APPENDIX A INTERSECTION NUMBERS AND EXISTENCE RESULTS FOR

TWO-POINT BOUNDARY VALUE PROBLEMS

OF SINGULARLY PERTURBED SYSTEMS

The analysis of Example 3.5 above leads to the following existence result.

A.1 Theorem. Assume a: [0, 1] → ]0, 1[ is continuous, satisfies a(λ) ≠ 1/2 for λ = 0, 1, and has the property that (a(x) − \frac{1}{2}) changes sign at least once over the interval [0, 1]. Then there exists ε∗ > 0 and a family of solutions {v_{ε}}_{0<ε≤ε∗} of the two-point boundary value problem

\begin{matrix}ε^{2}v′′ + v(1 − v)(v − a(x)) = 0 \\ v′(0) = 0 = v′(1) \end{matrix}

with the properties that for each x ∈ [0, 1], 0 < v_{ε}(x) < 1 and

lim_{ε→0} v_{ε}(x) = \{ \begin{matrix}0, & if a(x) > \frac{1}{2} \\ 1, & if a(x) < \frac{1}{2}. \end{matrix}

Remark. The hypothesis “(a(x) − \frac{1}{2}) changes sign at least once over [0, 1]” ensures that the solutions found are not the constant solutions v_{ε} ≡ 0 or v_{ε} ≡ 1.

If the zero set of (a(x) − \frac{1}{2}) consists of an infinite discrete set at each point of which (a(x) − \frac{1}{2}) changes sign together with finitely many limit points (e.g., a(\frac{1}{2}) = \frac{1}{2} and a(x) = \frac{1}{2} + (x − \frac{1}{2})^{2} sin (π(2 − 4x)^{−1}) for x ≠ \frac{1}{2}), then near the limit points the family exhibits “chattering”; i.e., in any given neighborhood of a limit point the number of oscillations from near zero to near one exhibited by v_{ε} increases as ε ↓ 0.

If it is assumed that (a(x) − \frac{1}{2}) has only finitely many zeros at each of which (a(x) − \frac{1}{2}) changes sign, then the theorem implies that the family of solutions v_{ε}, 0 < ε ≪ 1, exhibits an interior transition layer at each such zero. It follows from the proof of Theorem A.1 given below that in these layers (v_{ε}(x) − \frac{1}{2}) changes sign in opposite direction from (a(x)− \frac{1}{2}). Hence, if in fact a is C^{1} with non-zero derivative at each zero of (a(x) − \frac{1}{2}), then according to [AnMP] the non-constant stationary solution v_{ε} is a stable equilibrium of the Fisher equation with Neumann boundary conditions. Using the results of [K7] one can also find many unstable stationary states which together with the results of [K6] yield all the stationary states of the Fisher equation. Details will appear elsewhere.

The proof of Theorem A.1 is based on an existence theorem for singularly perturbed, vector two-point boundary value problem in R^{m} on the finite interval [a, b] of the form

εu′ = F(u, x, ε) (A.1a) u(a) ∈ E(a), u(b) ∈ E(b) (A.1b)

where differentiation is with respect to x ∈ R and where E(λ) for λ = a, b is the locus of points in R^{m} satisfying the endpoint condition at λ. The following hypotheses are made on F and on the endpoint sets:

Hypotheses BVP.

(1) F is continuous on an open, connected subset of R^{m+1}×R^{+}, and for λ ∈ W, an open interval containing [a, b], the family of fast systems u̇ = F(u, λ, 0) is defined and generates a continuous family of flows {φ_{λ}}_{λ∈W} in R^{m} and the family of extended systems εu′ = F(u, x, ε), x′ = 1 is defined and generates a continuous family of flows {ψ_{ε}}_{0<ε≪1} in R^{m} × R;

(2) there exists a continuous section x ↦ (S(x), x) of S(φ) defined on an open interval containing [a, b];

(3) there exists a nested index pair (N_{1}, N_{0}) for S(a) relative to the forward time flow and a singular chain c with support in E(a) that is a relative cycle in N_{1} modulo N_{0} representing α ∈ H̃_{∗}C (S);

(4) there exists a nested index pair (N∗_{1}, N∗_{0}) for S(b) relative to the reverse time flow and a singular chain c∗ with support in E(b) that is a relative cycle in N∗_{1} modulo N∗_{0} representing γ ∈ H̃_{∗}C∗(S);

(5) C (β)_{∗}α # γ ≠ 0 where β is the path class in S(φ) from S(a) to S(b) defined by the section x ↦ (S(x), x); i.e., β(t) = (S((1 − t)a + tb), (1 − t)a + tb).

