Geometric Singular Perturbation Theory 2

2.1.1 Basic Setting

The equations under consideration are of the form {x^′ = f (x, y, ε),

y^′ = εg(x, y, ε), (1)

where ^′ = d

dt, x ∈ Rⁿ, y ∈ R^m and ε ∈ R is a small parameter (0 < ε ≪ 1) which gives the equations a singular character.

The following statement is one of the basic assumptions:

Hypothesis 1. The functions f and g are both assumed to be C^r on a set U × I, where U ⊂ R^n+m is open and I is an open interval, containing 0 and r≥ 1.

From (1), let s = εt, then equation (1) is transformed to the following form {ε ˙x = f (x, y, ε),

y = g(x, y, ε),˙ (2)

where ˙ = d

ds. The time scale given by s is said to be slow whereas that for t is fast.

In fact, as long as ε ̸= 0 the two systems are equivalent. Thus we call (1) the fast system and (2) the slow system.

Letting ε−→ 0, that the limiting systems of (1) and (2) are, respectively, {x^′ = f (x, y, 0),

y^′ = 0, (3)

and {

f (x, y, 0) = 0,

y = g(x, y, 0).˙ (4)

Let N ={ (x, y) ∈ U | f(x, y, 0) = 0}. In view of system (3), each point of N is a fixed point and the system (3) has nontrivial dynamical behaviors on U\N. On the other hand, system (4), defined only on N , usually has nontrivial dynamical behaviors on N . We next define the concept of a set being normally hyperbolic

Definition 1. Let M₀ ⊂ N. The set M0 is said to be normally hyperbolic if Dxf (˜x, ˜y, 0) has exactly n eigenvalues λ with Re(λ)̸= 0 for any (˜x, ˜y) ∈ M0.

Hypothesis 2. Let M₀ ⊂ N. The set M0 is assumed to be a compact manifold and is normally hyperbolic.

The set M₀ will be called the critical manifold.

Since from Hypotheses 2, we have that Dxf (˜x, ˜y, 0) is invertible for any (˜x, ˜y) ∈ M₀. By the Implicit Function Theorem, x can locally be solved for y, that is, x = h⁰(y) for some smooth function h⁰. Since the set which is a graph of some smooth function is easier to manipulate, we want to decompose M₀ into a finite number of its subset that is a graph of some smooth function and satisfies Hypotheses 2. (i.e.

M0 = M₀⁽¹⁾ ∪ M0⁽²⁾∪ · · · ∪ M0^(k) where M₀⁽ⁱ⁾ is a graph of some smooth function and satisfies Hypotheses 2 for i=1,2,. . . , k.

In order to simplify discussion, we consider that M₀ is a graph of function satisfying the following two hypotheses.

Hypothesis 3. M₀ is normally hyperbolic and there are a compact subset K ofR^m and a function h⁰ defined on K such that M₀ ={ (x, y) ∈ U | x = h⁰(y)}.

Hypothesis 4. h⁰ is a smooth function on K and K is simply connected domain whose boundary is an (m− 1)-dimensional C^∞ submanifold.

In addition, we also assume that the following hypothesis holds.

Hypothesis 5. Dxf (˜x, ˜y, 0) has ℓ eigenvalues λ with Re(λ) < 0 and k(= n − ℓ) eigenvalues λ with Re(λ) > 0 for any (˜x, ˜y) ∈ M0.

2.1.2 Fenichel’s Theorems

We give some definitions and some notations in order to state Fenichel’s Theorems.

Notation. The notation x· t is used to denote the application of a flow after time t to the initial condition x. Similarly, V · t denote the application of a flow after time t to a set V , and x· [t1, t₂] is the resulting trajectory if the flow is applied over the interval [t₁, t₂].

Definition 2. A set M is locally invariant under the flow from (1) if there exists a neighbourhood V of M so that for all x∈ M, x · [0, t] ⊂ V implies that x · [0, t] ⊂ M, similarly with [0, t] replaced by [t, 0] when t < 0.

Theorem 1. (Fenichel’s First Theorem, see e.g., [24])

Under the Hypotheses 1 and 2. Then for ε > 0 and sufficiently small, there exists a manifold M_ε that lies within O(ε) of M₀ and is diffeomorphic to M₀. Moreover it is locally invariant under the flow of system (1), and C^r, including in ε, for any r < +∞.

The manifold Mε will be called the slow manifold.

