Theory and Applications of Categories, Vol. 7, No. 13, 2000, pp. 239–262.

SOLUTION MANIFOLDS FOR SYSTEMS OF DIFFERENTIAL EQUATIONS

JOHN F. KENNISON

Transmitted by Michael Barr

ABSTRACT. This paper defines a solution manifold and a stable submanifold for a system of differential equations. Although we eventually work in the smooth topos, the first two sections do not mention topos theory and should be of interest to non-topos theorists. The paper characterizes solutions in terms of barriers to growth and defines solutions in what are called filter rings (characterized as C^{∞}-reduced rings in a paper of Moerdijk and Reyes). We examine standardization, stabilization, perturbation, change of variables, non-standard solutions, strange attractors and cycles at infinity.

Introduction

We explore what is meant by a solution of a system of differential equations. Although the approach is based on solving equations in the smooth topos, the material in the first two sections does not depend on topos theory and should be of independent interest. One of our results analyzes limit cycles of autonomous differential equations. In section 3, we define a solution manifold and a stabilization operation. Section 4 contains examples.

Conceptually, this paper is related to [3] which dealt with differential equations for a single function. We have had to change our technical approach considerably to accommodate systems of equations. So the reading of [3] is not a prerequisite for this paper.

In section 1, we characterize solutions of a system of differential equations as n-tuples of functions which respect certain “barriers to growth”. This fact enables us, in section 2, to use barriers to define solutions in filter rings (the C^{∞}-reduced rings of [5], [6]). Solutions in a filter ring are, in effect, parameterized solutions, such as solutions of parameterized differential equations, or solutions parameterized by initial conditions, see Theorem 2.12 and its corollary. The filter sometimes gives us non-standard real parameters, and transfinite cycles as in Theorem 2.20. Filter ring solutions often reflect the behavior of nearby solutions which accounts for their effectiveness in examining issues such as stability.

In section 3, we examine solutions in a smooth topos. The solution manifold is defined as the subobject of (R^{n})R where the barrier conditions are satisfied and where R is (the manifold corresponding to) the real line. We work in the topos of sheaves on the category V of filter rings, and use the finite open cover topology, see [4]. Our approach allows us to define the submanifolds of stable and of asymptotically stable solutions and the quotient manifold of standard solutions. For most of our theorems, we need to impose a boundedness condition due to the fact that R is not Archimedean in this topos.

The author thanks the University of Southern Colorado for providing a sunny, friendly and stimulating environment during his year there in 1998-99.

Received by the editors 1999 April 21 and, in revised form, 2000 October 9.

Published on 2000 October 16.

2000 Mathematics Subject Classification: 18B25, 58F14, 26E35.

Key words and phrases: smooth topos, differential equation.

\begin{matrix}c \\ ◯ \end{matrix} John F. Kennison, 2000. Permission to copy for private use granted.

Notation. We often use x = (x_{1}, …, x_{n}) to denote an element of R^{n}. In this case, ∥x∥ denotes Max{|x_{i}|}, the ℓ_{∞} norm. (Although any reasonable norm will usually do.)

1. Barriers for Systems of Differential Equations

We consider systems of the following type:

(∗) \frac{dx_{i}}{dt} = W_{i}(t, x_{1}, …, x_{n}) for i = 1, …, n

We assume that each W_{i} is a continuous, real-valued function defined on all of R × R^{n}. Letting x = (x_{1}, …, x_{n}) and W = (W_{1}, …, W_{n}), system (∗) becomes:

(∗) \frac{dx}{dt} = W (t, x)

We say that f is a solution of (∗) iff f is an n-tuple of differentiable functions from R to R such that, for 1 ≤ i ≤ n, we have f′_{i}(t) = W_{i}(t, f(t)) for all t ∈ R.

Our approach is to study solutions of (∗) by finding barriers to their growth.

1.1. Definition. Let (∗) be as above. Then a C^{∞}-function B(t, x) = B(t, x_{1}, …, x_{n}) is a barrier function for (∗) over [a, b] if a < b and if whenever B(t, x) = 0 for t ∈ [a, b] then ∂B/∂t + ∑(∂B/∂x_{i})W_{i} < 0 at the point (t, x).

1.2. Lemma. Let g(t) be a differentiable function on [a, b] (where a < b) with the property that if g(c) = 0 for any c ∈ [a, b] then g′(c) < 0. It follows that if g(a) ≤ 0 then g(b) < 0.

Proof. Case 1: Assume g(a) < 0. Suppose g(b) ≥ 0. Let c be the smallest element of [a, b] for which g(c) = 0. By choice of c, we have g(t) < 0 for a ≤ t < c which implies that g′(c) ≥ 0 by calculating g′(c) as t approaches c from the left. This contradicts the hypothesis that g′(c) < 0 since g(c) = 0. Case 2: Assume g(a) = 0. Then, by hypothesis, g′(a) < 0 so there clearly exists a_{0} with a < a_{0} < b such that g(a_{0}) < 0. The argument given in case 1, applied to the interval [a_{0}, b], now leads to the result that g(b) < 0.

1.3. Definition. A system f = (f_{1}, …, f_{n}) satisfies the strong barrier condition for (∗) if whenever B(t, x) is a barrier over some [a, b] then B(a, f(a)) > 0 or B(b, f(b)) < 0. Also f satisfies the weak barrier condition for (∗) if under the same assumptions on B, B(a, f(a)) ≥ 0 or B(b, f(b)) ≤ 0.

1.4. Lemma. If f = (f_{1}, …, f_{n}) is a solution of (∗) then f satisfies the strong barrier condition for (∗).

Proof. Let B(t, x) be a barrier over [a, b] and define g(t) = B(t, f(t)). Then apply Lemma 1.2 to g which directly leads to the result.

1.5. Theorem. Let f = (f_{1}, …, f_{n}) be an n-tuple of functions from R to R (with no differentiability or even continuity assumed). Then the following are equivalent:

(1) Each f_{i} is differentiable and f is a solution of (∗).

(2) f satisfies the strong barrier condition for (∗).

(3) f satisfies the weak barrier condition for (∗).

Proof. (1)⇒(2): By Lemma 1.4. (2)⇒(3): Obvious. (3)⇒(1): Let t_{0} be given and let x^{0} = f(t_{0}) where x^{0} = (x_{1}^{0}, …, x_{n}^{0}) and x_{i}^{0} = f_{i}(t_{0}). Let m = W (t_{0}, x^{0}), with m = (m_{1}, …, m_{n}). We need to show that f′_{i}(t_{0}) exists and equals m_{i} for i = 1, …, n.

Let ϵ > 0 be given. Choose an open neighborhood U of (t_{0}, x^{0}) in R × R^{n} such that for (t, x) ∈ U, with x = (x_{1}, …, x_{n}), we have, for i = 1, …, n:

(m_{i} − ϵ) < W_{i}(t, x) < (m_{i} + ϵ)

Choose δ > 0 so that (t, x) ∈ U whenever |t − t_{0}| < δ and |x_{i} − x_{i}^{0}| < δ for all i. Let η be any real number with 0 < η ≤ δ/2. Let h_{i} : R→R be the straight line function with slope m_{i} + ϵ for which h_{i}(t_{0}) = x_{i}^{0} + η. Let ℓ_{i} : R→R be the straight line function with slope m_{i} − ϵ for which ℓ_{i}(t_{0}) = x_{i}^{0} − η. Choose t_{1} > t_{0} such that whenever t ∈ [t_{0}, t_{1}] then ℓ_{i}(t) and h_{i}(t) are within δ of x_{i}^{0}. (It suffices to do this for η = δ/2.) It follows that if x = (x_{1}, …, x_{n}) and ℓ_{i}(t) ≤ x_{i} ≤ h_{i}(t) and t ∈ [t_{0}, t_{1}] then (t, x) ∈ U. Now, using the notation exp(r) for e^{r}, we define:

b_{i}(t, x_{i}) = exp[K(h_{i}(t) − x_{i})(ℓ_{i}(t) − x_{i})]

where K > 0 is to be chosen. To continue the proof, we need:

1.6. Lemma. Using the construction in the above proof, and assuming t ∈ [t_{0}, t_{1}] and x = (x_{1}, …, x_{n}), we have:

(1) If b_{i}(t, x_{i}) ≤ 1, then ℓ_{i}(t) ≤ x_{i} ≤ h_{i}(t).

(2) If b_{i}(t, x_{i}) ≤ 1 for all i, then ∂b_{i}/∂t + ∂b_{i}/∂x_{i}W_{i} < 0.

(3) B(t, x) = (∑ b_{i}(t, x_{i})^{2}) − 1 is a barrier function for (∗) over [t_{0}, t_{1}].

Proof. (1) If b_{i}(t, x_{i}) ≤ 1, then (h_{i}(t)−x_{i})(ℓ_{i}(t)−x_{i}) ≤ 0 which implies ℓ_{i}(t) ≤ x_{i} ≤ h_{i}(t).

