Notes on Homoclinic Bifurcation Theory

Ale Jan Homburg KdV Institute for Mathematics University of Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam The Netherlands [email protected]

Version 20051027

1 Introduction

These notes contain a proof of the disappearance of an isolated periodic orbit in a homoclinic bifurcation with a homoclinic loop to a hyperbolic saddle equilibrium. The notes include a treatment of the implicit function and the stable manifold theorem.

2 Implicit function theorem

At several places we will ﬁnd the need to solve equations on Banach spaces. We include the statement of the implicit function theorem allowing us to do so. Implicit function theorems on Banach spaces were developed by Hildebrandt and Graves [12]. We follow the treatment in [7]. See [25] for an account of the history of implicit functions theorems.

2.1 Contraction theorem

The Banach contraction theorem says that a contraction on a complete metric space has a unique ﬁxed point. Let S be a metric space with metric d. Recall that T : S → S is a contraction if for some θ ∈ [0, 1),

d(T (x), T (y)) ≤ θd(x, y). (2.1)

The constant θ is called a contraction constant.

1 Note that in an equivalent metric d˜, this gives d˜(T (x), T (y)) ≤ Cθnd˜(x, y) Theorem 2.1. If (S,d) is a complete metric space and T : S → S is a contraction, then n there is a unique ﬁxed point x¯ of T in S. Also, for any x0 ∈ S, T x0 → x¯ as n →∞.

2.2 Uniform contraction theorem

For a contraction depending on a parameter, also the ﬁxed point will depend on a parameter. A contraction is called uniform if the contraction constant can be chosen independent of the parameter. Theorem 2.2. Let U, V be open sets on Banach spaces X,Y , let U¯ be the closure of U, T : U¯ × V → U¯ a uniform contraction on U¯ and let g(y) be the unique ﬁxed point of T (·,y) in U¯. If T ∈ Ck(U¯ × V,X), 0 ≤ k < ∞ then g(·) ∈ Ck(V,X).

DxT (g(y),y)Dg(y)+ DyT (g(y),y)= Dg(y). So Dg(y) should be a solution M(y) to

DxT (g(y),y)M(y)+ DyT (g(y),y)= M(y). (2.2)

This equation can be solved since DxT (g(y),y) gives a contraction. To establish that M(y) equals the derivative of g, we must show that |g(y + h) − g(y) − M(y)h| = o(|h|) as h → 0. Write γ(y)= g(y + h) − g(y). Now γ(y) = T (g(y)+ γ(y),y + h) − T (g(y),y)

= DxT (g(y),y)γ(y)+ DyT (g(y),y)h + R, (2.3)

2 where R = T (g(y)+ γ(y),y + h) − T (g(y),y) − DxT (g(y),y)γ(y) − DyT (g(y),y)h. Since T is diﬀerentiable, for any ǫ > 0 there is δ > 0 with |R| < ǫ(|γ(y)| + |h|) if |γ(y)|, |h| < δ. Since γ is continuous in h we may further restrict δ so that this estimate holds for |h| < δ. −1 From (2.3) we get γ(y)=(I − DxT (g(y),y)) (DyT (g(y),y)h + R), so that |γ(y)| < C|h| for some C > 0, if |h| < δ. This implies |R| < ǫ(1 + C)|h| for |h| < δ.

Equation (2.2) shows that (I − DxT (g(y),y))M(y) = DyT (g(y),y). With (2.3) this yields (I − DxT (g(y),y))(γ(y) − M(y)h)= R. We obtain |γ(y) − M(y)h| < Kǫ|h| for some K > 0, if |h| < δ. This proves that M(y) equals the derivative Dg(y). Higher order derivatives are treated by induction. Assume that T is of class Ck and g has been shown to be Cj, j

2.3 Implicit function theorem

The implicit function theorem is a corollary of the uniform contraction theorem, by combining it with a Newton type algorithm. Theorem 2.3. Suppose X,Y,Z are Banach spaces, U ⊂ X,V ⊂ Y are open sets, F : U × V → Z is continuously diﬀerentiable, (x0,y0) ∈ U × V , F (x0,y0)=0 and DxF (x0,y0) has a bounded inverse. Then there is a neighborhood U1 × V1 ∈ U × V of (x0,y0) and a function f : V1 → U1, f(y0) = x0 such that F (x, y)=0 for (x, y) ∈ U1 × V1 if and only if k k x = f(y). If F ∈ C (U × V,Z), k ≥ 1, then f ∈ C (V1,X).

−1 Proof. If L = [DxF (x0,y0)] , G(x, y) = x − LF (x, y), then the ﬁxed points of G are the solutions of F = 0. The map G has the same smoothness properties as F , G(x0,y0) = 0, DxG(x0,y0) = 0. Therefore, |DxG(x, y)| ≤ θ < 1 for all x, y is a neighborhood U1 × V1 of (x0,y0). It is easy to choose the neighborhood so that G : U1 × V1 → U1. The result now follows from Theorem 2.2.

Remark 2.4. Note that the above reduction to an application of the uniform contraction theorem works if y 7→ F (x, y) is continuous, as long as x 7→ F (x, y) is continuously diﬀer- entiable and DxF (x0,y0) has bounded inverse. See [5].

3 Invariant manifolds

Consider a flow φt for a differential equationx ˙ = f(x) on a compact manifold M. A submanifold W ⊂M is invariant if φt(W )= W , t ∈ R. It is positive invariant if this holds for t > 0. It is locally invariant if f(x) ∈ TxW for x ∈ W . Invariant manifolds play an important role in the qualitative study of flows. There exists a substantial theory on robust invariant manifolds, for which perturbed flows have perturbed invariant manifolds.