Remark. In the context of hypotheses BVP, Corollary 6.3 and the non-singular-ity of the pairing ^{#}𝔏 (hence of L) guaranteed by Corollary 9.6 (assuming coefficients taken in a field) imply that given a left endpoint set supporting non-zero homology in H̃_{∗}C (S(a)), then there exist right endpoint sets so that hypotheses BVP are satisfied. More importantly, to satisfy hypotheses BVP, Corollary 9.6 implies that E(a) and E(b) must support dual classes relative to L modulo continuation along β.

A.2 Theorem. Under the assumption of hypotheses BVP, there exists a solution u_{ε} to the boundary value problem (A.1) for 0 < ε ≪ 1. Furthermore, if O is open in R^{m} × R, if S(x) × {x} ⊂ O for each x ∈ [a, b], and if N_{1} × {a} and N∗_{1} × {b} are subsets of O, then u_{ε} can be chosen so that its graph, i.e., the point set {(u_{ε}(x), x) : x ∈ [a, b]}, also lies in O.

Before giving the proof, it is convenient to make the following definitions and notational conventions: (1) A singular chain c′′ is called a subchain of a singular chain c′ if c′′ is obtained from an iterated barycentric subdivision of c′ by deleting some of the summands in the subdivision. (2) If c′ is a singular chain in R^{m}, then for x ∈ R, let c′ × {x} denote the singular chain in R^{m} × {x} induced by the identification of R^{m} with R^{m} × {x}. (3) For (u, x, t) in the domain of ψ_{ε}, define

(u, x) ·^{ε}t := ψ_{ε}(u, x, t) .

A.3 Proof of Theorem A.2. The proof is based on Theorem 3.1 of [K4] which takes hypotheses BVP(1)–(3) as hypotheses, but also requires that an index pair for S(b) relative to the forward time flow be given. Let us give one by using Lemma 5.12 to choose ⟨N_{1}(b), N_{0}(b)⟩ to be an index pair for S(b) relative to the fast flow φ_{b} with the property that ((N_{1}(b), ∩N_{0}(b)), (N∗_{1}, N_{0}^{∗ −s})) is an object of AIP(S(b)) for some sufficiently large s > 0; see Remark 5.12.2.

By [K4, Theorem 3.1] there exists ε̄ > 0 with the property that if 0 < ε ≤ ε̄, then there exists a subchain c_{ε} of c whose image under the chain map induced by the time (b − a) flow map of ψ_{ε} mapping singular chains in R^{m} × {a} to singular chains in R^{m} × {b} is a relative cycle in N_{1}(b) modulo N_{1}(b) ∩ N_{0}(b) representing C (β)_{∗}α. That is, where c̄_{ε} is defined by

c̄_{ε} × {b} := c_{ε} × {a} ε ; · ; ♯(b − a) ,

c̄_{ε} is a relative cycle in N_{1}(b) modulo N_{0}(b) representing C (β)_{∗}α. It follows immediately from Proposition 2.7 and hypotheses BVP(4) and BVP(5) that there exists (u_{b ε}, b) ∈ (|c̄_{ε}| \ N_{0}(b) ∩ |c∗| \ N_{0}^{∗ −s}) × {b}. It follows that the curve u_{ε} defined by

(u_{ε}(x), x) := (u_{b ε}, b) ·^{ε}(x − b) for a ≤ x ≤ b

is a solution of the boundary value problem (A.1).

Furthermore, given O open in R^{m} × R with S(x) × {x} ⊂ O for x ∈ W, Theorem 3.1 of [K4] also ensures that c_{ε} will satisfy |c_{ε}| × {a} ·^{ε}[0, b − a] ⊂ O whence u_{ε} as defined will have graph contained in O. □

A.4 Proof of Theorem A.1. Throughout the proof rather than work with the second-order equation, we instead work with the equivalent first-order non-autono-mous system

(A.2_{ε})  \begin{aligned}εv′ & = w \\ εw′ & = −v(1 − v)(v − a(x)) . \end{aligned}

In fact because the argument is conveniently phrased in terms of invariant regions, we work with the flow in R^{3} generated by system (A.2_{ε}) extended by the equation of the independent variable, x′ = 1. Call the extended system, system (A.2_{ε}^{+}).