Theorem 2. (Fenichel’s First Theorem for graph version, [24])

Under the Hypotheses 1, 3and 4. Then for ε > 0 and sufficiently small, there exists a function h^ε, defined on K, so that Mε = { (x, y) | x = h^ε(y), y ∈ K} is locally invariant under the flow of system (1). Moreover h^ε is C^r, for any r < +∞, jointly in y and ε.

Remark. The diffeomorphism between Mε and M0 follows easily in this formulation through the diffeomorphism of the graph to K.

Flow on M_ε:

y^′ = εg(h^ε(y), y, ε). (5)

In the alternative slow scaling we recast (5) as

y = g(h˙ ^ε(y), y, ε). (6)

which has distinct advantage that a limit exists as ε → 0, given by g(h⁰(y), y, 0) which naturally describes a flow on the critical manifold M₀, and is exactly the second equation in (2). Using this theorem and this resulting equation (6), the problem of studying (1), at least on M_ε is reduced to a regular perturbation problem.

Example. Consider the system





















˙u₁ = u₂

˙u₂ = u₃ ε ˙u₃ = u₄ ε ˙u₄ = u₅ ε ˙u₅ = u₆

ε ˙u₆ =−Au5− u3− cu2− f(u1), where A > 0, f (ϕ) = ϕ(ϕ− a)(1 − ϕ), a < ¹₂, c: constant.

Let x =







 u₃ u₄ u₅ u₆







, y = [ u₁

u₂ ]

then x is the fast variable and y is the slow variable.

The critical manifold M₀ can be taken as any compact subset of {u4 = u₅ = u₆ = 0, u₃ =−cu2− f(u1)} and will shall be large enough to contain any of the dynamics of interest.

The eigenvalues of linearization at any point of M₀, other than the double eigenvalue at 0 are seen to be solutions of quartic equation µ⁴+ Aµ²+ 1 = 0 which are not pure imaginary if 0 < A < 2.

The equation for the slow flow on the critical manifold M₀are given by

{ ˙u₁ = u₂,

˙u₂ =−cu2− f(u1).

The slow manifold M_ε, which exists by virtue of theorem 1,is given by the equations (u₃, u₄, u₅, u₆) = h^ε(u₁, u₂) = (−cu2 − f(u1), 0, 0, 0) + O(ε)

and the equations on M_ε are

{ ˙u1 = u2,

˙u₂ =−cu2− f(u1) + O(ε).

We know nothing of the flow off the slow manifold and this must now be addressed.

The slow manifold possesses accompanying stable and unstable manifolds that are perturbations of the corresponding manifolds when ε = 0.

Theorem 3. (Fenichel’s Second Theorem, see e.g., [25])

Under the Hypotheses 1 and 2. Then for ε > 0 and sufficiently small, there exists manifolds W^s(M_ε) and W^u(M_ε) that are O(ε) close and are diffeomorphic to W^s(M₀) and W^u(M₀) respectively, and they are each locally invariant under the flow of system (1), and C^r, including in ε, for any r <∞.

We want to state Fenichel’s Second Theorem for graph version but we need a detailed analysis and more notations. So we postpone this after Fenichel’s third Theorem. The following is talking about a motivation for Fenichel’s third Theorem.

First, we observe that

W^s(M₀) = ∪

v0∈M0

W^s(v₀), W^u(M₀) = ∪

v0∈M0

W^u(v₀)

In other words, the manifolds W^s(v₀) and W^u(v₀) form collections of fibers for W^s(M₀) and W^u(M₀), respectively, with base points v₀ ∈ M0.

Second, a natural question to ask now, is whether the individual stable and unstable manifolds W^s(v₀) and W^u(v₀) perturb to analogous objects ? —— Fenichel’s third Theorem answers this question. In order to avoid difficulties we restrict ourselves to a neighbourhood D of Mεin which the linear terms of (1) are dominant, and consider only trajectories in W^u(M_ε) that have not left D in forward time (yet), and trajecto-ries in W^s(M_ε) that have not left D in backward time. To facilitate this discussion, we need a definition.

Definition 3. The forward evolution of a set V ⊂ D restricted to D is given by the set

V ·Dt≡ { x · t | x ∈ V and x · [0, t] ⊂ D }.

Figure 1: A diagram for Fenichel’s Third Theorem.

Now, we sate Fenichel’s Third Theorem.