(2) If b_{i}(t, x_{i}) ≤ 1 for all i, then, by (1), we have ℓ_{i}(t) ≤ x_{i} ≤ h_{i}(t), for all i, and by choice of t_{1}, we see that (t, x) ∈ U. Now let W_{i}(t, x) = \overline{m_{i}}. then (m_{i} −ϵ) < \overline{m_{i}} < (m_{i} +ϵ), by definition of U. We readily find that:

∂b_{i}/∂t = Kb_{i}[(m_{i} + ϵ)(ℓ_{i}(t) − x_{i}) + (m_{i} − ϵ)(h_{i}(t) − x_{i})]
∂b_{i}/∂x_{i} = Kb_{i}[−(ℓ_{i}(t) − x_{i}) − (h_{i}(t) − x_{i})]

So, collecting these terms and using \overline{m_{i}} = W_{i}(t, x), we get:

∂b_{i}/∂t + ∂b_{i}/∂x_{i}W_{i} = Kb_{i}[ℓ_{i}(t)(m_{i} + ϵ − \overline{m_{i}}) + h_{i}(t)(m_{i} − ϵ − \overline{m_{i}}) + x_{i}(2\overline{m_{i}} − 2m_{i})] Since m_{i} − ϵ < \overline{m_{i}} < m_{i} + ϵ, we see that the coefficient of ℓ_{i}(t) is positive while the coefficient of h_{i}(t) is negative. Also Kb_{i} > 0. But, by (1), ℓ_{i}(t) ≤ x_{i} ≤ h_{i}(t) (and at least one of these inequalities is strict, ℓ_{i}(t) < x_{i} ≤ h_{i}(t) or ℓ_{i}(t) ≤ x_{i} < h_{i}(t).) So, if we replace ℓ_{i}(t) and h_{i}(t) by x_{i}, we get a strictly larger expression, which, as can be readily seen, simplifies to 0:

∂b_{i}/∂t + ∂b_{i}/∂x_{i}W_{i} < 0

(3) Let B(t, x) = (∑ b_{i}(t, x_{i})^{2})−1. Suppose t ∈ [t_{0}, t_{1}] and that B(t, x) = 0. Then, clearly, b_{i}(t, x_{i}) ≤ 1 for all i, so (t, x) ∈ U. We calculate that:

∂B/∂t + ∑ ∂B/∂x_{i}W_{i} = ∑ 2b_{i}[∂b_{i}/∂t + ∂b_{i}/∂x_{i}W_{i}] < 0

(as follows from (2).) This shows that B is a barrier function for (∗).

Proof of theorem 1.5, continued. By the lemma, B(t, x) is a barrier function over [t_{0}, t_{1}] for any positive choice of K. By choosing K sufficiently large, we can make each b_{i}(t_{0}, x_{i}^{0}) small enough so that B(t_{0}, x^{0}) < 0. Since f satisfies the weak barrier condition, it follows that B(t_{1}, f(t_{1})) ≤ 0. But this means that each b_{i}(t_{1}, f_{i}(t_{1})) < 1 so f_{i}(t_{1}) is caught between ℓ_{i}(t_{1}) and h_{i}(t_{1}). Moreover, this argument works as we let η→0 (noting that the same value of t_{1} works for all η). We can also repeat the same argument for any t ∈ [t_{0}, t_{1}]. This is enough to show that the differential quotient (f_{i}(t)−f_{i}(t_{0}))/(t−t_{0}) approaches m_{i} as t approaches t_{0} from the right. An entirely analogous argument works for approaching t_{0} from the left. (In fact, it could be formalized by systematically replacing t by −t, see the first part of the proof of 2.23, and then showing that we get no genuinely new barrier conditions on f).

1.7. Proposition. A pointwise limit of solutions of (∗) is again a solution of (∗). Moreover, if S is a family of solutions of (∗) and f is an n-tuple of functions such that for every a, b and ϵ > 0, there exists s ∈ S with ∥s(a) − f(a)∥ < ϵ, ∥s(b) − f(b)∥ < ϵ then f is a solution of (∗).

Proof. Let B(t, x) be any barrier function for (∗) over [a, b]. We can readily show that f satisfies the strong barrier condition at B by approximating f sufficiently closely by a member of S at a, b.

2. Solutions in Filter Rings

To study non-standard solutions, stability and cycles at infinity, we use the barrier conditions to define solutions in what are called filter rings (or the reduced rings of [5], [6]).

By a proper filter on R^{m} we mean a non-empty collection, F, of non-empty subsets of R^{m} closed under finite intersection and supersets (meaning that if F_{1}, F_{2} ∈ F then F_{1} ∩ F_{2} ∈ F and if F ∈ F, F ⊆ G then G ∈ F). The collection of all subsets of R^{m} is called the improper filter on R^{m}. We say that B is a base for F, or that B generates F, when F ∈ F iff there exists B ∈ B with B ⊆ F.

2.1. Definition. By a closed filter on R^{m} we mean a filter with a base of closed subsets of R^{m}.

We let C^{∞}(R^{m}) denote the ring of all C^{∞}-functions from R^{m} to R. If F is a closed filter on R^{m}, then I(F) is the ideal of all f ∈ C^{∞}(R^{m}) which vanish on a member of F.

By a filter ring, we mean a ring of the form C^{∞}(R^{m})/I(F).

Note: If F is the improper filter, generated by the empty set, which is closed, then C^{∞}(R^{m})/I(F) will be the trivial ring consisting of just one element.

The point of defining a filter ring A is to be able to define what is meant by “a solution of (∗) with parameters in A”. To do this we define the “ring of real-valued maps with parameters in A”. Note that if u = (u_{1}, …, u_{n}) represents an n-tuple of generators of A then it is plausible that α(u, t) represents “a function of t parameterized by A ”. For technical reasons, we sometimes need to impose a boundedness condition. (See also section 3 for a more theoretical approach to these definitions.)

2.2. Definition. For A = C^{∞}(R^{m})/I(F) define Map(A) as C^{∞}(R^{m} × R)/I(π^{∗}F) where π^{∗}F is the closed filter generated by sets of the form F ×R for F ∈ F. We also define Map_{0}(A) as the subring of Map(A) consisting of the “semi-bounded” elements, where w ∈ C^{∞}(R^{m} × R) is semi-bounded with respect to F if for every closed bounded interval J, there exists F ∈ F such that the restriction of w to F × J is bounded.

If F is understood, then C_{sb}^{∞}(R^{m} × R) denotes the subring of all maps in C^{∞}(R^{m}×R) which are semi-bounded with respect to F.

Also, n-Map(A) is the product of n copies of Map(A) and n-Map_{0}(A) is the product of n copies of Map_{0}(A).

Remark. If w denotes an element of Map_{0}(A), then, through intentional abuse of notation, we let w also denote a representative function in C_{sb}^{∞}(R^{m} × R).

C^{∞}-maps can be evaluated in filter rings. If (w_{1}, …, w_{n}) is a k-tuple of elements of the filter ring A, and if λ ∈ C^{∞}(R^{k}), then λ(w_{1}, …, w_{n}) makes sense. For if we regard each w_{i} as a function (see above remark) then λ(w_{1}, …, w_{n}) is a well-defined composite function, and can readily be shown to represent an element of A. It remains to check that the indicated construction is independent of the actual functions used to represent the elements (w_{1}, …, w_{n}) of A, and this is straightforward.

(A ring for which C^{∞}-maps can be evaluated in a reasonable way is called a C^{∞}-ring. See [4] for details.)

A consequence of evaluating C^{∞}-functions is that we can use the barrier functions to define whether an element of Map(A) satisfies the weak and strong barrier conditions, and so be a “non-standard” solution of (∗).

2.3. Definition. Let A be a filter ring. Then σ ∈ n-Map(A) is an A-solution of (∗) if it satisfies the strong barrier condition for (∗) which means that whenever B(t, x) is a barrier over some [a, b] then there exists F ∈ F such that for all u ∈ F either B(a, σ(u, a)) > 0 or B(b, σ(u, b)) < 0. We say that σ is a semi-bounded solution of (∗) if σ is a solution in n-Map_{0}(A)

The weak barrier condition for (∗) is defined analogously, using ≥, ≤ instead of >, <, in the manner of Definition 1.3.

We will show that the strong and weak barrier conditions are equivalent for semibounded solutions. First we need:

2.4. Definition. A closed filter on R^{m} is maximally closed if it is a maximal element of the family of all proper closed filters on R^{m}.

Remark. Since every closed subset of R^{m} is a zero-set, the closed filters and maximally closed filters defined above coincide with the filters generated by the z-filters and the z-ultrafilters as defined in [1]. Note that a z-ultrafilter does not necessarily generate an ultrafilter, see [1], page 152.

Notation. If F is a proper filter on R^{m} and f : R^{m}→R then L, “the limit of f along F”, denoted by L = lim_{F} f, is defined by the condition that for each ϵ > 0 there exist F ∈ F such that |L − f(u)| < ϵ for all u ∈ F. Clearly L is unique, if it exists.

2.5. Definition. Let A = C^{∞}(R^{m})/I(F) be a non-trivial filter ring and let σ(u, t) ∈ Map_{0}(A) be given. Whenever M is a maximally closed extension of F, we define σ_{M} so that, for each fixed t, σ_{M}(t) = lim_{M} σ(u, t).

Similarly, if σ = (σ_{1}, …, σ_{n}) , then σ_{M} = (σ_{1}_{,M}, …, σ_{n}_{,M}).

A straightforward compactness argument shows that σ_{M}(t) exists and is unique. Note that by semi-boundedness, there exists F ∈ F such that σ is bounded on F × {t}. By the smooth Tietze theorem, we may as well assume that σ(u, t) is bounded when restricted to R^{m} × {t}.

2.6. Lemma. Let A = C^{∞}(R^{m})/I(F) be a non-trivial filter ring and assume that σ(u, t) ∈ n-Map_{0}(A) satisfies the weak barrier condition for (∗). Then, whenever M is a maximally closed extension of F, we have that σ_{M} is a solution of (∗).

Proof. By Theorem 1.5, it suffices to show that σ_{M} satisfies the weak barrier condition. Let B(t, x) be a barrier function over [a, b] and suppose that neither B(a, σ_{M}(a)) ≥ 0 nor B(b, σ_{M}(b)) ≤ 0. So, B(a, σ_{M}(a)) < 0 and B(b, σ_{M}(b)) > 0. Since B is continuous, there exists a neighborhood L_{1} of σ_{M}(a) (in R^{n}) such that B(a, x) < 0 for x ∈ L_{1}, and a neighborhood L_{2} of σ_{M}(b) such that B(b, x) > 0 for x ∈ L_{2}. By the convergence of σ(u, a) to σ_{M}(a) we can find U ∈ M such that σ(u, a) ∈ L_{1} for u ∈ U. Similarly we can find V ∈ M such that σ(u, b) ∈ L_{2} for u ∈ V.. But σ(u, t) satisfies the weak barrier condition, so there exists F ∈ F such that for u ∈ F we have either B(a, σ(u, a)) ≥ 0 or B(b, σ(u, b)) ≤ 0. It follows that U ∩ V ∩ F = ∅ so ∅ ∈ M, which is a contradiction.