Before entering into the details of the proofs of existence and the requisite limiting properties, let us first show that a non-constant solution v_{ε} of the boundary value problem must necessarily satisfy 0 < v_{ε}(x) < 1 for each x ∈ [0, 1]. We begin by partitioning R^{3} into five closed regions W_{I}, W_{II} , …, W_{V} as follows:

(v, w, x) ∈ W_{I} if, and only if, v ≤ 0 and w ≥ 0;

(v, w, x) ∈ W_{II} if, and only if, v ≤ 0 and w ≤ 0 ;
(v, w, x) ∈ W_{III} if, and only if, 0 ≤ v ≤ 1;
(v, w, x) ∈ W_{IV} if, and only if, v ≥ 1 and w ≥ 0;
(v, w, x) ∈ W_{V} if, and only if, v ≥ 1 and w ≤ 0.

These sets are illustrated in cross-section in Figure 8 where the arrows indicate the direction of flow across the boundary of these regions, but observe that the points (0, 0, x) and (1, 0, x) correspond to constant solutions of ε^{2}v′′+v(1−v)(v−a(x)) = 0. In particular, as is easily checked by computing derivatives along trajectories at boundary points, regions W_{II} and W_{IV} are positively invariant relative to the flow of system (A.2_{ε}^{+}) while regions W_{I} and W_{V} are negatively invariant.

Figure 8. Cross-sections at x ∈ R of the closed regions W_{I}, W_{II}, W_{III}, W_{IV}, W_{V}.

Since v′_{ε}(0) = 0 and v_{ε} is not a constant solution, it is evident from Figure 8 that if v_{ε}(0) < 0 (respectively, v_{ε}(0) > 1), then (v_{ε}(x), εv′_{ε}(x), x) lies interior to W_{II} (respectively W_{IV}) for 0 < x ≤ 1 contradicting that v′_{ε}(1) = 0. Thus 0 < v_{ε}(0) < 1. Similarly, interpreting Figure 8 in backwards time, it follows that 0 < v_{ε}(1) < 1. Thus z_{ε}(λ) := (v_{ε}(λ), εv′_{ε}(λ), λ) lies in the interior of W_{III} for λ = 0, 1. Figure 8 again shows that the orbit segment through z_{ε}(0) with 0 ≤ x ≤ 1 cannot exit W_{III} and then reenter. It follows that 0 < v_{ε}(x) < 1 for x ∈ [0, 1].

For the existence and limit proofs, the notation established in Example 3.5 will be used. For definiteness, it will be assumed that a(0) > \frac{1}{2} > a(1); the other possibilities are dealt with similarly.

First, let us see that the locus of points in the phase space of the fast system at λ satisfying the Neumann endpoint condition supports a relative 1-cycle in B(λ) modulo its exit (entrance) set representing α_{0} (respectively, γ_{1}) when λ equals zero (respectively, one). As drawn in Figure 1, B(λ) is a square with one pair of diametrically opposite vertices lying on the line w = 0. As these vertices are in the intersection of the exit and entrance sets of B(λ), the line segment of w = 0 between these two vertices is an arc having one endpoint in each component of the exit set of B(λ) and also having one endpoint in each component of the entrance set of B(λ). Hence as observed in Example 3.5, this line segment when regarded as the homeomorphic image of the standard 1-simplex can on the one hand, when λ = 0, be taken as a geometric representative of the homology class α_{0} ∈ H̃_{1}C (S_{a}(0)) and on the other, when λ = 1, can be regarded as a geometric representative of the homology class γ_{1} ∈ H̃_{1}C∗(S_{a}(1)).

These geometric representations of α_{0} and γ_{1} and statement (3.7), viz.,

C (β)_{∗}α_{0} # γ_{1} ≠ 0 ,

allow Theorem A.2 to be applied to system (A.2_{ε}) yielding a family of solutions of the boundary value problem. However, for the family to possess the requisite limiting properties some care must be taken in applying Theorem A.2.

Accordingly, set H := \{x ∈ [0, 1] : a(x) = \frac{1}{2}\} and set

gr(σ_{a}) := ⋃_{λ∈[0,1]} S_{a}(λ) × {λ} ⊂ R^{2} × [0, 1] .