Theorem 4. [Fenichel’s Third Theorem]¹

Under the Hypotheses 1, 2 and 5. Then if ε > 0 and sufficiently small then for every v_ε ∈ Mε there are an ℓ-dimensional manifold W^s(v_ε) ⊂ W^s(M_ε) and an k-dimensional manifold W^u(vε) ⊂ W^u(Mε), that are O(ε) close and diffeomorphic to W^s(v₀) and W^u(v₀) respectively.

Moreover, they are C^r for any r, including in v and ε.

The families { W^s(vε)| vε ∈ Mε} and { W^u(vε)| vε∈ Mε} are invariant in the sense that

W^s(v_ε)·Dt ⊂ W^s(v_ε· t) if v_ε· s ∈ D for all s ∈ [0, t] (t > 0) and

W^u(vε)·Dt ⊂ W^u(vε· t) if v_ε· s ∈ D for all s ∈ [t, 0] (t < 0).

The idea is, that the order in which the flow after time t is applied to a base point and the fiber of a base point is constructed does not matter, as depicted in this figure (1).

In the unperturbed setting (1) with ε = 0, the decay in forward time of points in W^s(M₀) to M₀ is clearly to the base point v₀ of their fiber, where the decay rate as t→ ∞ is exponential, since all associated eigenvalues have nonzero real part.

The fibers give a very useful matching between the points in W^s(M_ε) and partners they have in M_ε. One can then see that the decay of points in W^s(M_ε) to M_ε is actually to the base point of the fiber, this gives a decay result with ”asymptotic phase”; similarly for points in W^u(M_ε).

Preparation for statement of Fenichel’s Second Theorem for graph version Suppose that M₀satisfies the Hypotheses1,3and4. Without loss of generality, we can assume that h⁰(y) = 0 for all y∈ K. Indeed, we can replace x by ˆx = x − h⁰(y)

1we may refer to [25], P. 376

and recompute the equations. For each y ∈ K, there are subspaces S(y) and U(y), corresponding, respectively, to stable and unstable eigenvalues. Since the eigenvalues are bounded uniformly away from the imaginary axis over K, the dimensions of S(y) and U (y) are independent of y. Let dim S(y) = ℓ and dim U (y) = k. Since K is simply connected by hypothesis4, we can smoothly choose bases for S(y) and U (y).

Changing the coordinates to in terms of these new bases, we can set x = (a, b), where a∈ R^ℓ and b ∈ R^k, so that our equations have the form









a^′ = A(y)a + F₁(x, y, ε), b^′ = B(y)b + F₂(x, y, ε), y^′ = εg(x, y, ε),

(7)

where σ(A(y)) ⊂ {λ | Re(λ) < 0} and σ(B(y)) ⊂ {λ | Re(λ) > 0}. Both F1 and F₂ consist of any higher order terms in x, y for each ε; to be precise, we have the estimates |Fi| ≤ γ(|x| + ε), i = 1, 2 and γ can be taken to be as small as desired by restricting to a set with|a| and |b| small.

With this notation established, we can determine W^s(M_ε) and W^u(M_ε) as graphs and give the following restatement of theorem 3.

Theorem 5 (Fenichel’s Second Theorem for graph version). ² If ε > 0, but sufficiently small, then ,for some ∆ > 0,

(a) there is a function a = h_s(b, y, ε) defined for y ∈ K and |b| ≤ ∆, so that the graph W^s(M_ε) = {(a, b, y) | a = hs(b, y, ε)} is locally invariant under (7).

Moreover, h_s(b, y, ε) is C^r in (b, y, ε) for any r < +∞.

(b) there is a function b = h_u(a, y, ε) defined for y ∈ K and |b| ≤ ∆, so that the graph W^u(M_ε) = {(a, b, y) | b = hu(a, y, ε)} is locally invariant under (7).

Moreover, hu(a, y, ε) is C^r in (a, y, ε) for any r < +∞.

These theorems also apply when ε = 0 and provide the stable and unstable man-ifolds of the known critical manifold, the existence of which is also guaranteed by the usual stable and unstable manifold theorems at critical points. These latter two theorems then assert that these manifolds, W^s(M_ε) and W^u(M_ε), are perturbed from W^s(M₀) and W^u(M₀) respectively. Theorem 1 can be concluded from Theorem 3 by taking the intersection of W^s(M_ε) with W^u(M_ε). Locally, the Implicit Function The-orem gives the intersection as a graph, and these functions can be patched together since K is a compact set. Moreover, we need only give the stable manifold, as that of the unstable manifold follows immediately by a reversal of time.