3 We will see a glimpse of the general theory by studying dynamically deﬁned invariant manifolds near equilibria.

3.1 Stable manifold theorem

Consider a differential equation x˙ = f(x) (3.1) on a smooth compact manifold M. We assume that X is a smooth, that is C∞, map. Write (t, x0) 7→ φt(x0) for the flow of X giving the solution to (3.1) equal to x0 at time t = 0. Standard theory of ordinary differential equations shows that φt(x0) depends smoothly on (t, x0). Suppose (3.1) has an equilibrium p ∈M, f(p) = 0. Suppose further that the origin is a hyperbolic equilibrium, i.e. the spectrum of Df(p) is disjoint from the imaginary axis. s u s u There is a Df(p) invariant splitting E ⊕ E of TpM where Df(p) restricted to E (E ) has spectrum with negative (positive) real part. We refer to Es as the stable subspace of Df(p) and to Eu as the unstable subspace of Df(p). Define the stable set

s W (p)= {x ∈M| lim φt(x)= p}. t→∞

u The unstable set W (p) is deﬁned as the stable set for φ−t. There are several proofs of the stable manifold theorem (for a hyperbolic equilibrium) stated below. We give a variant of Perron’s proof [29], see also [19, 20, 21, 32, 16]. A more geometric approach to stable manifolds is with graph transforms. This approach originates with [11]. An extensive use of graph transforms is in [13]. We quote D.S. Anosov [2]:

Every five years or so, if not more often, someone discovers the theorem by Hadamard and Perron, proving it by Hadamard’s method of proof or by Perron’s.

s s s Proposition 3.1. W (p) is an injectively immersed smooth manifold with TpW (p)= E .

Proof. We construct a local stable manifold deﬁned as

s Rn Wloc(p)= {x ∈ | d(x, p) < σ, lim φt(x)= p} t→∞

s s for a sufficiently small σ > 0. Then W (p) = ∪t≤0Wloc(p). Using a local coordinate system we may assume that p equals the origin in Rn and that we are working with a differential equationx ˙ = f(x) on Rn. Using a test function we may assume that X is globally close to Df(0). Indeed, let ψ : Rn → R be a test function, i.e. a smooth nonnegative function such that ψ = 0 outside a neighborhood of 0 and ψ = 1 near 0. Consider the rescaled version ψǫ(x) = ψ(x/ǫ). Then ψǫX + (1 − ψǫ)Df(0) converges together with first order derivatives to Df(0) as ǫ → 0.

4 Consider the time 1 map F = φ1. This is a smooth map. We will construct the local stable manifold for F and show that this is also invariant for φt. Write B for the space of bounded sequences N → Rn, equipped with the supnorm. Note that B is a Banach space. s u n n i Take coordinates x =(xs, xu) ∈ E × E on R . Let πi : R → E , i = s,u, be the canonical coordinate projections. Define Γ : Es × B → B by (π F (ν(i − 1)), π F −1(ν(i + 1))), i> 0, Γ(a , ν)(i)= s u s (a , π F −1(ν(1))), i =0. s u It is straightforward to show that Γ is smooth, see [32]. Here we note that a map G : B → B, G(ν)(i) = g(ν(i)), for a given smooth map g : Rn → Rn, has derivative DG(ν)ζ(i) = Dg(ν(i))ζ(i). −1 By a linear coordinate change, we may assume that Df(0)| Es and DX (0)| Eu are contractions. From this it follows that for fixed as, ν → Γ(as, ν) is a contraction on B. for X sufficiently close to Df(0) (assured by choosing ǫ small enough). This map possesses a unique fixed point Γ(as,ζ)= ζ. Since Γ is a smooth map, the fixed point depends smoothly on as by the uniform contraction theorem. Write ζas for the fixed point of Γ(as, ·). Note that π1(ζas (0)) = as. We claim that s Wloc = ∪as∈Es ζas (0) (3.2) We will first show that the right hand side equals the local stable manifold of f. The fixed point ζas satisfies

πsζas (1) = πsf(ζas (0)), −1 πuζas (0) = πuf (ζas (1)).

From this one concludes that ζas (1) = F (ζas (0)). Continuing one sees that ζas (i +1) =

F (ζas (i)), i ≥ 0. Thus ζas is a positive orbit of f. It is easily seen that ζas (i) converges to 0 as i →∞. Namely, since ζas (i) lies on a smooth curve, π1ζas (i + 1) lies at least a ﬁxed factor

λ for some λ < 1 closer to 0 than π1ζas (i) (this is trivial for the linear map x 7→ DF (0)x and therefore holds for F as well, if F is close enough to x 7→ DF (0)x). This shows that the right hand side of (3.2) is indeed the local stable manifold of f. As

φtζas (i +1)= φt(F (ζas (i))) = F (φt(ζas (i))), N {φtζas (i)}, i ∈ , lies on the local stable manifold of f as well. Uniqueness implies that the right hand side of (3.2) is the local stable manifold of the ﬂow φt.

We sketch an alternative construction directly for the flow φt. Write A = Df(0) and f(x)= Ax + g(x). Let πs and πu be coordinate projections as before. Use coordinates x =(xs, xu) in Es × Eu. An orbit {z(t)}, t ≥ 0 in W s(0) satisfies t ∞ At A(t−v) A(t−v) z(t)= e z(0) + e (πsg(z(v)), 0)dv − e (0, πug(z(v)))dv. Z0 Zt 5 0 Rn The right hand side gives a contraction on the space Cb ([0, ∞), ) of bounded continuous orbits. Its fixed point, for t = 0, gives a point on the stable manifold with prescribed s coordinate πsz(0) on E .