2.7. Lemma. Let A = C^{∞}(R^{m})/I(F) be a non-trivial filter ring and let σ(u, t) be in n-Map_{0}(A). Assume that σ_{M} is a solution of (∗) whenever M is a maximally closed extension of F. Then σ(u, t) satisfies the strong barrier condition for (∗).

Proof. Let B(t, x) be a barrier function over [a, b] and let:

E = {u ∈ R^{m} : Either B(a, σ(u, a)) > 0 or B(b, σ(u, b)) < 0}

We have to find an F ∈ F with F ⊆ E. Assume no such F exists. Then we can find a maximally closed extension M of F such that E^{c} ∈ M (where the closed set E^{c} is the complement of E). By hypothesis, σ_{M} is a solution of (∗) and, by Theorem 1.5, it satisfies the strong barrier condition so:

Either B(a, σ_{M}(a)) > 0 or B(b, σ_{M}(b)) < 0

Case 1: Assume B(a, σ_{M}(a)) > 0. By continuity, there exists a neighborhood L_{1} of σ_{M}(a) with B(a, x) > 0 for every x ∈ L_{1}. Since σ(u, a) converges to σ_{M}(a) there exists U ∈ M such with σ(u, a) ∈ L_{1} for all u ∈ U. It follows that U ∩ E^{c} = ∅, a contradiction. Case 2: Assume B(b, σ_{M}(b)) < 0. This case uses a similar contradiction.

2.8. Proposition. Let A = C^{∞}(R^{m})/I(F) be a non-trivial filter ring and let σ(u, t) be an n-tuple of elements of Map_{0}(A). Then the following statements are equivalent:

(1) σ is an A-solution of (∗) (i.e. σ satisfies the strong barrier condition).

(2) σ satisfies the weak barrier condition for (∗).

(3) Whenever M is a maximally closed extension of F, then σ_{M} is a solution of (∗).

Proof. (1)⇒(2): Obvious. (2)⇒(3): By Lemma 2.6. (3)⇒(1): By Lemma 2.7.

2.9. Definition. Let A = C^{∞}(R^{m})/I(F). Then I_{PtCn}, the ideal of pointwise convergence, is the set of all α ∈ Map_{0}(A) such that for every fixed real t_{0} and every ϵ > 0 there exists F ∈ F with |α(u, t_{0})| < ϵ whenever u ∈ F. It follows by semi-boundedness that I_{PtCn} is an ideal of Map_{0}(A) and α ∈ I_{PtCn} iff α_{M} = 0 for every maximally closed extension M of F.

2.10. Proposition. Assume σ ∈ n-Map_{0}(A) for some filter ring A. Then σ represents an A-solution of (∗) iff σ + α does for every n-tuple α of members of I_{PtCn}.

Proof. Straightforward, using Proposition 2.8.

Perturbed differential equations. A differential equation, (∗_{u}) dx/dt = V (u, t, x), is said to be a perturbation of (∗) if V (u, t, x) equals W (t, x) for a particular value of the parameter u. More generally, we will allow V (u, t, x) to approach W (t, x) for a “limiting value” of u in the sense of the following definition. In this case, parameterized solutions of dx/dt = V (u, t, x) not only approximate solutions of (∗), they actually form an A-solution of (∗) for suitable A.

2.11. Definition. The differential equation (∗_{u}) dx/dt = V (u, t, x), with parameter u ∈ R^{m} is a perturbation of (∗) with respect to a closed filter F on R^{m} if W (t, x) = lim_{F} V (u, t, x) for fixed t and x.

2.12. Theorem. Suppose that (∗) is perturbed by an equation (∗_{u}) dx/dt = V (u, t, x) with respect to F a filter on R^{m}. If the semi-bounded σ(u, t) is a parameterized solution of (∗_{u}), then σ represents an A-solution of (∗) where A = C^{∞}(R^{m})/I(F).

Proof. Let B(t, x) be a barrier function on [a, b] for (∗). By semi-boundedness, we may as well assume that for some M, we have ∥σ(u, t)∥ ≤ M for (u, t) ∈ R^{m} × [a, b]. Let:

K = {(t, x) : t ∈ [a, b], ∥x∥ ≤ M, B(t, x) = 0}

Clearly K is compact and, since ∂B/∂t + ∑ ∂B/∂x_{i}W_{i} < 0 for all (t, x) ∈ K, there is a maximum value m of ∂B/∂t + ∑ ∂B/∂x_{i}W_{i} (on K) and m is negative. Since each ∂B/∂x_{i} is bounded on K there exists ϵ > 0 such that if ∥V (u, t, x) − W (t, x)∥ < ϵ then ∂B/∂t + ∑ ∂B/∂x_{i}W_{i} < m/2 < 0 on K. By hypothesis, each (t_{0}, x_{0}) ∈ K has a neighborhood L with some F ∈ F for which V (u, t, x) is within ϵ of W (t, x) for (t, x) ∈ L and u ∈ F. By covering K with finitely many of these L’s and taking the corresponding finite intersection of the F ’s we can find F_{0} ∈ F such that V (u, t, x) is within ϵ of W (t, x) for (t, x) ∈ K and u ∈ F_{0}. Therefore, for each u ∈ F_{0} we see that B(t, x) is a barrier function for the perturbed equation (∗_{u}) and since σ(u, t) is a solution of this equation, we see that σ satisfies the required barrier condition to be an A-solution of (∗).

2.13. Corollary. Let σ(u, t) be a solution of (∗) for each fixed value of u. For example, σ(u, t) might be a solution satisfying an initial condition which depends on u. Then σ(u, t) represents an A-solution of (∗) whenever σ(u, t) ∈ Map_{0}(A).

Proof. Let V (u, t, x) = W (t, x) for all u.

The associated map and non-standard solutions. As shown in section 1, a standard solution of (∗) is an n-tuple of functions f = (f_{1}, …, f_{n}), with f_{i} : R→R, which satisfies the weak barrier conditions. This same definition can be interpreted in the internal language of a smooth topos and, as will be shown in the next section, the resulting notion relates to A-solutions.

One advantage of the internal definition is that we can discuss solutions of (∗) in which R is replaced by a non-standard version of the reals. We will illustrate this possibility by working with ultrapowers of R, which can be used to show that some differential equations have non-standard cyclic-like solutions with transfinite periods.

2.14. Definition. Let N denote the positive integers and let U be an ultrafilter on N. We say that two sequences, x, y ∈ RN are equivalent modulo U iff {i : x_{i} = y_{i}} ∈ U. Then R_{U}, the resulting set of equivalence classes is called an ultrapower of R.

The above are the only ultrapowers we will consider but, in general, the set N can be replaced by any index set. It is well-known that any ultrapower of R has all the relations and operations that R has and satisfies the same first-order properties. Such ultrapowers are also filter rings, as shown by:

2.15. Lemma. Every ultrapower of R, arising from an ultrafilter on N, is a filter ring.

Proof. The ultrafilter U on N is clearly the base of a closed filter on R, which we will also denote by U. It is then readily seen that R_{U} = C^{∞}(R)/I(U).

2.16. Definition. Let R_{U} and R_{V} be ultrapowers of R. Then an n-tuple f = (f_{1}, …, f_{n}) with f_{i} : R_{U}→R_{V} is a non-standard solution of (∗) if the weak barrier conditions are satisfied.

2.17. Definition. If σ(u, t) ∈ Map(A), then the associated map σ̂ : A→A is defined so that if α ∈ A is represented by α(u), then σ̂(α) is represented by σ(u, α(u)). Similarly, if σ(u, t) ∈ n-Map(A) with σ = (σ_{1}, …, σ_{n}) , then σ̂ = (σ̂_{1}, …, σ̂_{n})

2.18. Proposition. Let R_{U} be an ultrapower of R and let σ ∈ n-Map(R_{U}) be given. Then σ is an R_{U}-solution of (∗) iff σ̂ is a non-standard solution of (∗).

Proof. σ is an R_{U}-solution iff there exists U ∈ U such that for u ∈ U we have either B(a, σ(u, a)) ≥ 0 or B(b, σ(u, b)) ≤ 0. But since U is an ultrafilter it is a prime filter and this means that the above is true iff either {u|B(a, σ(u, a))} ≥ 0 is in U or {u|B(b, σ(u, b))} ≤ 0 is in U. This is easily seen to be equivalent to saying that σ̂ is a non-standard solution of (∗). (Note: Not every non-standard solution is the associated map of a member of n-Map(R_{U}).)

Autonomous equations and the pseudo-cycle at ∞. An equation of the form dx/dt = W (x), where W depends only on x and not on t, is said to be autonomous. In this case, if f is a solution of (∗) then f(u + t) is also a solution for any constant u.

We will assume that f is a solution of the autonomous equation (∗) and is bounded meaning that the set {f(t) : t ∈ R} is a bounded subset of R^{n}. We aim to study the limit points L^{+}(f), or points that f(t) gets close to as t→∞ (see below).

We will investigate the behavior of f(t) as t→∞ by, in effect, considering a solution of the form f(γ + t) where γ is a transfinite constant. In order for this to make sense, we need to replace f by a non-standard solution with domain R_{U} where U is a free (i.e. non-principal) ultrafilter, so that R_{U} has transfinite elements.

Construction. (1) We are given f : R→R^{n} a solution with bounded range of the autonomous differential equation (∗). (As will be noted below, this construction extends to the case where we only assume that {f(t)|t > 0} is a bounded subset of R^{n}.)

(2) We assume that (∗) satisfies a uniqueness condition which means that any two ordinary solutions which agree at a single value of t must then agree at all (finite) values of t. Mild conditions on W, such as being a C^{1} function, are sufficient to guarantee the uniqueness condition, see [2], page 542.

(3) We let L^{+}(f) be the set of all points P ∈ R^{n} such that f(t) comes arbitrarily close to P for arbitrarily large values of t. So P ∈ L^{+}(f) iff for every ϵ > 0 and every t_{0} there exists t > t_{0} such that ∥f(t) − P ∥ < ϵ. If n = 2, the Poincaré-Bendixson theorem says that, barring any limit equilibrium points, the set L^{+}(f) is a cyclic orbit. But L^{+}(f), for n > 2, can be a “strange attractor” with weird properties, see [2].

