chapter 1, follows from the same construction.

Corollary 5.3 (Connectedness of the omega-limit set). The set is connected.

Proof. The set is connected: it is a product of the connected sets , , and . For each , the set

is the image of the connected set under the continuous map . So is connected, and therefore its closure is connected as well.

Because the family is nested downward, the intersection over all agrees with the countable intersection : if belongs to every , then for any we choose an integer , and nestedness gives .

The sets are therefore nonempty, compact, connected, and nested downward. We invoke a topological fact proved in Engelking (General Topology, Theorem 6.1.18) and posed as one of the exercises to Section 26 of Munkres (Topology): the intersection of a nested sequence of nonempty compact connected subsets of a Hausdorff space is itself nonempty, compact, and connected. The nonemptiness and compactness parts we have already established in Lemma 5.1 and Lemma 5.2 by direct arguments. The connectedness part now gives is connected.

This corollary adds geometric information beyond the attractor definition. Once the phase space and absorber are connected, the eventual-behavior set cannot split into disconnected islands.

Verification in the Chapter 0 example. The three equilibria found in Chapter 0 form a compact set, but not a connected one. Since the absorber is connected, Corollary 5.3 shows that cannot consist of only those three points. If all three equilibria lie in the attractor, some additional invariant material must connect them.

Compactness still leaves one major gap. The attractor must be invariant under the semiflow, and for a set defined through late-time limits that means two things:

1. if we start inside and move forward, we stay in ;
2. every point of also has a full past inside .

The first is positive invariance. The second upgrades inclusion to equality.

We also record the complete-trajectory consequence explicitly.

Definition (Complete trajectory). A complete trajectory of the semiflow is a continuous map such that

A complete trajectory is therefore an orbit defined for all past and future times, extending the forward orbit to all of .

Theorem 5.4 (Invariance and complete trajectories). For every ,

Moreover, every point lies on a complete trajectory with .

The proof has three parts. First we prove . Next we prove the reverse inclusion. Finally we use that surjectivity to construct a complete trajectory through an arbitrary point of .

Proof. Fix .

Step 1: positive invariance. Let . By definition of , there exist and such that

By continuity of the semiflow,

Since and each belongs to , the limit point belongs to . Therefore

Step 2: reverse inclusion. Let . Again choose and such that

After discarding finitely many terms, assume for all . Set

Because , , and is forward invariant, we have for every . By compactness of , a subsequence of converges to some ; we pass to that subsequence and keep the index name . Since and , the limit point belongs to .

Now continuity of the semiflow gives

Thus . Since was arbitrary,

Combining the two inclusions yields

Step 3: complete trajectories. Fix . From the equality , for each integer we may choose a point such that

Define

Then , and because and is positively invariant, we have for every .

Let be the vector field of system (L1) from Chapter 2. By Chapter 3, is continuous. Since and is compact, there exists

Fix . For every , the trajectory is defined on , takes values in the compact set , and satisfies

Therefore the family is uniformly bounded and equi-Lipschitz on . We now use the finite-dimensional Arzela-Ascoli theorem on the compact interval : a uniformly bounded equicontinuous family of curves with values in the compact set has a uniformly convergent subsequence. Because equi-Lipschitz implies equicontinuity, the family has a subsequence converging uniformly on . A diagonal argument over yields a function and a subsequence, still denoted , such that uniformly on every compact interval of .

Because is closed and each lies in whenever is large enough that , we have for every . Also because for all .

It remains to verify the trajectory identity. Fix and . Choose so large that both and lie in . For all sufficiently large ,

Passing to the limit and using continuity of ,

So is a complete trajectory in through . This proves the theorem.

Verification in the Chapter 0 example. Each equilibrium produces a constant complete trajectory. Theorem 5.4 says that every other point of the attractor must also lie on a complete trajectory contained in . This is the precise form of the Chapter 0 intuition that late-time behavior is organized by complete bounded motions rather than by isolated snapshots.

An invariant compact set is not yet a global attractor. It must attract every bounded subset of the phase space.

The proof separates the two mechanisms cleanly. First we show directly from the omega-limit definition and compactness of that attracts the absorber . Then we use the fact that every bounded set eventually enters .

Recall from Chapter 1 the one-sided Hausdorff distance: the semidistance from to is

In particular, if , then .

Proposition 5.5 (Attraction of the absorbing set and of bounded sets). The set attracts :

Consequently, attracts every bounded subset :

In particular,

We prove attraction of by contradiction, then transfer to bounded sets via the absorbing property of from Chapter 4.

Proof. We begin with .

Assume for contradiction that does not attract . Then there exist , times , and points such that

Because is forward invariant, each point still lies in . Since is compact, the sequence has a convergent subsequence; write

Because and , the limit point belongs to . Hence along the subsequence, contradicting the lower bound . Therefore

Now let be bounded. By Proposition 4.5, there exists a time such that

For every ,

By monotonicity of the one-sided set distance,

The right-hand side tends to as , so attracts .