3.2 Center stable manifolds

In the bifurcation study below we will assume that stable and unstable manifolds of a hyperbolic equilibrium contain orbits moving with different exponential speeds to the equilibrium as time goes to plus or minus infinity. The general setup is as follows. Suppose that the stable subspace Es of Df(p) admits a Df(p) invariant splitting Ews ⊕ Ess, where the spectrum of Df(p)| Ess has real parts lying to the left of the real parts of the spectrum of Df(p)| Ews . We call Ews principal stable directions (also weak stable directions) and Ess strong stable directions. There are invariant manifolds, invariant under the flow of X, associated to these Df(p) invariant directions. In this context we distinguish

s s s • the stable manifold W (p) with TpW (p)= E ,

ss ss ss • the strong stable manifold W (p) with TpW (p)= E ,

ws,u ws,u ws u • the center unstable manifold W (p) with TpW (p)= E ⊕ E ,

Unstable manifolds, strong unstable manifolds and center stable manifolds, assuming appro- priate Df(p) invariant splittings, have analogous definitions. We will briefly indicate the constructions of these manifolds, using variants of Perron’s method of proof for stable manifolds. Several details are omitted. The stable manifold has been dealt with in Proposition 3.1. Graph transform techniques, which we do not consider here, form perhaps a more consistent approach to the construction of invariant manifolds of different kinds.

ss ss ss Proposition 3.2. W (p) is an injectively immersed smooth manifold with TpW (p)= E .

Proof. We indicate the diﬀerence with the construction of W ss,s(p) in Proposition 3.1. ss s,u,uu n n i Take coordinates x = (xss, xs,u,uu) ∈ E × E on R . Let πi : R → E , i = ss or i = s,u,uu, be the canonical coordinate projections. Deﬁne Γ : Ess × B → B by

(π f(ν(i − 1)), π f −1(ν(i + 1))), i> 0, Γ(a , ν)(i)= ss s,u,uu ss (a , π f −1(ν(1))), i =0. ss s,u,uu Let the real number r be such that the line of complex numbers with real part r divides the r spectra of Df(p)| Ess and Df(p)| Ews,u . Note that λ < 0. Write λ = e , which is smaller than 1. Deﬁne A on B by Aν(i)= λ−iν(i).

6 N Rn −i Let Bλ = {ν : → | supi kν(i)kλ < ∞}. Then Bλ equipped with the norm kνk = −i ¯ ss supi kν(i)kλ . is a Banach space. Deﬁne Γ : E × Bλ → Bλ by −1 Γ(¯ ass, ν)= AΓ(ass, A ν).

This is a smooth map for λ < 1. For fixed ass, ν → Γ(¯ ass, ν) is a contraction on Bλ. The −i i−1 i unique fixed point ζass satisfies ζass (i) = λ f(λ ζass (i − 1)), so that λ ζass (i) is an orbit.

ws,u 1 ws,u Proposition 3.3. W (p) is an injectively immersed C manifold with TpW (p) = Ews,u. It is not uniquely deﬁned, but its tangent bundle along W u(p) is smooth and uniquely deﬁned.

Proof. By reversing the direction of time, we need to construct a center stable manifold W s,wu(0). The construction in Proposition 3.2 works for center stable manifolds. Note that now λ > 1. We remark that center stable manifolds are continuously differentiable, but in general not smooth. The difficulty being that the map Γ¯ is not smooth, see [20, 22, 10]. We do not include a proof of differentiability here. Uniqueness and smoothness of the tangent bundle T W ws,u(p) along W u(p) is shown as fol- n n lows. Let Gws,u(R ) be the Grassmannian manifold of linear subspaces in R with dimension ws,u n n dim E . Lift the time one map F to a map F¯ on R × Gws,u(R ): F¯(x, α)=(F (x), DF (x)α). Observe that this is a skew product map. Clearly, (p, Ews,u) is a fixed point of F¯. A direct computation shows that (p, Ews,u) is a hyperbolic fixed point and the unstable directions n include the tangent space of the fiber {0} × TEws,u Gws,u(R ). The computation can be done using local coordinates in Rn ×L(Ews,u, Ess) near (p, Ews,u) In these local coordinates, F¯ takes the form F¯(x, v)=(F (x),w), with graph w = DF (x)graph v. To compute eigenvalues in the fiber over the origin, it suffices to consider the map v 7→ w, graph w = DF (0)graph v. See [14]. The tangent bundle T W ws,u(p) along W u(p) is the unstable manifold of the fixed point (p, Ews,u) of F¯. It is therefore unique and smooth.

3.3 Families of diﬀerential equations

Consider a family x˙ = fµ(x) (3.3)

of diﬀerential equations with (µ, x) 7→ fµ(x) smooth. If for µ = µ0, (3.3) has a hyperbolic equilibrium p, then it has a hyperbolic equilibrium pµ that continuous p for µ near µ0. Also µ 7→ pµ is smooth.

7 By incorporating the parameter µ in the construction of stable manifolds above, one sees s that W (pµ) depends smoothly on µ. The same holds for strong stable manifolds. Center stable manifolds depend continuously diﬀerentiable on µ. Note that can consider an ex- tended diﬀerential equationx ˙ = fµ(x), µ˙ = 0 and construct a center stable manifold of the equilibrium (pµ0 ,µ0).

4 Homoclinic bifurcations

For a smooth compact manifold M, let X (M) be the set of C∞ vector ﬁelds on M equipped with the C∞ topology. Consider a system of diﬀerential equationsx ˙ = f(x), X ∈ X (M) such that X has an equilibrium p: f(p) = 0. We will assume that p is hyperbolic. Note that, by the implicit function theorem, p persists perturbations of X. There is a Df(p) invariant 1 2 1 2 splitting E ⊕ E of TpM where Df(p) restricted to E (E ) has spectrum with negative (positive) real part. We distinguish the following cases.