(4) We let U be any free ultrafilter on N and let R_{U} be the resulting filter ring. By Corollary 2.13, we see that σ(u, t) = f(u + t) is an R_{U}-solution of (∗).

(5) Let σ̂ be the resulting non-standard solution of (∗) as in Proposition 2.18. Note that if τ ∈ R_{U} is represented by the sequence (τ_{1}, …, τ_{i}, …) then σ̂(τ) is represented by the sequence (σ(1, τ_{1}), …, σ(i, τ_{i}), …).

(6) We let Lim_{U} σ̂ : R_{U}→R^{n} be the function obtained by taking limits along U. Since the range of f is bounded, this limit exists and is unique. Since limits preserve the weak barrier conditions, it follows that Lim_{U} σ̂ is still a non-standard solution of (∗).

(7) We let R_{U}^{+} denote the positive elements of R_{U}. We also let \overline{f} : R_{U}^{+}→R^{n} be the restriction of Lim_{U} σ̂ to R_{U}^{+}. Note that \overline{f} is essentially the limit of f(γ + t) where γ is represented by the identity sequence (1, 2, …, i, …). In effect this map shows us what happens to f(u + t) as u “goes to infinity” along U.

2.19. Lemma. Let (∗) and f be as above. Assume that (∗) satisfies the uniqueness condition. Let γ = {γ_{k}} be such that γ_{k}→∞ (along U) and let ϵ = {ϵ_{k}} be such that Lim_{U}(ϵ_{k}) = 0. Then Lim_{U}f(γ_{k}) = Lim_{U}f(γ_{k} + ϵ_{k}).

Proof. Since f(t) is bounded for t > 0, we see that f′(t) = W (f(t)) is also bounded for t > 0. So there exists M > 0 such that ∥f′(t)∥ < M for all t > 0. (Recall that we are using the ℓ_{∞} norm on R^{n}.) From the mean-value theorem we now get that ∥f(γ_{k} + ϵ_{k}) − f(γ_{k})∥ < M∥ϵ_{k}∥ and the lemma follows.

2.20. Theorem. Let (∗) be an autonomous equation where each W_{i}(x) is a C^{∞}- function. Let f(t) be a bounded solution for (∗). Let \overline{f} : R_{U}^{+}→R^{n} and L^{+}(f) be as defined above. Then:

(1) The range of \overline{f} is precisely L^{+}(f).

(2) \overline{f} is “locally cyclic” in the sense that for each α ∈ R_{U}^{+} there exists ρ ∈ R_{U}^{+} such that \overline{f}(α) = \overline{f}(α + ρ).

(3) If \overline{f}(α) = \overline{f}(α + ρ), then \overline{f}(β) = \overline{f}(β + ρ) whenever β − α is bounded.

(4) \overline{f} imposes no order on its range L^{+}(f) in the sense that if \overline{f}(α) = P_{1} and P_{2} is any other point in L^{+}(f) then there exists ρ ∈ R_{U}^{+} such that \overline{f}(α + ρ) = P_{2}.

Proof. (1) It is obvious that the range of \overline{f} is contained in L^{+}(f). Suppose P ∈ L^{+}(f). Then we can find {t_{k}} such that P is the classical limit of f(t_{k}) by choosing t_{k} > k + 1 with f(t_{k}) within 1/k of P. Let τ = \{τ_{k}\} where τ_{k} = t_{k} − k then P = \overline{f}(τ).

(2) Let α = \{α_{k}\} ∈ R_{U}^{+} be given and let \overline{f}(α) = Lim_{U}f(k + α_{k}) = P. Clearly we can find an increasing sequence of positive integers m_{k} such that ∥P − f(m_{k} + α_{m_{k}})∥ < 1/k. We can clearly further require that m_{k} > α_{k} + k. Define ρ = {ρ_{k}} where ρ_{k} = (m_{k} + α_{m_{k}}) − (k + α_{k}). Then it easily follows that \overline{f}(α + ρ) = \overline{f}(α).

(3) Assume \overline{f}(α + ρ) = \overline{f}(α) and that β − α = λ = \{λ_{k}\} is bounded. This means that Lim_{U}(λ_{k}) = t_{0} where t_{0} ∈ R. Write λ_{k} = t_{0} + ϵ_{k} then Lim_{U}(ϵ_{k}) = 0.

For each t, let g(t) = Lim_{U}f(k + α_{k} + t). Since, for fixed k, f(α_{k} + k + t) is a solution of (∗), it follows from Proposition 1.7 that g(t) is a solution of (∗) (in the usual sense, as a map from R to R^{n}). Similarly, let h(t) = Lim_{U}f(k + α_{k} + ρ_{k} + t). By the same argument, h(t) is also a solution of (∗). But g(0) = h(0). Since W is C^{∞}, it follows that (∗) satisfies the uniqueness condition, see [2], and therefore g(t) = h(t), ∀t ∈ R.

Now, \overline{f}(β) = Lim_{U}(k + β_{k}) = Lim_{U}(k + α_{k} + λ_{k}) = Lim_{U}(k + α_{k} + t_{0} + ϵ_{k}) = Lim_{U}(k + α_{k} + t_{0}) (by Lemma 2.19). But this is g(t_{0}). A similar calculation shows that \overline{f}(β + ρ) = h(t_{0}), so the proof follows from the fact that g(t_{0}) = h(t_{0}).

(4) Since \overline{f}(α) = P_{1}, we can write P_{1} = Lim_{U}f(k + α_{k}). Since P_{2} ∈ L^{+}(f), we can, for each k, find t_{k} > k + α_{k} such that f(t_{k}) is within 1/k of P_{2}. Let ρ_{k} = t_{k} − (k + α_{k}), then ρ = {ρ_{k}} has the required property.

Remarks. (1) Part (4) of the above theorem says, in effect, that “everything comes around again, infinitely often”. This is illustrated in Example 4.8.

(2) If W (t, x) is not sufficiently smooth, we can usually use C^{∞}-approximation and Theorem 2.12.

(3) The above theorem applies even if we only assume {f(t) : t > 0} is bounded. In this case, let {f(t) : t > 0} be contained in a closed ball B_{1} in R^{n}. Let B_{2} be a larger closed ball such that B_{1} is entirely contained in the interior of B_{2}. Then modify the definition of W (x) so that it has the same values for x ∈ B_{1} but is redefined on the boundary of B_{2} so that as we trace f(t) for negative values of t it cannot leave B_{2}. (Since t is going backwards, this means redefining W as, say, the unit vector normal to the surface of B_{2} and pointing outward.) We can redefine W by the smooth Tietze theorem and f will exist (as it cannot “go off to ∞”) and f(t) will be unchanged for t > 0 as W is unchanged on B_{1}.

(4) Note that the above theorem has no hypothesis about the absence of equilibrium points, so, for n = 2, it applies even when Poincaré-Bendixson might not because of limiting equilibrium points. In this case, L^{+}(f) may consist of equilibrium points and orbits between them (that is, orbits which tend to equilibria as t→ ± ∞). The orbit based on R_{U}^{+} hits all of these equilibria and traverses all of the orbits between them. This example shows there is no hope of proving that once \overline{f}(α) = \overline{f}(α+ρ), then \overline{f}(β) = \overline{f}(β+ρ) for all β. When the non-standard orbit hits an equilibrium point at, say, \overline{f}(α), then \overline{f}(α) = \overline{f}(α +ρ) for any finite ρ, and, by (3) of the above theorem, any such finite ρ works as a period until sometime “infinitely later” when, by (4) of this theorem, the orbit must leave the equilibrium point to hit the other points in L^{+}(f). See Example 4.8.

Change of variables. We conclude this section by examining the behavior of (∗) under a change of variables. We use the following notation:

(1) Members of R^{n} will be thought of as column vectors, that is, as n × 1 matrices.

(2) If f : R^{n}→R^{m} is a C^{1}-function, then Df is the m × n matrix with ∂f_{i}/∂x_{j} in row i, column j. Recall that the chain rule says that if h = fg then Dh = DfDg (matrix product).

(3) If x : R→R^{n} is a solution of (∗) and if W is regarded as a column vector of functions, then (∗) can be written in the form:

(∗_{1}) Dx = W

These conventions make the following easier to state:

2.21. Theorem. Let (∗_{1}) be as above and consider the change of the dependent variable from x to y suggested by the equations:

y = ϕ(x) and x = θ(y)

where ϕ, θ are C^{∞}-inverses of each other. Then (∗_{1}) is transformed into:

(∗_{2}) Dy = DϕW (t, θ(y)).

For any filter ring A, there is a one-to-one correspondence between A-solutions of (∗_{1}) and of (∗_{2}) under which the (∗_{1})-solution σ(u, t) corresponds to the (∗_{2})-solution ϕ(σ(u, t)). This correspondence takes semi-bounded solutions to semi-bounded solutions.

Proof. It is readily shown that that B(t, x) is a barrier function for (∗_{1}) iff B(t, θy) is a barrier for (∗_{2}). The result then follows. The preservation of semi-bounded solutions is obvious and follows from Proposition 2.8 anyway.

Having proven this theorem, we immediately use it to obtain a more general version:

2.22. Theorem. More generally, let (∗_{1}) be as above and consider the change of the dependent variable from x to y suggested by the equations:

y = ϕ(t, x) and x = θ(t, y)

where ϕ, θ are C^{∞}-inverses of each other. Then (∗_{1}) is transformed into:

(∗_{2}) Dy = ∂ϕ/∂t + DϕW (t, θ(y)).

For any filter ring A, there is a one-to-one correspondence between A-solutions of (∗_{1}) and of (∗_{2}) under which the (∗_{1})-solution σ(u, t) corresponds to the (∗_{2})-solution ϕ(t, σ(u, t)). This correspondence takes semi-bounded solutions to semi-bounded solutions.