Finally, let . Then there exist and such that

Since attracts , we have . Because is compact and hence closed, every limit point of such a sequence lies in . Therefore . This proves .

The two-step structure carries the whole asymptotic argument. The absorber captures every bounded orbit after a transient, and is the part of that remains visible at arbitrarily late times.

Verification in the Chapter 0 example. Once a trajectory enters the box , Proposition 5.5 says that its later evolution approaches the single compact invariant set . The proposition does not yet decide which equilibrium or connecting behavior is selected by a given initial condition; it says only that every such outcome must come from the same attractor.

The set is a compact invariant set attracting every bounded subset of , so is a global attractor. The next proposition shows it is the only one.

Proposition 5.6 (Uniqueness of the global attractor). Let be a compact invariant set attracting every bounded subset of . Then

Proof. We prove the two inclusions separately.

Step 1: . Since is bounded and attracts every bounded set,

Let . By definition of , there exist and such that

Because ,

For each , choose such that

Then . Since is compact, the sequence has a convergent subsequence; write

Because and , it follows that . Thus .

Step 2: . Proposition 5.5 says that attracts every bounded subset of , hence in particular the bounded invariant set . Therefore

But is invariant, so for every . Hence

If there were a point , then because is compact and therefore closed, we would have . This would force , a contradiction. So .

Combining the two inclusions gives .

The definite article in “the” global attractor is justified only after Proposition 5.6.

Verification in the Chapter 0 example. In the Chapter 0 model, Proposition 5.6 confirms that the attractor containing the three equilibria and any connecting invariant material is the only compact invariant set attracting every bounded subset of . There is no competing invariant set that could serve as an alternative attractor.

We can now collect the chapter into the prescribed-closure attractor theorem.

Theorem 5.7 (Existence of the prescribed-closure global attractor). Assume the standing hypotheses of Chapters 2—4: the system of Definition 2.10 is well-defined with prescribed closure map , the solution map is the continuous semiflow of Proposition 4.6, and is the compact absorbing set of Proposition 4.5. Define

Then:

1. is nonempty and compact.
2. is invariant: for every .
3. attracts every bounded subset of .
4. is the unique global attractor of system (L1).

In particular, this carries out, for system (L1), the program announced as Theorem 1.6 in Chapter 1.

Proof. Item 1 follows from Lemma 5.1 and Lemma 5.2. Item 2 is Theorem 5.4. Item 3 is Proposition 5.5. Item 4 is Proposition 5.6.

Two additional geometric conclusions were proved along the way and will be useful later. First, Corollary 5.3 shows that is connected because the absorber is connected. Second, Theorem 5.4 shows that every point of lies on a complete trajectory contained in .

Theorem 5.7 is already a complete attractor theorem for the prescribed-closure system. Nothing in the proof used the controlled-chain mixing results of Chapter 6. The price of that modularity is that the closure map is still abstract at this stage; Chapter 6 will identify the physically relevant choice.

For the reinforcement-learning reader. Theorem 5.7 says that the deterministic mean dynamics has a single well-defined eventual-behavior set. Different initializations may approach different parts of it, but they cannot create asymptotic behavior outside . This is more informative than the assertion that “the algorithm converges,” because the attractor can contain multiple equilibria and connecting trajectories, each corresponding to a different possible long-run outcome.

The worked example from Chapter 0 now has a fully justified attractor theorem. Its three equilibria all belong to : if is any equilibrium in , then

so .

The connectedness corollary now adds a useful piece of information that Chapter 0 only hinted at. If the attractor contains three distinct equilibria, then it cannot consist of only those three isolated points, because a finite set with more than one point is disconnected. So the attractor must also contain additional invariant material. The most natural candidates are connecting trajectories, but the existence theorem does not classify the geometry any further.

Theorem 5.7 is the end of the prescribed-closure existence theory, but it is not the end of the asymptotic story.

What the theorem does determine is the correct asymptotic object. There is one compact invariant set that attracts every bounded deterministic initialization. Every late-time deterministic mean regime lives inside , and every point of belongs to a complete trajectory. For the reinforcement-learning reader, this is the mathematically precise replacement for the vague question “what can the algorithm do in the long run?”

What the theorem does not determine is the internal structure of . A compact invariant attracting set may contain equilibria, heteroclinic trajectories, periodic orbits, or more complicated recurrent pieces. The theorem does not tell us which of these occur in the present model. It also does not identify basins of attraction, convergence rates, or the stability type of individual equilibria.