• p is of saddle-saddle type: the eigenvalues of Df(p)| E1 and Df(p)| E2 closest to the imaginary axis are real and simple.

• p is of saddle-focus type: the eigenvalues of Df(p)| E1 closest to the imaginary axis are complex conjugate, the eigenvalue of Df(p)| E2 closest to the imaginary axis is real,

• p is of bifocus type: the eigenvalues of Df(p)| E1 and Df(p)| E2 closest to the imaginary axis are complex conjugate.

The eigenvalues nearest to the imaginary axis are called the principal eigenvalues or weak eigenvalues.

4.1 Blue sky catastrophe

A blue sky catastrophe is a bifurcation in which a periodic orbit developes inﬁnite period as the parameter moves to the bifurcation value, after which the periodic orbit disappears [28]. Homoclinic bifurcations, treated in this section, lead to this phenomenon. There are other bifurcations, not involving an equilibrium, that show this phenomenon as well, see e.g. [18]. Some authors reserve the phrase blue sky catastrophe for bifurcations not involving an equilibrium. A homoclinic orbit (to p) is a solution ofx ˙ = f(x) converging to p for t → ±∞. We will consider an equilibrium p of saddle-saddle type. There is thus a Df(p) invariant splitting ss ws wu uu E ⊕ E ⊕ E ⊕ E of TpM in strong stable, one dimensional weak stable, one dimensional weak unstable and strong unstable directions. Notation for invariant manifolds is as introduced above. Let Γ = {γ(t) | t ∈ R} be a homoclinic orbit to p. A number of open assumptions will be made.

8 xss Γ

Figure 1: The orbit ﬂip.

(H1) T W s(p) intersects T W u(p) along Γ only in Rγ˙ , (H2) Γ ∈ W s(p)\W ss(p) ∩ W u\W uu, (H3) Γ ∈ W s,wu(p) ⋔ W u(p) and Γ ∈ W s(p) ⋔ W ws,u(p), (H4) λs + λu =6 0.

Here A ⋔ B stands for a transverse intersection of manifolds A and B. Violation of (H 2) is called an orbit ﬂip [30], of (H 3) an inclination ﬂip [15]. A homoclinic orbit with λs + λu =0 is called a resonant homoclinic orbit [6].

Denote by XH the set of vector ﬁelds f with a homoclinic orbit to a saddle-saddle type equilibrium as above. In particular the open and dense conditions (H1), (H2), (H3), (H4) are assumed to hold. Consider a family of diﬀerential equationsx ˙ = fµ(x), depending on a real parameter µ, with f0 ∈ XH . For µ near µ0, fµ possesses an equilibrium pµ near p = p0. Invariant manifolds, such as the stable manifold W s(p), have a continuation in µ. We will s write W (pµ) and similarly for the other invariant manifolds.

Take a local cross section Σ transverse to the ﬂow of f0 at a point in the homoclinic orbit ss,s u,uu Γ. Write d(µ) for the signed distance between the intersections of W (pµ) and W (pµ) with Σ. Then d is a smooth function of µ and d(0) = 0. Recall condition (H 4). By changing the direction of time, if necessary, we may assume that λs + λu < 0 at µ = 0. The following theorem is due to Shil’nikov [31]. A two dimensional version is in [1].

Theorem 4.1. Let x˙ = fµ(x) be a family of diﬀerential equations, depending on a real ∂ parameter µ, with f0 = f as above. Assume that ∂µ d(µ) =6 0 at µ = 0. The family has a

9 W s

ss x Γ

W s

ss x Γ

Figure 2: The inclination ﬂip. The geometry of the convergence of the stable manifold to the equilibrium, for negative times, depends on eigenvalues.

10 hyperbolic periodic orbit for µ > 0 that converges in the Hausdorﬀ topology to Γ as µ → 0. Its period converges to inﬁnity as µ → 0.

5 Planar vector ﬁelds

We will present a proof of Theorem 4.1 for families of diﬀerential equations in the plane.

Lemma 5.1. In smooth local coordinates x =(xs, xu) near p, and after a multiplication by a smooth positive function, the vector ﬁeld X is given by the set of diﬀerential equations

s s x˙ s = λ xs + f (x), u x˙ u = λ xu, where f s(x) satisﬁes

s 2 f (x) = O(|xs| ), as kxsk→ 0.

Proof. We may take smooth coordinates x =(xs, xu) near p, so that s W (p) = {xu =0}, (5.1) u W (p) = {xs =0}, (5.2) In such coordinates we have

s s x˙ s = λ xs + f (x), u u x˙ u = λ xu + f (x),

s u s u for quadratic and higher order terms f , f with f (x)= O(|xs|kxk) as xs → 0 and f (x)= u u O(|xu|kxk) as xu → 0. Dividing the vector ﬁeld by the positive function λ + f (x)/xu gives u s an expression with f = 0 and still f (x)= O(|xs|kxk). 2 Write f s(x , x )= kx x + gs(x , x ) with ∂ gs(x , x )=0in(x , x )=(0, 0). A direct s u s u s u ∂xs∂xu s u s u computation shows that a polynomial coordinate change removes the term kxsxu from the diﬀerential equation for xs. Indeed, let ys = xs + axsxu and yu = xu. Theny ˙u = λuyu and

y˙s = λsys +(k + aλu)ysyu + ...

2 where the dots stand for terms O(|ys| ) and terms of order three and higher. As λu =6 0, we can choose a such that k + aλu = 0. This does the trick. s 2 2 We may now assume f (xs, xu) = O(|xs| )+ O(|xs|kxk ). To ﬁnish the proof, we must remove terms O(xsxu) from the equation for xs. We follow the reasoning from [26]. Consider a coordinate change

ys = xs + p(xu)xs,

yu = xu

11 for a function xu 7→ p(xu) that vanishes at xu = 0. Write the diﬀerential equations in the new coordinates as

y˙s = λsys + P (ys,yu)ys,

y˙u = λuyu.