Proof. This is really the same theorem as the above if we imagine using an additional variable x_{0} which plays the role of t. So we redefine W (x_{0}, x) as W (t, x) and W_{0} = 1. The equations of the previous theorem then reduce to the ones given above. We also have to impose an initial condition that x_{0}(0) = 0 and it is easily verified that the transformations preserve solutions which satisfy this condition.(Alternatively, we could argue as in the previous proof.)

2.23. Theorem. Let (∗_{1}) be dx/dt = W (t, x) and consider the change of the independent variable from t to s suggested by the equations:

s = ϕ(t) and t = θ(s)

where ϕ, θ are C^{∞}-inverses of each other. Then (∗_{1}) is transformed into:

(∗_{2}) dx/ds = W (θ(s), x)θ′(s).

For any filter ring A, there is a one-to-one correspondence between A-solutions of (∗_{1}) and of (∗_{2}) under which the (∗_{1})-solution σ(u, t) corresponds to the (∗_{2})-solution σ(u, θs)). This correspondence takes semi-bounded solutions to semi-bounded solutions.

Proof. First, consider the transformation s = −t and t = −s. Then B(t, x) is a barrier function for (∗_{1}) over [a, b] iff −B(s, x) is a barrier function over [−b, −a] for (∗_{2}) and the result follows in this special case. In general, notice that ϕ′(t) can never be 0, as ϕ has a differentiable inverse. We may as well assume that ϕ′(t) > 0 for all t (otherwise, first change t to −t, using the above case.) It is then readily shown that B(t, x) is a barrier function for (∗_{1}) over [a, b] iff B(θs, x) is a barrier for (∗_{2}) over [ϕa, ϕb] (and conversely). The result follows. The preservation of semi-boundedness is obvious.

3. Solution Manifolds in a Smooth Topos

Our goal is to define and examine the “manifold” of all solutions of the system (∗) using a generalized notion of manifold which, in effect, allows for non-standard solutions. Typically a smooth topos is regarded as a category of generalized manifolds. The advantage of working in a topos is that it has good categorical properties, such as the existence of power objects, M^{N}, which conceptually is the “manifold” of all smooth maps from M to N. Also, in a topos, we can define subobjects using the internal language, as discussed below.

By a smooth topos we mean a topos which fully contains the category M of C^{∞}- manifolds and smooth (i.e. C^{∞}) maps. We follow the approach in [4] and extend M in several stages. First, each manifold M ∈ M gives rise to C^{∞}(M), the finitely presented C^{∞}-ring of all smooth maps from M to R. This embeds M fully into V the dual of f.g. reduced C^{∞}-rings (where “f.g.” means finitely generated and “reduced”, defined algebraically in [5], is equivalent to being a filter ring. See also [4] where these rings are described as being of the form C^{∞}(R^{m})/I where I is a C^{∞}-radical ideal.)

By PreSh(V) we mean the category of functors from V^{op} to Sets, equivalently, the category of functors from filter rings to Sets. Note that if A = C^{∞}(R^{m})/I(F) and B = C^{∞}(R^{k})/I(G), then a C^{∞}-homomorphism from h : A→B is given by a smooth map η : R^{k}→R^{m} for which η^{−1}(F) ∈ G whenever F ∈ F. Then h is defined by h(α)(u) = α(η(u)), see [4].

A presheaf is a sheaf with respect to the finite open cover topology if it maps every covering sieve to a limit diagram (see [4], pages 350 and 364 for details). This topology is subcanonical which means that every filter ring A determines a sheaf [A, ], which is the representable hom functor which assigns hom[A, B] to the filter ring B.. Following [4], we let V_{fin} denote the topos of sheaves. If we trace the embedding of M into V_{fin}, we see that the real line R is mapped to the underlying set functor from filter rings to Sets.

The internal language. As with any topos, V_{fin} has an internal language, see [4] pages 353-361. We use this language to define subsheaves much as subsets can be defined by conditions. In terms of this language, R is a ring with a compatible order. This follows since for each object A, the set R(A) = A has such a structure, and this structure is preserved by the maps. The ring structure on A is obvious and, for α, β ∈ A, represented by α(u), β(u), we define α < β iff {u : α(u) < β(u)} ∈ F.

We next define x, y in R to be infinitesimally close if for each ordinary positive integer n, we have −1/n < (x − y) < 1/n. It is important to note that here x, y are internal variables associated with the sheaf R, while n is an external variable (or an ordinary integer). We could, alternatively, use the natural number object, but we will not go into that. In summary:

(1) The relation < has been defined on R. (It is a subsheaf of R × R.)

(2) The relation of being infinitesimally close on R (another subsheaf of R × R) has been defined by the conditions −1/n < (x−y) < 1/n for each ordinary positive integer n. By a convenient abuse of notation, we sometimes write this condition as |x−y| < 1/n, even though there is no actual absolute value operation from the sheaf R to itself. Similarly, we define infinitesimally close on (R)^{n} by using the projections to R. The defining conditions can be abbreviated to ∥x − y∥ < 1/n for each ordinary positive integer n.

(3) The object R in V_{fin} lacks nilpotents, so the elegant Kock-Lawvere definition of the derivative is not available. Instead we define solutions to (∗) using the barrier conditions. This is, in effect, a non-standard analysis approach and it is useful for analyzing stability when we move infinitesimally away from an equilibrium point. See Examples 4.1, 4.2.

3.1. Definition. The solution manifold for (∗), denoted by Sol, is defined internally as the subobject of all f ∈ (R^{n})R for which B(a, f(a)) > 0 or B(b, f(b)) < 0 whenever B is an ordinary barrier function over [a, b] for (∗).

It will follow from the results of section 2 and the lemma below that Sol is the functor which assigns to A the set of all A-solutions of (∗). We need to describe the functor RR from filter rings to Sets. Let A = C^{∞}(R^{m})/I(F) be a filter ring. Then, by the Yoneda lemma, RR(A) = n.t.([A, ], RR), where “n.t.” stands for the set of all natural transformations, and “=” means “naturally isomorphic”.

3.2. Lemma. Let A = C^{∞}(R^{m})/I(F). Then RR(A) = Map(A) as defined in section 2. It follows that the functor (R^{n})R is n-Map(A) and that Sol(A) is the set of all A-solutions of (∗).

Proof. The main steps are that RR(A) is naturally isomorphic to Map(A) are: n.t.([A, ], RR) = n.t.(R × [A, ], R) = n.t.([Map(A), ], R) = Map(A). The remaining details are straightforward, but note we do have to use a covering

argument: Let σ(u, t) be an A-solution for some A and let B be a barrier over [a, b]. Then {U, V } is a finite cover where U = {u : B(a, σ(u, a)) > 0} and V = {u : B(b, σ(u, b)) < 0}. We can now readily show that σ ∈ Sol(A).

One drawback of the topos V_{fin} is that the real line (i.e. the underlying set functor, or the object which corresponds to the manifold of reals) is not Archimedean. For technical reasons (perhaps because our techniques are not good enough) we need to deal with the “bounded reals”.

3.3. Definition. The bounded reals, R_{0}, is defined internally as the subobject of all x ∈ R for which there exists an ordinary integer n with −n < x < n.

3.4. Definition. (1) The sheaf of R_{0}-preserving maps is defined internally as the subsheaf of RR of all f ∈ RR for which f(t) ∈ R_{0} whenever t ∈ R_{0}.

(2) The submanifold of semi-bounded solutions for (∗), denoted by Sol_{0}, is defined internally as the subobject of all f ∈ Sol which are R_{0}-preserving (meaning that all components of f are R_{0}-preserving).

(3) The standardization manifold of (∗), denoted by Stnd, is obtained internally from Sol_{0} by setting two solutions, f and g, to be equivalent if f(t) is infinitesimally close to g(t) for each ordinary real number t.

We will show that the sheaf of R_{0}-preserving maps coincides with the functor Map_{0} of semi-bounded maps, that Sol_{0} is the functor which assigns to A the set of all semibounded A-solutions of (∗), and that Stnd(A) = Sol_{0}(A)/I_{PtCn}. To do this, we need to examine R_{0}, R^{R^{0}} and (R)R_{0}_{0}.

3.5. Definition. Let J be the set of all closed, bounded intervals of the ordinary reals. Let J ∈ J be given. Let A be a C^{∞}-ring. Then a ∈ A is J-bounded if λ(a) = 0 whenever λ ∈ C^{∞}(R) vanishes on J. We way that a ∈ A is bounded if there exists J ∈ J for which a is J-bounded.

3.6. Lemma. The object R_{0}, regarded as a functor from V^{op} to Sets, has the following properties:

(1) R_{0}(A) is the set of all bounded elements of A.

(2) R_{0} is the filtered colimit of the intervals J, for J ∈ J. (Recall that each such J is identified with the representable functor [C^{∞}(J), ].)

(3) Let A = C^{∞}(R^{m})/I(F) and let J ∈ J. Define A_{J} = C^{∞}(R^{m} × J)/I(π^{∗}F). The functors R_{0} × [A, ] and Colim[A_{J}, ] are then naturally equivalent.

Proof. (1) If x ∈ A satisfies −n < x < n, then x is obviously [−n, n]-bounded. Conversely, if x is [−m, m]-bounded, for any m > n, then x satisfies −n < x < n. It is easy to show that the set of all bounded elements is functorial and a sheaf and the result readily follows.

(2) Each x ∈ A is the image of the identity under a unique homomorphism h : C^{∞}(R)→A. It follows that x is J-bounded iff h(λ) = λ(x) = 0 whenever λ is in the kernel of the restriction map from C^{∞}(R) to C^{∞}(J) and the result follows.

(3) R_{0} × [A, ] = (Colim[J, ]) × [A, ] = Colim([J, ] × [A, ]). It is easily shown that [J, ] × [A, ] = [A_{J}, ] and the result follows.

The following lemma describes R^{R^{0}}.

3.7. Lemma. Let A = C^{∞}(R^{m})/I(F). An element of R^{R^{0}}(A) is then represented by a filtered family {α_{J} : J ∈ J } where α_{J} ∈ C^{∞}(R^{m} × J) and where the family is filtered in the sense that if J ⊆ K then there exists F ∈ F such that α_{J}(u, t) = α_{K}(u, t) whenever u ∈ F and t ∈ J.