Because the set map λ ↦ S_{a}(λ) varies upper semi-continuously with λ ∈ [0, 1] as a consequence of the continuity of σ_{a}, it follows that gr(σ_{a}) is a compact subset of R^{3}. Hence, choose a sequence {O_{n}}_{n=1}^{∞} of relatively compact open subsets of R^{3} satisfying

(1) if a(λ) ≠ \frac{1}{2} and |λ − H| ≥ (2n)^{−1}, then O_{n} ∩ R^{2} × \{λ\} has two components each containing a component of S_{a}(λ) (n = 1, 2, …),

(2) cl (O_{n+1}) ⊂ O_{n} (n = 1, 2, …),

(3) ⋂_{n=1}^{∞} O_{n} = gr(σ_{a}).

For λ = 0, 1, the blocks B(λ) and C(λ) illustrated in Figure 1 depend implicitly on a parameter r > 0 where as r shrinks to zero B(λ) ∪ C(λ) shrinks to S_{a}(λ). To see this, let us make this dependence on r explicit in the notation and write B(λ, r) and C(λ, r) from hereon instead of B(λ) and C(λ). Then regarded as subsets of R^{2} ≃ R^{2} × {λ}, for λ = 0, 1,

B(λ, r) := {(v, w) : |v − λ| + |w| ≤ r}

and

C(λ, r) := {(v, w) : (|v − (1 − λ)| + |w| ≤ r and (−1)^{λ}(v + λ − 1) ≥ 0)
or (H(v, w, λ) ≤ H(1 − λ, r, λ) and 0 ≤ v ≤ 1)}

where r is chosen so small, say 0 < r < r_{0}, that B(λ, r) and C(λ, r) are disjoint. From these descriptions and the definition of S_{a}(λ), it follows easily that B(λ, r) ∪ C(λ, r) shrinks to S_{a}(λ).

Then choose a sequence of positive reals {r_{n}}_{n=1}^{∞} within the open interval ]0, r_{0} [ and strictly decreasing to zero so that for λ = 0, 1, (B(λ, r_{n})∪C(λ, r_{n}))×{λ} ⊂ O_{n}. Also, let α_{n 0} be the maximal segment of the line w = 0 contained in B(0, r_{n}) and let γ_{n 1} be the maximal segment of w = 0 contained in B(1, r_{n}). It follows that α_{n 0} is a relative cycle in B(0, r_{n}) modulo its exit set representing α_{0} and that γ_{n 1} is a relative cycle in B(1, r_{n}) modulo its entrance set representing γ_{1}.

Application of Theorem A.2 to the family of systems (A.2_{ε}) yields for each positive integer n an ε_{n} > 0 and a family of solutions {v_{ε n}}_{0<ε≤ε_{n}} of the boundary value problem with the property that

(v_{ε n}(x), εv′_{ε n}(x), x) ∈ O_{n} for x ∈ [0, 1].

We can, without loss of generality, assume that the sequence {ε_{n}}_{n=1}^{∞} strictly decreases to zero. Then set ε∗ := ε_{1} and for ε ∈ ]0, ε∗] if ε_{n+1} < ε ≤ ε_{n}, set v_{ε} := v_{ε n}.

The family of solutions {v_{ε}}_{0<ε≤ε∗} has the desired limiting properties. This follows immediately from the choice of the sequence of sets O_{n} once it is realized that if a(x) ≠ \frac{1}{2} and if O_{n} ∩ R^{2} × \{x\} has two components each containing a component of S_{a}(x) × {x}, then (v_{ε}(x), εv′_{ε}(x), x) always lies in the component of O_{n} ∩ R^{2} × {x}, call it C_{n}(x), that does not contain (a(x), 0, x). Hence, as ε decreases and n therefore increases forcing O_{n} to squeeze down to gr(σ_{a}), the point (v_{ε}(x), εv′_{ε}(x), x) is forced to limit on (0, 0, x) if a(x) > \frac{1}{2}, but on (1, 0, x) if a(x) < \frac{1}{2}.

The placement of (v_{ε}(x), εv′_{ε}(x), x) in C_{n}(x) is a consequence of the proof of [K4, Theorem 3.1] which is invoked to prove Theorem A.2. In particular, the component of S_{a}(x) × {x} containing (a(x), 0, x) has, as observed in Example 3.5 above, trivial Conley homotopy index in the forward time direction so that its Conley index has zero reduced homology in all dimensions. The singular chains selected in the proof of [K4, Theorem 4.1] always evolve so as to give the continuation of a non-zero homology class in the index so that at any point of the evolution any subchain lying in a component of O_{n} ∩ R^{2} × {x} whose closure isolates an invariant set with trivial index can be discarded as a consequence of the fact that the homology of the Conley index of an isolated invariant set that is the union of two disjoint isolated invariant sets is the direct sum of the homology modules of the Conley indices of the disjoint pieces which follows trivially from a Meyer-Vietoris sequence using an index pair for the whole isolated invariant set that is the union of disjoint index pairs for the pieces.