Treating p as a variable, compute

s s y˙s =x ˙ s +px ˙ s + px˙s = λsxs + f (xs, xu) +px ˙ s + pλsxs + pf (xs, xu)= λsys +py ˙ s + h.o.t.,

where h.o.t. are terms of quadratic and higher order involvingp,p,y ˙ s,yu. Along the unstable manifold ys = 0, P (0,yu) =p ˙+h.o.t., where the higher oder terms are of quadratic and higher order in yu,p. We look for a function p so that P (0,yu) vanishes. This yields diﬀerential equations

p˙ = h.o.t.

y˙u = λuyu

We obtain p as the one dimensional unstable manifold for this system of diﬀerential equations.

Remark 5.2. One can smoothly linearize the vector ﬁeld restricted to the stable manifold s λst W (0). This is done through the inverse of the map e 7→ xs(t), or

1 λ ln z s s z 7→ z 1+ f (xs(v), 0)dv . Z0 ! One can estimate that this is a smooth map. This brings

s 2 f (x)= O(|xs| |xu|).

Take a cross section Σ transverse to the homoclinic orbit.

in out Proposition 5.3. Let Ψ : Σ → Σ be the first return map given by the flow φt of X. Write β = −λs/λu. In coordinates near p as in Lemma 5.1, There exists φ =6 0, ω > 0, so that as xu → 0, xu 7→ Ψ(xu) satisfies

dk Ψ(x ) − φxβ ≤ C |x |β+ω−k, dxk u u k+l u u

for positive constants Ck+ l.

12 in out Proof. Let Σ ⊂{xs = δ}, Σ ⊂{xu = δ} be cross-sections close to p. By a rescaling we may assume δ = 1. Note that in the normal form

s s x˙ s = λ xs + f (x), u x˙ u = λ xu,

s 2 this leaves us with f (x)= δO(|xs| ). It suffices to consider the first hit map Sin : Σin → Σout as the first hit map Sout : Σout → Σin is a local diffeomorphism by the flow box theorem [27]. Write

β = −λs/λu.

We will show that Sin satisﬁes

k in in in β in β+ω−k D in S (x ) − φ(x ) ≤ C |x | , xu u u k+l u

for positive constants C k+l.

Write down the variation of constants formula for an orbit (xs(t), xu(t)) starting in a point (1, x ) ∈ Σin. Observe that this orbit reaches Σout at time τ = − 1 ln x . Thus x (t) = u λu u u λut λu(t−τ) e xu = e t s s λ t λ (t−v) s λu(v−τ) xs(t)= e + e f (xs(v), e )dv. (5.3) Z0 We mimick Picard iteration on a weighted Banach space to construct the orbit and at the same time obtain asymptotics for it. Let

s λst B = {xs : [0, τ] → E | sup |xs(t)|e < ∞}. t∈[0,τ]

−λst Equipped with the norm kxsk = supt∈[0,τ] |xs(t)|e , B is a Banach space. Write Γ : B → B for the map given by the right hand side of (5.3). We claim that there is a ball U ⊂ B so that Γ(U) ⊂ U. λst Suppose |xs| ≤ Re and compute (we adopt the analyst convention that C is a constant that may change from line to line but does not depend on t)

t s s λ t λ (t−v) s λu(v−τ) Γxs = e + e f (xs(v), e )dv Z0 t λst λs(t−v) 2 ≤ e + e Cδ|xs(v)| 0 Z t s s s ≤ eλ t + eλ (t−v)CδR2e2λ v dv 0 s Z λ t 2 ≤ e (1 + CδR ),

13 for some C > 0, using Lemma 5.1. It follows that, for δ small enough, there exists R> 0 so that for kxsk ≤ R, also kΓxsk ≤ R. We see that the orbit xs satisﬁes a bound

λst |xs(t)| ≤ Ce . (5.4)

s s u u Writing f (xs, xu)= g (xs)+ g (xs, xu) with g (xs, 0) = 0, compute

t t λst λs(t−v) s λs(t−v) u λu(v−τ) |xs − e − e g (xs(v))dv| = e g (xs(v), e )dv 0 0 Z Z t ≤ eλs(t−v)Cδe2λsveλu(v−τ)dv. Z0

∞ −λsv s With (5.4) this implies the lemma for xs with φ =1+ 0 e g (xs(v))dv. Bounds for derivatives are obtained similarly by diﬀerentiR ating the variation of constants formula and estimating as before.

Remark 5.4. If one also smoothly linearizes the vector ﬁeld on W s(0), then φ =1.

With the asymptotic formula for Ψ it is now straightforward to prove Theorem 4.1.

6 Higher dimensional analysis

The generalization of the method of proof in Section 5 involves some new ideas, as deriving asymptotic expansions for a ﬁrst hit map near the equilibrium is complicated by the exis- tence of several unstable directions. We start again with a normal form result bringing the expression for the diﬀerential equations close to a linear form.

6.1 Normal forms

We formulate a normal form result for single vector ﬁelds. An analogous result holds for families.

Lemma 6.1. There is a smooth local coordinate system x =(xss, xws), xu =(xwu, xuu) near p in which the vector ﬁeld X is given by the set of diﬀerential equations

ss ss x˙ ss = A xss + f (x), s ws x˙ ws = λ xws + f (x), u wu x˙ wu = λ xwu + f (x), uu uu x˙ uu = A xuu + f (x),

14 where the quadratic and higher order terms f ss(x),...,f uu(x) satisfy

ss 2 f (x) = O(kxsskkxk)+ O(|xws| ), ws 2 f (x) = O(kxsskkxwu,uuk)+ O(|xws| kxwu,uuk), as kxss,wsk→ 0 and

wu 2 f (x) = O(kxuukkxss,wsk)+ O(|xwu| kxss,wsk), uu 2 f (x) = O(kxuukkxk)+ O(|xwu| ), as kxwu,uuk→ 0.