Moreover, if α ∈ RR(A) is represented by α(u, t) ∈ C^{∞}(R^{m} × R) then the restriction of α to R^{R^{0}} is represented by the family of restrictions, {α_{J}}, of α to R^{m} × J.

Proof. R^{R^{0}}(A) = n.t.([A, ], R^{R^{0}}) = n.t.(R_{0} × [A, ], R) = n.t.(Colim[A_{J}, ], R) = lim n.t.([A_{J}, ], R) = lim A_{J} which leads to the above results.

The question arises as to when a filtered family {α_{J}} can be patched together to be equivalent to the restrictions of some α ∈ C^{∞}(R^{m}×R). The answer is “always”, as shown below. Although this proposition is not needed in our development, it seems interesting on its own.

3.8. Proposition. The restriction map from RR to R^{R^{0}} is onto (but not always oneto-one).

Proof. Let A = C^{∞}(R^{m})/I(F) be given. Let α_{J} ∈ C^{∞}(R^{m} × J) be filtered (as in the above lemma). We need to find α ∈ C^{∞}(R^{m} ×R) such that the restriction of α to R^{m} ×J is equivalent to α_{J} for each J ∈ J. In other words, there must exist, for each J, an F ∈ F such that α(u, t) = α_{J}(u, t) for all u ∈ F and t ∈ J.

For each positive integer n, let α_{n} = α_{[−n,n]}. Let F_{n} ∈ F be a closed set such that α_{n}(u, t) = α_{n+1}(u, t) for u ∈ F_{n} and |t| ≤ n. We may as well assume that the F_{n}’s are nested, or that F_{n} ⊇ F_{n+1} for all n, otherwise replace F_{n} by F_{1} ∩ … ∩ F_{n}. We observe that if n < m and u ∈ F_{m} then α_{n}(u, t) = α_{m}(u, t) for |t| ≤ n.

For each u ∈ R^{m} let deg(u) = max{k : u ∈ F_{k}}. If u ∉ F_{1} let deg(u) = 0 and if u ∈ F_{k}, ∀k let deg(u) = ∞. (There need not be any u with deg(u) = ∞.) Let:

Q = {(u, t) ∈ R^{m} × R : |t| + 2 ≤ deg(u)}

We claim that Q is closed. Suppose (u_{0}, t_{0}) ∉ Q. We need to find a neighborhood of (u_{0}, t_{0}) which misses Q. Clearly |t_{0}| + 2 > deg(u_{0}) since (u_{0}, t_{0}) ∉ Q. Let n = deg(u_{0}) then u_{0} ∉ F_{n+1} and |t_{0}| > n−2. Since F_{n+1} is closed, the set {(u, t) : u ∉ F_{n+1}, |t| > n−2} is the required neighborhood of (u_{0}, t_{0}).

Now define α(u, t) for (u, t) ∈ Q as α_{n}(u, t) provided that u ∈ F_{n}, |t| ≤ n. Such an n must exist, for example, n = min{k : |t| < (k − 0.5)}, as can be readily verified. On the other hand, if u ∈ F_{n}, |t| ≤ n and u ∈ F_{m}, |t| ≤ m then α_{n}(u, t) = α_{m}(u, t) because if n < m this follows from the definition and nested property of F_{m}.

It remains to show that α can be smoothly extended from Q to all of R^{m}×R, as it then readily follows that α has the required restrictions. By using partitions of unity, it suffices to find a neighborhood of each (u, t) ∈ Q to which α can be smoothly extended. But given (u, t) ∈ Q let n = min{k : |t| < (k−0.5)}, let U be the set where (n−1.4) < |t| < (n−0.5), then α on Q ∩ U extends smoothly to α_{n} on U.

That the restriction from RR to R^{R^{0}} is not always one-to-one follows from the example in the following remark.

Remark. Let A = C^{∞}(R)/I(F) where F is the “filter at infinity” meaning the filter generated by sets of the form [u, ∞). We can readily find α, β ∈ Map(A) having equivalent sets of restrictions to members of J. For example, we can let α be 0 on precisely {(u, t) : |t| ≤ u} while β is identically 0.

On the other hand, if A = C^{∞}(R^{m}) or even C^{∞}(M) for M a classical C^{∞}-manifold, then restriction from RR(A) to R^{R^{0}}(A) is one-to-one and onto, as is easily seen.

We need to describe RR_{0}_{0} to show that the semi-bounded maps correspond to those which are “R_{0} preserving”. First we need:

3.9. Lemma. Let A = C^{∞}(R^{m})/I(F) be a given filter ring. Then RR_{0}_{0}(A) ⊆ R^{R^{0}}(A) is represented by the filtered families {α_{J}} ∈ R^{R^{0}}(A) for which each α_{J} is bounded, meaning that for each J there exists B > 0 and F ∈ F such that |α_{J}(u)| < B for all u ∈ F.

Proof. Using the same method as in the proofs of 3.2 and 3.7, we see that RR_{0}_{0}(A) = lim n.t.([A_{J}, ], R_{0}) = lim(R_{0}(A_{J})) which readily leads to the stated result.

3.10. Proposition. α ∈ RR(A) is R_{0}-preserving iff α is represented by α(u, t) ∈ C^{∞}(R^{n} × R) which is semi-bounded with respect to F.

Proof. Clearly, α is R_{0}-preserving iff the restriction of α to R^{R^{0}}(A) is actually in RR_{0}_{0}(A). But this restriction is represented by the set of restrictions, {α_{J}} of α to R^{n} × J for J ∈ J. It is readily shown that this set of restrictions satisfies the condition of the above lemma iff α is semi-bounded with respect to F.

Behavior near a given equilibrium point. In what follows, we assume that (∗) is autonomous, meaning it is of the form dx/dt = W (x) where W depends only on x, not on t. We further assume that \overline{x} is an equilibrium point of (∗) meaning that W (\overline{x}) = 0, so the function which is constantly equal to \overline{x} is a solution of (∗). The stable solutions are those which, in the internal language, stay infinitesimally close to \overline{x}. We also define the asymptotically stable solutions, even though the definition does not seem readily expressible in the internal language.

3.11. Definition. The submanifold of stable solutions of (∗), denoted by Stab, (we assume \overline{x} and (∗) are understood) is defined internally as the subfunctor of Sol_{0} of all solutions f with f(t) infinitesimally close to \overline{x} for each ordinary non-negative t.

3.12. Definition. The submanifold of asymptotically stable solutions of (∗), denoted by Asym, is defined as the following subfunctor of Stab: If A = C^{∞}(R^{m})/I(F) then Asym(A) is the set of all σ(u, t) ∈ Stab(A) for which ∃F ∈ F having the property that ∀ϵ > 0, ∃t_{0} such that u ∈ F and t > t_{0} imply ∥σ(u, t) − \overline{x}∥ < ϵ. It readily follows that Asym is functorial, and a sheaf.

If A = C^{∞}(R^{m})/I(F), then Stab(A) is readily shown to be the set of all σ(u, t) ∈ Sol_{0}(A) such that for every ϵ > 0, ∃F ∈ F with ∥σ(u, t) − \overline{x}∥ < ϵ whenever u ∈ F and t ≥ 0.

Classically, we assume that the smooth map ϕ(x_{0}, t) is the unique solution of (∗) with initial value x_{0}. Then \overline{x} is a stable equilibrium point if for every ϵ > 0 there is a δ > 0 such that ∥x_{0} − \overline{x}∥ < δ implies ∥ϕ(x_{0}, t) − \overline{x}∥ < ϵ for all t ≥ 0. If, in addition, there exists r > 0 such that ∥x_{0} − \overline{x}∥ < r implies ∥ϕ(x_{0}, t) − \overline{x}∥→0 as t→∞, then \overline{x} is asymptotically stable. See [2]. The connection between these conditions is given by:

3.13. Lemma. Let ϕ(x_{0}, t) be as above and let A = C^{∞}(R^{n})/I(F) where F is the filter of neighborhoods of \overline{x}. Note that ϕ(x_{0}, t) is clearly semi-bounded, as F has bounded members. Then:

(1) ϕ(x_{0}, t) ∈ Stab(A) iff \overline{x} is a stable equilibrium point.

(2) ϕ(x_{0}, t) ∈ Asym(A) iff \overline{x} is asymptotically stable.

Proof. (1) Suppose \overline{x} is stable. Let ϵ > 0 be given. Then, by stability, there exists δ > 0 such that ∥x_{0} − \overline{x}∥ < δ implies that, for all t ≥ 0, we have ∥ϕ(x_{0}, t) − \overline{x}∥ < ϵ. Let F ∈ F be the δ-neighborhood of \overline{x} then F has the required property which shows that ϕ ∈ Stab(A). The proof of the converse is similar.

(2) If \overline{x} is asymptotically stable there exists r > 0 such that ∥x_{0} − \overline{x}∥ < r implies ∥ϕ(x_{0}, t) − \overline{x}∥→0 as t→∞. Let F be the r-neighborhood about \overline{x} and then F ∈ F has the required property. Again, the converse is similar.

Stabilization. Given A and σ ∈ Sol_{0}(A), we find the “best extension” ν : A→A_{σ} of A in which σ becomes stable, meaning that Sol_{0}(ν)(σ) ∈ Stab(A_{σ}).

3.14. Definition. Given σ ∈ Sol_{0}(A), then h : A→B makes σ stable if Sol_{0}(h)(σ) ∈ Stab(B).

3.15. Theorem. Let σ ∈ Sol_{0}(A). There exists ν : A→A_{σ} which makes σ stable and such that whenever h : A→B makes σ stable then h factors as h = h_{0}ν for a unique h_{0}.