In more detail, at stage n when Theorem A.2 is applied to choose ε_{n}, Theorem 3.1 of [K4] is implicitly invoked. As applied to the current problem, the first step in the proof of the latter chooses a partition P of [0, 1] so that C (β) = F (σ_{a}, P) where F (σ_{a}, P) is a composite of equivalences between Conley indices as in (5.4) above. It can be assumed that P contains sufficiently many partition points so that each partition interval of P satisfies at least one of two properties: (i) each point of the partition interval lies within distance n^{−1} of H; (ii) each point of the partition interval is at least distance (2n)^{−1} from H. For a partition interval J satisfying the second, O_{n} ∩ R^{2} × J has components C_{n} and D_{n} where the labeling is such that C_{n}(x) = C_{n} ∩ R^{2} × \{x\} for each x ∈ J. Thus, if a(x) − \frac{1}{2} > 0 for each x ∈ J, then (0, 0, x) ∈ C_{n} and (1, 0, x) ∈ D_{n} for each x ∈ J, but if a(x) − \frac{1}{2} < 0 for each x ∈ J, then (1, 0, x) ∈ C_{n} and (0, 0, x) ∈ D_{n} for each x ∈ J.

Let us consider one such partition interval J, write J = [y, z], and note J ⊂ [0, 1]. As applied to the family of systems (A.2_{ε}), the proof of [K4, Theorem 3.1] is carried out by iteratively applying Sublemma(**) of Theorem 3.1 in [K4] to the family of autonomous stretched systems associated to the systems (A.2_{ε}) restricted to successive partition intervals of P. For present purposes, we need not specify these stretched systems, but instead rephrase the relevant parts of the proof of Sublemma(**) in terms of the flow of system (A.2_{ε}^{+}).

As it applies in the present context and using the notation established in its proof, Sublemma(**) assumes given a singular chain c representing C (β^{y}) α_{0} relative to an index pair ⟨N_{1 1}(y), N_{2 1}(y)⟩ of S_{a}(y) and various index pairs for S_{a}(z)—the only one of interest to us here is denoted there by ⟨N_{1 3}(z), N_{2 3}(z)⟩ and contains a perturbable index pair whence (N_{1 3}(z), ∩N_{2 3}(z)) has thick exit set. For sufficiently small ε > 0 (here sufficiently small means ε ≤ ε_{n}) Sublemma(**) exhibits a subchain ĉ_{ε} of c so that regarding ĉ_{ε} as having support in R^{2} × {y}, its image under the chain map induced by following the flow of system (A.2_{ε}^{+}) for time (z − y) is a chain c̄_{ε} with support in R^{2} ×{z} ≃ R^{2} and is a relative cycle in (N_{1 3}(z), ∩N_{2 3}(z)) representing C (β^{z})_{∗} α_{0}. Further, for each time x ∈ [0, z − y], the image of ĉ_{ε} under the chain map induced by following the flow generated by (A.2_{ε}^{+}) for time x has support in O_{n} ∩ R^{2} × {y + x}. Thus, to complete the proof it will suffice to show that for x ∈ J, the image of ĉ_{ε} obtained by following it under the flow for time x actually has support in C_{n}(x).

The selection of ĉ_{ε} is actually made by selecting c̄_{ε} as a subchain of the image of c under a certain chain map between index spaces using the barycentric subdivision operator on chains, but the details of this selection and how ĉ_{ε} is produced from c̄_{ε} are not relevant to us here. What is important here is that the selected chain c̄_{ε} is a relative cycle in (N_{1 3}(z), ∩N_{2 3}(z)) representing C (β^{z})_{∗} α_{0}. Because O_{n} ∩ R^{2} × {x} = C_{n}(x) ∪ D_{n}(x) we can, for i = 1, 2, write N_{i 3}(z) as the union of two disjoint sets M_{i C} and M_{i D}, the former a subset of C_{n}(z), the latter a subset of D_{n}(z). It follows that (M_{1 C}(z), ∩M_{2 C}(z)) is an index pair for that part of S_{a}(z) contained in C_{n}(z) and that (M_{1 D}(Z), ∩M_{2 D}(Z)) is an index pair for that part of S_{a}(z) contained in D_{n}(z). Further, both these index pairs have thick exit set because (N_{1 3}(z), ∩N_{2 3}(z)) does. Then as that part of S_{a}(z) contained in D_{n}(z) has trivial Conley index, it follows that H_{∗}(M_{1 D}(z), ∩M_{2 D}(z)) ≃ 0. Hence, as