Proof. Recall that near p there exist a (local) stable manifold W ss,ws(p) and a (local) unstable manifold W wu,uu(p). The stable and unstable manifold are both smooth mani- ws,wu,uu folds. Further, there is a codimension-qss center unstable manifold W (p) with tan- ws wu uu ss,ws,wu gent space E ⊕ E ⊕ E at p, and a codimension-quu center stable manifold W (p) ss ws wu ss,ws,wu with tangent space E ⊕ E ⊕ E at p. The tangent bundles TW ss,ws(p)W (p) ws,wu,uu and TW wu,uu(p)W (p) are smooth bundles. We may take thus smooth coordinates x =(xss, xws, xwu, xuu) near p, so that

ss,ws W (p) = {xwu, xuu =0}, (6.1) wu,uu W (p) = {xss, xws =0}, (6.2) ss,ws,wu W (p) ∩W ss,ws(p) {xuu =0}, (6.3) ws,wu,uu W (p) ∩W wu,uu(p) {xss =0}. (6.4)

ss The notation W ∩qV means that W is tangent to V at q. From (6.2) it follows that f and ws wu uu f are O(kxss,wskkxk). Similarly, from (6.1) we get that f and f are O(kxwu,uukkxk). With in addition (6.4) and (6.3) we get

ss 2 f = O(kxsskkxk)+ O(|xws| ), ws f = O(kxss,wskkxk), wu f = O(kxwu,uukkxk), uu 2 f = O(kxuukkxk)+ O(|xwu| ).

The following coordinate changes are as in the proof of Lemma 5.1. A polynomial coordinate change removes quadratic terms xwsxwu from the diﬀerential equations for xws. Consider next a change of coordinates of the form

yss = xss,

yws = xws + pws(xwu, xuu)xws,

ywu = xwu,

yuu = xuu,

15 for pws that vanishes along xwu, xuu = 0. Write the diﬀerential equations in the new coordinates y as ss y˙ss = A yss + gss(y), s y˙ws = λ yws + Pws(y)yws + gws(y)yss, u y˙wu = λ ywu + gwu(y), uu y˙uu = A yuu + guu(y).

Along the unstable manifold yss,ws = 0 we have

Pws(0,ywu,uu) =p ˙ws + h.o.t.,

As before, considerp ˙ws as a variable and construct the local unstable manifold tangent to {pws =0} at the origin for the resulting diﬀerential equations,

p˙ws = h.o.t., u y˙wu = λ ywu + gwu(y), uu y˙uu = A yuu + guu(y) ws along the unstable manifold yss,yws = 0. The resulting coordinate change will transform f to a map with asymptotics ws 2 f (x)= O(kxsskkxk)+ O(|xws| ). wu As the coordinate change left ywu unaltered, a similar coordinate change for f can be performed. The final coordinate changes are restricted to the local stable and unstable manifolds. Con- sider the local stable manifold. There is a smooth strong stable foliation on it extending the strong stable manifold. By a smooth coordinate change we may assume that the strong stable foliation is affine {xws = const.}. Then we smoothly linearize the vector field in the xws coordinates.

6.2 Linear vector ﬁelds

Under additional eigenvalue conditions, one can ﬁnd Ck local coordinates near p in which the vector ﬁeld is linear for all parameter values near the bifurcation value. The following result is due to Takens [33].

Proposition 6.2. Given Dfµ0 (p) and k > 0, there exists N(k) so that the following holds.

Suppose the real parts of the eigenvalues of Dfµ0 (p) do not satisfy resonance conditions of order N(k) or less, that is,

Re λj =6 aiRe λi, i X k for ai ≥ 0 with 1 < i ai ≤ N(k). Then there are C local coordinates near p, also depending k C on µ, in which the vector ﬁeld is linear for µ close to µ0. P 16 in out Let Σ ⊂ {xws = δ}, Σ ⊂ {xwu = δ} be cross-sections close to p. By a rescaling we in in in in in may assume δ = 1. Write x = (xss, xwu, xuu) for the coordinate system on Σ inherited out out out out from the coordinates near p. Write similarly (xss , xws , xuu ) for coordinates on Σ . A straigthforward computation shows that Ψ : Σin → Σout is given by

ss uu in in in in −A /λu in in −λs/λu in −A /λu in Ψ(xss, xwu, xuu)=( xwu xss, xwu , xwu xuu) (6.5) 6.3 Shilnikov variables

We consider a single vector field. Families are treated analogously. in out Let Σ ⊂{xws = δ}, Σ ⊂{xwu = δ} be cross-sections close to p. By a rescaling we may in in in in in assume δ = 1. Write x =(xss, xwu, xuu) for the coordinate system on Σ inherited from the out out out out coordinates near p. Write similarly (xss , xws , xuu ) for coordinates on Σ . The following result discusses the first hit map Ψ : Σin → Σout in Shil’nikov variables, see [8, 9] and [17] for the specific formulation used here. In its statement, one encounters functions that contain fractional powers of a real positive variable x. These functions can be differentiated, losing a power in x with each derivative with respect to x. We will use the following notation for this smoothness property. Notation 6.3. Let (x, y) 7→ f(x, y) with x ∈ R, be a function that depends smoothly on y α k l and on x for x > 0. When writing f(x, y) = O(x ), we will assume that kDy Dxf(x, y)k ≤ α−l Ck+lx for positive constants Ck+l. Proposition 6.4. Let Ψ : Σin → Σout be the first hit map given by the flow of X. Write s u in in in out out out β = −λ /λ . In coordinates near p as in Lemma 6.1, Ψ(xss, xwu, xuu)=(xss , xws , xuu ) satisfies the following. There exists a map S so that we can write