Proof. For each ϵ > 0 let

Eϵ = \{u ∈ R^{m} : ∥σ(u, t) − \overline{x}∥ ≤ ϵ, ∀t ≥ 0\}

Let F_{σ} be the filter generated by adding all sets Eϵ to F. Let A_{σ} = C^{∞}(R^{m})/I(F_{σ}) (If F_{σ} is the improper filter, then the C^{∞}-ring A_{σ} is trivial.) Let ν : A→A_{σ} be the obvious homomorphism, (induced by the identity on R^{m}). It is then readily shown that ν makes σ stable. Suppose h : A→B also makes σ stable, where B = C^{∞}(R^{k})/I(G). Let h be determined by a smooth map η : R^{k}→R^{m} for which η^{−1}(F) ∈ G whenever F ∈ F. Then, as h makes σ stable, there exists Gϵ ∈ G such that ∀t ≥ 0, v ∈ Gϵ we have ∥σ(ηv, t) − \overline{x}∥ ≤ ϵ. This implies that Gϵ ⊆ η^{−1}(Eϵ) which shows that η defines a map h_{0} : A_{σ}→B. The remaining details are straightforward.

The topos of presheaves. The definition of Sol depended on the fact that we were using the finite open cover topology as indicated in the proof of Lemma 3.2. If we interpreted the same condition in the topos of presheaves, we would get a presheaf which would miss many solutions. But, even in this topos, we could define Sol by the condition that “If B(a, f(a)) ≤ 0, then B(b, f(b)) < 0” (which need not be equivalent to “B(a, f(a)) > 0 or B(b, f(b)) < 0” because the internal logic is intuitionistic). The presheaf defined by the “If-then” condition would actually be the sheaf of solutions, Sol. The definitions of R, R_{0}, RR, R^{R^{0}}, RR_{0}_{0}, Sol_{0}, Stnd, Stab can all be interpreted in the topos of presheaves and the presheaves so defined would coincide with the sheaves defined above.

4. Examples

This section fleshes out the theory with several examples. All of these examples are assumed to be equations for a single function (in other words, n = 1) unless the contrary is explicitly noted.

4.1. Example. Comparison of barrier conditions with the internal derivative.

There is an internal notion of a derivative (based on nilpotents) for maps in a smooth topos which contains the topos we are working in. Obviously, this notion can be used to define the object of solutions of (∗). In general, our definition, based on barrier conditions, allows for a larger solution object than the definition based on the internal derivative. But for C^{∞}-rings of the form C^{∞}(F), where F ⊆ R^{m} is a closed subset, the two definitions for the solution object agree. (This case includes rings of the form C^{∞}(M) for M a C^{∞}-manifold.) In these cases, the solution object at C^{∞}(F) contains the maps σ(u, t) ∈ C^{∞}(F ×R) which are smooth parameterized solutions over F in the sense that σ is smooth and for each fixed u ∈ F, we have f(t) = σ(u, t) is a (smooth) classical solution of (∗). For other C^{∞}-rings, the internal definition still leads to the object of maps that can be represented by smooth parameterized solutions. The barrier conditions definition leads to a larger solution object as shown by the next example.

4.2. Example. The differential equation dx/dt = 0.

Classically, the solutions of this equation are the constant functions. In filter rings, the semi-bounded solutions are “nearly constant” in the sense indicated below. We will show that:

(1) If g ∈ C^{∞}(R) and σ(u, t) is a solution of dx/dt = 0, then so is g(σ(u, t)).

(2) If f ∈ C^{∞}(R), then B(t, x) = f(x) − t is a barrier function over any interval [a, b], for which a < b.

(3) Internally, an R_{0}-preserving function f is a solution of dx/dt = 0 iff, for every pair of ordinary reals a, b, f(a) is infinitesimally close to f(b). Externally the semi-bounded σ(u, t) is a solution iff, for all a, b and ϵ > 0, ∃F ∈ F such that |σ(u, a) − σ(u, b)| < ϵ for all u ∈ F.

(4) In general, if σ(u, t) is a solution (not necessarily semi-bounded) then, given a, b, arctan σ(u, a) must be infinitesimally close to arctan σ(u, b) (for all u in some member of the filter) but not conversely. In fact σ(u, t) can be infinitesimally close to a constant (even uniformly so) without being a solution. So the semi-bounded hypothesis (that σ ∈ n-Map_{0}(A)) cannot be eliminated from Proposition 2.10.

Proof. (1) Clearly B(t, g(x)) is a barrier whenever B(t, x) is and the result follows. (2) Note that ∂B/∂t = −1 < 0 which is more than enough.

(3) If σ satisfies the given condition, then for any maximal closed filter M which extends F it is clear that σ_{M}(a) = σ_{M}(b) for all a, b. So σ_{M} is a solution of (∗) which implies that σ is an A-solution, by Proposition 2.8.

Conversely, if there exist a, b, ϵ such that E = {u : |σ(u, a) − σ(u, b)| < ϵ} is not in F, then we can extend F to a maximally closed filter M which contains E^{c}, the closed set which is the complement of E. It follows that |σ_{M}(a) − σ_{M}(b)| ≥ ϵ which shows that σ_{M} cannot be a solution of (∗), which contradicts Proposition 2.8.

(4) The first sentence follows from (1), using g = arctan, and (3), as arctan σ(u, t) is obviously semi-bounded. As for the second part, let A = C^{∞}(R)/I(F) where F is the filter generated by sets of the form [u, ∞) for u ∈ R. Let θ(u, t) ∈ C^{∞}(R × R) be such that θ(u, t) = u + (t/u) for u ≥ 1. Then θ is not an A-solution: Let B(t, x) = f(x) − t, where f ∈ C^{∞}(R) is such that f(n) < 0 for all integers n and f(n + (1/n)) > 1 for all n ≥ 2. Then B(0, θ(n, 0)) < 0 and B(1, θ(n, 1)) > 0 for all integers n ≥ 2. So θ does not even satisfy the weak barrier condition. In fact, defining θ(u, t) = u + (sin t)/u we get a non-solution which is uniformly close to constants and to actual solutions.

4.3. Example. The differential equation dy/dt = y.

This equation can be obtained from dx/dt = 0 by the transformation y = e^{t}x and x = e^{−t}y, using Theorem 2.22. So, σ(u, t) is a solution for dy/dt = y iff e^{−t}σ(u, t) is a solution for dx/dt = 0. The results of Example 4.2 can now be restated for this equation.

4.4. Example. High curves, low curves and barrier functions.

The notions of “high curve” and “low curve” are introduced in [3], using a different notation, and treating the equation x′ = W (t, x) for a single function x(t) (so n = 1). We recall that h(t) is a high curve (resp. ℓ(t) is a low curve) over [a, b] iff h′(t) > W (t, h(t)) (resp. ℓ′(t) < W (t, ℓ(t))) for all t ∈ [a, b]. It follows that B(t, x) = x − h(t) (resp. B(t, x) = ℓ(t) − x) is a barrier function over [a, b] (but not conversely.) Clearly f(t) satisfies the corresponding growth conditions (for these barriers, or as in [3]) iff either f(a) > h(a) or f(b) < h(b) (resp. iff either f(a) < ℓ(a) or f(b) > ℓ(b)). So if f(t) is transfinite for all t (which can happen in a filter ring, if f is not semi-bounded) then f satisfies all conditions arising from high and low curves, which is quite different from, for example, the restrictions and non-solutions mentioned in (4) of Example 4.2. (We note that the definition of “solution”, as given in [3], requires more than the satisfaction of each individual growth condition.)

4.5. Example. Smooth, global systems.

The system dx/dt = W (t, x), where W (t, x) = (W_{1}(t, x), …, W_{n}(t, x)), is smooth and global if each W_{i} is C^{∞} and if there exists a global solution with any given initial value (x_{1}(0), …, x_{n}(0)). In this case, as is well-known, there exists ϕ(c, t) = (ϕ_{1}(c, t), …, ϕ_{n}(c, t)) which is the unique solution with initial condition c, for c ∈ R^{n}. Moreover, each ϕ_{i} is known to be C^{∞}. Then:

(1) Let A = C^{∞}(R^{m})/I(F) be any filter ring. Let c(u) = (c_{1}(u), …, c_{n}(u)) represent an n-tuple of bounded members of A. Then ϕ(c(u), t) represents a semi-bounded Asolution.

(2) Conversely, for every semi-bounded A-solution σ, there is c(u), as above, such that for every t and every ϵ > 0 there exists F ∈ F such that ∥σ(u, t) − ϕ(c(u), t)∥ < ϵ for all u ∈ F.

(3) Stnd(A) is equivalent to the set of all n-tuples of bounded members of A. So the manifold Stnd is isomorphic to R_{0}^{n}. This indicates that the standardized solutions are parameterized by n-tuples of (bounded) reals, just as the classical solutions are.

Proof. (1) ϕ clearly satisfies the barrier conditions. Since ϕ is continuous, if we put a bound on the initial values c and if t varies in some bounded interval J, then the values of ϕ will be bounded. This shows that ϕ is semi-bounded.

(2) Let σ(u, 0) = (σ_{1}(u, 0), …, σ_{n}(u, 0)). Since σ is semi-bounded, there exists F ∈ F on which the maps σ_{i}(u, 0) are bounded, so we can define bounded maps c(u) where c_{i}(u) = σ_{i}(u, 0) for u ∈ F. The condition saying that σ is within ϵ of ϕ must be met, otherwise, arguing as in the proof of (3) in Example 4.2, we can find a solution σ_{M} which has the same initial conditions as ϕ_{M} but values differing by ϵ for some t.

(3) This follows from (2) when we factor out by I_{PtCn} to get Stnd(A).

4.6. Example. The differential equation dx/dt = 2\sqrt{|x|}.

In this case, the only smooth, classical solution is constantly 0. So, by Example 4.1, if A = C^{∞}(R) or C^{∞}(M) for a manifold M, then the only A-solution is identically 0. But, we can approximate 2\sqrt{|x|} by a smooth function V (u, t, x) as u→∞. So if A = C^{∞}(R)/I(F) where F is generated by {[u, ∞)}, we can get any ordinary, nonsmooth solution of dx/dt = 2\sqrt{|x|} in the form σ_{M} by choosing V with care. We can also choose V so that we get distinct A-solutions with the same initial conditions.