Section 5.7 already showed that the Chapter 0 attractor must extend beyond its three equilibria. Identifying what else it contains–heteroclinic connections, periodic orbits, or other invariant structure–requires phase-portrait or Lyapunov analysis that the existence theorem does not supply.

For the Chapter 0 model specifically, the phase space is three-dimensional (), so limit cycles are not ruled out by dimension alone. Even in dimension two the Poincare—Bendixson theorem permits periodic orbits: it confines planar limit sets to equilibria, periodic orbits, and cycles of connecting trajectories, and no analogous dimension-based constraint is available in dimension three. Whether the particular structure of the Chapter 0 vector field–with its decoupled relaxation mechanisms for the critic and the law variable–actually admits periodic orbits is a question that would require either a Lyapunov argument ruling them out (the construction of a function that decreases strictly along every non-equilibrium trajectory, which rules out periodic orbits because the function could never return to its starting value around a closed loop) or a numerical search finding one. The attractor theorem is silent on this point, which is exactly the kind of structural question it leaves open.

For the dynamical-systems reader. Chapters 2—4 prove dissipativity, and the present chapter turns the compact absorber into a global attractor by an explicit omega-limit construction. No controlled-chain mixing or infinite-dimensional compactness tools enter here. Finer structural tools (Conley decompositions, stable-manifold theory) become relevant only when classifying the internal geometry of , which is a separate question.

Bridge forward. The remaining gap is now modeling rather than asymptotic. Theorem 5.7 treats the closure map as prescribed data. Chapter 6 asks where that map should come from in the controlled-chain formulation and proves that, under uniform mixing, the canonical choice is the frozen invariant-law map .

Exercise 5.1 (Equilibria belong to the attractor). Let be an equilibrium of the semiflow : for all . Show directly from the definition of that . Apply this to the three equilibria identified in Chapter 0.

Exercise 5.2 (Single-point attractor). Suppose the global attractor has the form . Show that is a globally attracting equilibrium: for all and for every . (Optional: show that is also Lyapunov stable–every neighborhood of contains a forward-invariant neighborhood–hence globally asymptotically stable, using the fact that attracts a compact neighborhood of uniformly by Proposition 5.5.)

Exercise 5.3 (Connectedness and the absorber hypothesis).

(a) Use Corollary 5.3 and Exercise 5.1 to explain in one sentence why the Chapter 0 attractor cannot consist of only the three isolated equilibria.

(b) Corollary 5.3 used the fact that is connected. Construct a planar autonomous ODE with two disjoint compact forward-invariant sets , each containing one stable equilibrium, and show that consists of exactly the two equilibria and is therefore disconnected. Equivalently: the semiflow restricted to the disconnected phase space has a disconnected global attractor. Then explain why the global attractor of the same system on all of is nevertheless connected. What extra invariant material does it contain? (Hint: take , , and small closed boxes around the equilibria .)

(c) Identify the step in the proof of Corollary 5.3 that fails when the absorbing set is disconnected, and write down in one sentence the precise property of used at that step. Explain why that property fails when one factor of the product is disconnected.

Exercise 5.4 (A non-equilibrium global attractor). Consider the planar system in polar coordinates , .

(a) Solve the radial equation explicitly and show that the closed disk is absorbing: every bounded set enters it by time at the latest and never leaves. Show also that the closed unit disk, although forward invariant and attracting, is not absorbing: a solution starting at satisfies for every finite .

(b) Show that the global attractor is the closed unit disk : it contains the equilibrium at the origin, the circle , and the spiral trajectories connecting them, while every point with escapes every compact set in backward time. (Hint: the argument of Theorem 5.4 applies verbatim: every point of the attractor lies on a complete trajectory that stays inside a compact invariant set.)

(c) Show that the unit circle is the omega-limit set of every initial condition other than the origin. Conclude that the global attractor need not be a single equilibrium, and that it may strictly contain the set that individual trajectories converge to.

Exercise 5.5 (The asymmetric worked example). Return to the asymmetric Chapter 0 modification (Section 0.15, where the reward is varied). Explain which parts of the Chapter 5 proof are unchanged verbatim and which data from Chapter 4 need to be recomputed. Why does the global attractor theorem still apply without any new ideas?

Exercise 5.6 (What the theorem does not say). Give a one-paragraph answer to the following question: why does Theorem 5.7 not tell us which equilibrium a given initial condition converges to, even in the Chapter 0 model?

Exercise 5.7 (Forcing a single equilibrium attractor). State one additional hypothesis under which the global attractor would have to be a single equilibrium. A strict Lyapunov function is one natural answer (a continuous function that decreases strictly along every non-equilibrium trajectory; see Robinson, Infinite-Dimensional Dynamical Systems, Chapter 10, for the gradient-like consequence that the attractor then consists of equilibria and their connecting orbits). Explain briefly why that extra hypothesis is stronger than anything proved in Chapters 2—5.