out out in in in out (xss , xws , xuu)= S(xss, xwu, xuu )

ss ws uu There exists ω > 0, so that as xwu → 0, S =(S ,S ,S ) satisﬁes

ss in in out in β+ω S (xss, xwu, xuu ) = O xwu , ws in in out in β in β+ω S (xss, xwu, xuu ) = xwu + O xwu , uu in in out in 1+ω S (xss, xwu, xuu ) = O xwu . Proof. On a compact neighborhood of the origin that includes Σin and Σout, there is K > 0 so that

ss 2 kf (x)k ≤ Kδ(kxsskkxk + xws), ws 2 |f (x)| ≤ Kδ(kxsskkxwu,uuk + xwskxwu,uuk), wu 2 |f (x)| ≤ Kδ(kxuukkxws,ssk + xwukxws,ssk), uu 2 kf (x)k ≤ Kδ(xwu + kxuukkxk).

17 0 By the variation of constants formula, an orbit x(t) with xss,ws(0) = xss,ws and xwu,uu(τ)= τ xwu,uu, satisﬁes t Asst 0 Ass(t−v) ss xss(t) = e xss + e f (x(v))dv, (6.6) 0 t Z λst λs(t−v) ws xws(t) = e + e f (x(v))dv, (6.7) 0 Z τ λu(t−τ) λu(t−v) wu xwu(t) = e − e f (x(v))dv, (6.8) t Z τ Auu(t−τ) τ Auu(t−v) uu xuu(t) = e xuu − e f (x(v))dv. (6.9) Zt Let ω be a small positive number and consider the space E of functions x = {x(t), 0 ≤ t ≤ τ} for which −(λs−ω)t kxss(t)ke < ∞, −λst |xws(t)|e < ∞, −λu(t−τ) |xwu(t)|e < ∞, −(λu+ω)(t−τ) kxuu(t)ke < ∞. Equip E with the norm −(λs−ω)t −λst −λu(t−τ) −(λu+ω)(t−τ) kxk = sup kxss(t)ke , |xws(t)|e , |xwu(t − τ)|e , kxuu(t − τ)ke . 0≤t≤τ The right hand side of (6.6),. . . , (6.9) deﬁnes a map G on E. Write Br = {x ∈ E; kxk ≤ r} for the ball with radius r in E. We claim that G is a contraction on some ball in E: there exist r> 0 and 0 <λ< 1, so that

• G(Br) ⊂ Br.

• kG(x1) − G(x2)k ≤ λkx1 − x2k, for x1, x2 ∈ Br.

To check the ﬁrst item, consider the strong stable coordinate xss of x ∈ Br. Take ω small ss s enough so that λss <λs − 2ω. We may assume that keA tk ≤ e(λ −ω)t. In the following, C will denote a positive constant which can change from line to line, but does not depend on t. Now kGss(x)(t)k t (λs−ω)t 0 (λs−ω)(t−v) ss ≤ e kxssk + e kf (x(v))kdv 0 Z t (λs−ω)t 0 (λs−ω)(t−v) 2 ≤ e kxssk + e Cδ kxss(v)kkxws(v)k + kxss(v)kkxwu(v)k + kxws(v)k dv Z0 t (λs−ω)t 0 (λs−ω)(t−v) (λs−ω)v λsv (λs−ω)v λu(v−τ) 2λsv ≤ e kxssk + e Cδ e e + e e + e dv 0 s Z u ≤ e(λ −ω)t C + Cδ 1+ eλ (t−τ) . 18 Corresponding estimates for the other coordinates, xws(t), xwu(t), xuu(t) are obtained similarly. The ﬁrst item holds by taking r large enough and δ small enough. To prove the second item, take x1, x2 ∈ Br with x1,ss(0) = x2,ss(0) and x1,uu(τ) = x2,uu(τ). Restricting again to the strong stable coordinates,

kGss(x1(t)) − Gss(x2(t))k t (λs−ω)(t−v) ss ss ≤ e kf (x1(v)) − f (x2(v))kdv 0 Z t (λs−ω)(t−v) λsv λu(v−τ) λsv ≤ e Cδ (e + e )kx1,ss(v) − x2,ss(v)k + e kx1,ws(v) − x2,ws(v)k dv 0 Z (λs−ω)t (−λs+ω)v (λs−ω)t −λsv ≤ Cδe sup e kx1,ss(v) − x2,ss(v)k + Cδe sup e kx1,ws(v) − x2,ws(v)k v v (λs−ω)t ≤ Cδe kx1 − x2k.

0 τ Thus, given τ large and given xss, xuu, there is a unique solution to (6.6),. . . , (6.9), i.e. a 0 τ in unique orbit x(t) with xss(0) = xss, xuu(τ)= xuu, that needs time τ to move from {xws =1} out to {xwu =1}. Moreover, since kxk is bounded, for some C > 0,

(λs−ω)t kxss(t)k ≤ Ce , λst |xws(t)| ≤ Ce , λu(t−τ) |xwu(t)| ≤ Ce , (λu+ω)(t−τ) kxuu(t)k ≤ Ce .