H_{∗}(M_{1 C}(z), ∩M_{2 C}(z)) ⊕ H_{∗}(M_{1 D}(z), ∩M_{2 D}(z)) ≃ H_{∗}(N_{1 3}(z), ∩N_{2 2}(z))

with the isomorphism being given by the sum of inclusion induced homomorphisms, we can, without loss of generality, discard from the R-linear combination of simplexes giving c̄_{ε} those with support in D_{n}(z) and assume that c̄_{ε} has support in C_{n}(z). Because continuous maps preserve connectivity, it then follows by considering what happens to each simplex in the sum giving ĉ_{ε} that the image of the support of ĉ_{ε} over the time interval [0, z − y] under the flow of (A.2_{ε}^{+}) must lie entirely in C_{n}. Further, for each x ∈ J, the proof of Theorem A.2, and in particular, the conclusion of Theorem 3.1 of [K4], forces (v_{ε}(x), εv′_{ε}(x), x) to lie in the support of the image of the subchain ĉ_{ε} followed for time x under the flow generated by system (A.2_{ε}^{+}), i.e., it lies in C_{n}(x) as desired. □

A stronger form of Theorem A.2 proved in [K7] allows after suitable modification of hypotheses (3) and (4) the endpoint sets in (A.1b) to be replaced with endpoint sets E(λ, ε) that vary with ε and describes the limiting behavior of the solutions as ε decreases to zero. A key ingredient to this stronger result is Theorem 3.2. A brief discussion describing how results of the type of Theorem A.2 can be used to exhibit endpoint layers can be found in [K5, pp. 1129–1130]. For an explicit example exhibiting certain monotone equilibrium solutions of the above Fisher equation see [K6]; also see [K4, Example 1.4]. For a non-critical point section of the space of isolated invariant sets for a family of fast systems in R^{3} see [K4, Example 5.1]. Also examined in [K7] are situations where the family of fast systems associated to (A.1a) exhibits the breakup of saddle-saddle connections, and these lead to families of solutions exhibiting interior transition layers. A detailed description of how solutions with interior transition layers are obtained is beyond the scope of the current work. The underlying argument is similar but requires additional algebraic machinery, namely, the exact sequence on homology of a repellor-attractor pair and the splitting map of such a sequence, in conjunction with [K4, Corollary 3.2]. Also, see [K4, Example 5.2]. It was the desire to exhibit solutions with interior transition layers that cannot be found via methods employing lower and upper solutions which provided the impetus for the results found in [K7].

Another potentially interesting situation where a theorem of the above type might apply and be of interest in the applications would be for a section of isolated invariant sets of the fast systems where at each point the isolated invariant set would have a Conley index equivalent to that of a suspension of a sub-shift of finite type. Invariant sets of this type have been realized for Smale flows on the three sphere [dR], but their Conley indices have not been computed and can be zero— the invariant set of the Smale horseshoe map is a subshift of finite type and the invariant set of the flow obtained by suspending this map has trivial Conley index. However, recent work of M. Mrozek [Mr] which defines a cohomological Conley type index (the Leray reduction of the Alexander-Spanier cohomology of an index pair suitably defined and an associated degree zero endomorphism) for each isolated invariant set of a homeomorphism of a locally compact Hausdorff space shows (see Theorem 2.9 and Example 8.1 of [Mr]) that there are suspensions of sub-shifts of finite type with non-trivial Conley index and with non-trivial rational homology. One should therefore be able to use Theorem A.2 and generalizations to discover solutions whose behavior over small sub-intervals would appear quite complicated, but was in fact mimicking a sub-shift of finite type.

Further, with the notice of acceptance for publication of this work, the referee informed the author that Mrozek, McCord, and Mischaikow have given an explicit relation between Mrozek’s index for an isolated invariant set of a homeomorphism and the Conley index of the corresponding isolated invariant set of the suspended homeomorphism. The relation is expressed in terms of an exact sequence of cohomology modules; see [MMM].