4.7. Example. The differential equation dx/dt = 1 + x^{2}.

For this equation, B(t, x) = (t/2) − arctan(x) is a barrier function over any interval [a, b]. If a < −π and b > π then B(a, σ(u, a)) < 0 and B(b, σ(u, b)) > 0 for any σ. It follows that this equation has no (global) solutions. (A different approach in [3] allowed infinite solutions.) Note that 1 + x^{2} can be pointwise approximated by functions V (x) for which dx/dt = V (u, x) does have smooth global solutions. This shows that Theorem 2.12 cannot be extended from semi-bounded solutions to all solutions.

4.8. Example. The system dr/dt = (1 − r)^{2}, dθ/dt = cos^{2} θ + 1 − r.

Consider the solution f which starts at r = 1/2, θ = 0. Then r(t) = 1− \frac{1}{t+2} approaches 1 and θ steadily increases to ∞, as t→∞. Think of (r, θ) as polar coordinates and rewrite the system in terms of rectangular coordinates (x, y), for |r| ≥ 1/3. This region contains the orbit of f for t > 0. The resulting equations, defined smoothly on r ≥ 1/3, can be smoothly extended to the whole plane without affecting f(t) for t ≥ 0. The limit point set, L^{+}(f), is clearly the unit circle where r = 1. It has two equilibrium points, at (r, θ) = (1, π/2) and (1, 3π/2). There are also two orbits between these points, both moving counter-clockwise, one along the right side of the unit circle and the other along the left. Both take an infinite amount of time to get from one equilibrium point to the other. The non-standard \overline{f} : R_{U}^{+}→R^{2} makes L^{+}(f) cyclic as it traverses the entire circle, over and over again, stopping for only an “ordinary eternity” at each equilibrium point before moving on.

4.9. Example. Stabilization at 0.

Consider the following differential equations for a single function x(t). In each case, 0 is an equilibrium point and we consider the behavior of solutions near 0. Specifically, we consider A-solutions where A = C^{∞}(R)/I(F) and F is the filter of neighborhoods of 0. We let ϕ(u, t) be the unique solution with initial value u. (Such a C^{∞}-map, ϕ, exists in these cases.) We let C^{∞}(R)/I(G) be the stabilization of ϕ and we seek to describe G.

(1) dx/dt = −x. In this case, ϕ is already stable, so G = F

(2) dx/dt = x^{3}. Here ϕ is not stable and G is the trivial filter generated by {0}. But, surprisingly, the functor Stab is non-trivial for this equation, see Example 4.11.

(3) dx/dt = sin^{2} x. In this case, ϕ is not stable but G is not trivial, as G is the filter generated by (−ϵ, 0). Note that a solution will stay near 0 if it starts just below 0. But if it starts just above 0, it moves towards the next equilibrium point, at x = π.

4.10. Example. The system dx/dt = −x, dy/dt = y.

Consider the only equilibrium point, at (0, 0). Let (ϕ_{1}(x_{0}, y_{0}, t), ϕ_{2}(x_{0}, y_{0}, t)) be the solution with initial value (x_{0}, y_{0}). Let F be the filter of neighborhoods of (0, 0) in R^{2}. Let A = C^{∞}(R^{2})/I(F). As in the above example, let A = C^{∞}(R^{2})/I(G) be the stabilization of ϕ. Then G is generated by the intersections of members of F with the “x-axis”. This reflects the idea that ϕ is stable only in the x-direction.

4.11. Example. The Pitchfork Bifurcation.

(I read about this interesting example in [2].) Consider the equation dx/dt = x^{3} which has an unstable equilibrium point at x = 0. Consider also dx/dt = ux + x^{3}, which has equilibrium points when x = 0 or when u = −x^{2} (which form a “pitchfork” shaped subset of the (u, x) plane).

Let A = C^{∞}(R^{2})/I(F) where F is the filter of neighborhoods of (0, 0). Let ϕ(u, x_{0}, t) be the solution of dx/dt = ux+x^{3} with initial value x_{0}. Then, in view of Theorem 2.12, ϕ represents an A-solution of dx/dt = x^{3}. Note that (0, 0) is an unstable equilibrium point of dx/dt = x^{3} but some of the equilibrium points of dx/dt = ux + x^{3} are stable (when u = −x^{2} for u < 0), and ϕ does not wander far from 0 if u < −x_{0}^{2}. If G is the filter which defines the stabilization of ϕ, then G is generated by the following sets, defined for ϵ > 0:

{(u, x) : |x| < ϵ and − ϵ^{2} < u < −x^{2}} ∪ {(u, x) : x = 0 and |u| < ϵ}

Note that Stab for dx/dt = x^{3}, with respect to the equilibrium point 0, contains this stabilization of ϕ, so it is not, as one might expect, trivial.

On the definition of semi-boundedness. Suppose A = C^{∞}(R^{m})/I(F) and let λ ∈ C^{∞}(R^{m} × R) represent a member of Map(A). We want to find out if the definition of “semi-bounded” can be simplified. Say that λ is bounded on J if λ is bounded on R^{m} × J for every J ∈ J. Then clearly λ is semi-bounded with respect to any filter. Also ν ∈ C^{∞}(R^{m} × R) is semi-bounded with respect to F if there exists F ∈ F with ν = λ on F × R where λ is bounded on J. The converse however is false, as shown by the example below.

4.12. Example. The definition of semi-bounded cannot be simplified, as discussed above.

(I thank W. W. Comfort who pointed out the crucial role of P -points and non-P - points.) Let U be an ultrafilter on N which is a non-P -point of βN − N. See [1], particularly problems 4J,4K,4L,6S for details, and the existence of such points. It follows that U is an element of some G_{δ} which is not a neighborhood of U. So we can find a sequence {F_{n}} of members of U such that for no F ∈ U is the set-theoretic difference F − F_{n} finite for all n.

We may as well assume that the {F_{n}} are nested ( with F_{n} ⊇ F_{n+1} for all n) else replace each F_{n} by F_{1} ∩ … ∩ F_{n}. Also, we may as well assume that no integer lies in all F_{n}, else replace F_{n} by deleting the finite set {1, 2, …, n}. For each u ∈ N let deg(u) = max{n : u ∈ F_{n}}. (Assume F_{0} = N).

Now define ν ∈ C^{∞}(R × R) so that for u ∈ N we have ν(u, t) = t if |t| ≤ deg(u) and ν(u, t) = t + u if |t| ≥ 1 + deg(u). (Such a map ν can clearly be found, by the smooth Tietze theorem.) We claim that ν is semi-bounded with respect to U, regarded as a filter on R. Given a bounded interval J choose n so that |t| ≤ n for t ∈ J. Then, for u ∈ F_{n} and t ∈ J, we see that |ν(u, t)| ≤ n (for if u ∈ F_{n}, then deg(u) ≥ n, etc.

But if there exist F ∈ U and λ bounded on J such that ν = λ on F ×R, then for each n, there exists B_{n} such that λ(u, t) ≤ B_{n} for all t with |t| ≤ n and all u. So, if u ∈ F − F_{n} then it readily follows that n + u ≤ B_{n} so u ≤ B_{n} − n which shows that F − F_{n} is finite, contradicting the choice of the {F_{n}}.

References

[1] L.Gillman and M.Jerrison, Rings of Continuous Functions, D.Van Nostrand, Princeton, 1960.

[2] J.Hale and H. Koçak, Dynamics and Bifurcations, Springer-Verlag, New York, 1991

[3] J.Kennison. Synthetic solution manifolds for differential equations, J. for Pure and Applied Algebra, 143 (1999), 255-274.

[4] I.Moerdijk and G.E.Reyes, Models for Smooth Infinitesimal Analysis, Springer-Verlag, New York, 1991

[5] I.Moerdijk and G.E.Reyes, Rings of smooth functions and their localizations, I, J. of Algebra, 99(1986), 324-336.

[6] I.Moerdijk, N. van Quê and G.E.Reyes, Rings of smooth functions and their localizations, II, in: D.W. Kueker, E.G.K. Lopez-Escobar and C.H.Smith, (ed), Mathematical Logic and Theoretical Computer Science, Marcel Dekker, New York, 1987, pp. 277-Clark University, Worcester, MA, 01610, USA

Email: jkennison@clarku.edu

This article may be accessed via WWW at http://www.tac.mta.ca/tac/ or by anony-

mous ftp at ftp://ftp.tac.mta.ca/pub/tac/html/volumes/7/n13/n13.{dvi,ps}

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Editorial board.

John Baez, University of California, Riverside: baez@math.ucr.edu Michael Barr, McGill University: barr@barrs.org Lawrence Breen, Université Paris 13: breen@math.univ-paris13.fr Ronald Brown, University of North Wales: r.brown@bangor.ac.uk Jean-Luc Brylinski, Pennsylvania State University: jlb@math.psu.edu Aurelio Carboni, Università dell Insubria: carboni@fis.unico.it P. T. Johnstone, University of Cambridge: ptj@dpmms.cam.ac.uk G. Max Kelly, University of Sydney: maxk@maths.usyd.edu.au Anders Kock, University of Aarhus: kock@imf.au.dk F. William Lawvere, State University of New York at Buffalo: wlawvere@acsu.buffalo.edu Jean-Louis Loday, Université de Strasbourg: loday@math.u-strasbg.fr Ieke Moerdijk, University of Utrecht: moerdijk@math.uu.nl Susan Niefield, Union College: niefiels@union.edu Robert Paré, Dalhousie University: pare@mathstat.dal.ca Andrew Pitts, University of Cambridge: Andrew.Pitts@cl.cam.ac.uk Robert Rosebrugh, Mount Allison University: rrosebrugh@mta.ca Jiri Rosicky, Masaryk University: rosicky@math.muni.cz James Stasheff, University of North Carolina: jds@math.unc.edu Ross Street, Macquarie University: street@math.mq.edu.au Walter Tholen, York University: tholen@mathstat.yorku.ca Myles Tierney, Rutgers University: tierney@math.rutgers.edu Robert F. C. Walters, University of Insubria: walters@fis.unico.it R. J. Wood, Dalhousie University: rjwood@mathstat.dal.ca