out in The leading order terms in the exponential expansions of xws and xwu are computed as in the planar case. With r = e−λuτ , it follows that

out −λs/λu ss in out xss = r T (r, xss, xuu ), (6.10) out −λs/λu s in out −λs/λu ws in out xws = r φ (xss, xuu )+ r T (r, xss, xuu ), (6.11) in u in out wu in out xwu = rφ (xss, xuu )+ rT (r, xss, xuu ), (6.12) in uu in out xuu = rT (r, xss, xuu ). (6.13) Here φs,φu are smooth functions and T ss, T ws, T wu, T uu are smooth for r> 0, with

l k ∂ i in out ω/λu−l Dxin,xout T (r, xss, xuu ) ≤ Ck+lr , (6.14) ss uu ∂r

i = ss,ws,wu,uu. u in in out Because of φ (0, 0) =6 0 the function r can be solved from (6.12) as function of xss, xwu, xuu , in see also [24] for a calculation in three dimensions. This function is smooth for xwu > 0. From (6.12) and (6.14) (for i = u), one estimates that

k l in in out in in 1+ω−k D in D in out r(x , x , x ) − x ≤ C |x | . xwu xss ,xuu ss wu uu wu k+l+m wu

19 Putting this in the remaining formulas (6.10), (6.11), (6.13) proves the proposition.

Take a cross-section Σ transverse to Γ. Extend the coordinate system (xss, xws, xwu, xuu) near p to a coordinate system near {p} ∪ Γ. Suppose that

ss,ws,wu wu,uu W (p) ⋔Γ W (p), (6.15) ws,wu,uu ss,ws W (p) ⋔Γ W (p). (6.16)

The following result can be seen as giving a normal form for a return map on Σ, in Shil’nikov variables.

Proposition 6.5. Let Ψ : Σ → Σ be the ﬁrst return map given by the ﬂow of X, and let points x(j), x(j + 1) in Σ be such that x(j +1) = Ψ(x(j)). If (6.15), (6.16) holds, then a coordinate system x =(xss, xwu, xuu) on Σ can be chosen so that

ss,ws W (p) ∩ Σ = {xwu,uu =0}, (6.17) wu,uu W (p) ∩ Σ = {xss,wu =0}, (6.18) ss,ws,wu W (p) ∩ Σ ∩W ss,ws(p)∩Σ {xuu =0}, (6.19) ws,wu,uu W (p) ∩ Σ ∩W wu,uu(p)∩Σ {xss =0}. (6.20)

In these coordinates, x(j + 1) and Ψ(x(j)) are related by

(xss(j + 1), xwu(j + 1), xuu(j)) = T (xss(j), xwu(j), xuu(j + 1)),

for some map T , with the following asymptotics:

β+ω xss(j +1) = O(xwu(j) ), β β+ω xwu(j +1) = ϕxwu(j) + O(xwu(j) ), (6.21) 1+ω xuu(j) = O(xwu(j + 1) ),

where ϕ is a smooth nonvanishing function of xss(j + 1), xuu(j).

Proof. There are smooth vector bundles F ss and F s,wu along the stable manifold W s(p), ss,ws,wu ss ws wu ss ss invariant for the ﬂow of DX, with Fp = E ⊕ E ⊕ E and Fp = E . Similarly, there are smooth vector bundles F ss and F ss,ws,wu along the unstable manifold W wu,uu(p), ws,wu,uu ws wu uu uu uu with Fp = E ⊕ E ⊕ E and Fp = E . By changing the time parameterization ss uu of the ﬂow of X, we may assume that leaves Fx and Fx are contained in Σ for x ∈ Σ. By the transversality assumptions (6.15), (6.16), F ss, F uu, F ss,ws,wu and F ws,wu,uu extend to continuous vector bundles along {p} ∪ Γ, see e.g. [14]. The intersection of F ss,,ws,wu and ws,wu,uu ws,wu ws,wu ws wu F provides a continuous bundle F along Γ with Fp = E ⊕ E . Observe ss ws,wu uu that TΓM = F ⊕ F ⊕ F . It follows that coordinates (xss, xwu, xuu) on Σ satisfying (6.17),. . . ,(6.20) exist.

20 in in in in Recall that xss, xwu, xuu are coordinates on Σ . If xss, xwu, xuu denote coordinates on Σ, then in in in in (xss, xwu, xuu) = χ (xss, xwu, xuu) (6.22) in for some smooth local diffeomorphism χ . If the coordinates (xss, xwu, xuu) are chosen so ss,ws ss,ws,wu that Γ intersects Σ in (0, 0, 0) and W = {xwu,uu = 0}, TW ss,ws W = {xuu = 0}, then ∗ ∗ ∗ Dχin(0, 0, 0) = 0 ∗ ∗ (6.23)   0 0 ∗ and   2 xss O(kxk ) in in χ (xss, xwu, xuu) = Dχ (0, 0, 0) xwu + O(kxwu,uukkxk) .    2  xuu O(kxuukkxk + |xwu| ) By the implicit function theorem there is a map ψ (defined  on a small neighborhood) so that in in in (xss, xwu, xuu) = ψ(xss, xwu, xuu) (6.24) and ∗ ∗ ∗ Dψ(0, 0, 0) = 0 ∗ ∗ .   0 0 ∗ This yields   in in 2 xss xss O(k(xss, xwu, xuu) k ) in in in xwu = Dψ(0, 0, 0) xwu + O(kxwukk(xss, xwu) k) .    in   in in 2  xuu xuu O(kxuukk(xss, xwu) )k + |xwu| )      out By combining this and the asymptotics for S in Proposition 6.4 one solves (xws , xuu) as out functions of (xss, xwu ). This goes as follows. From the list of equations (6.25),. . . ,(6.30) in in in in below, use equations (6.27), (6.28), (6.29) to solve x = (xss, xwu, xuu) as functions of out (xss, xwu, xuu ) by the implicit function theorem. Put this in the remaining equations (6.25), (6.26), (6.30). out ss in in out xss = S (xss, xwu, xuu ), (6.25) out s in in out xws = S (xss, xwu, xuu ), (6.26) in uu in in out xuu = S (xss, xwu, xuu ), (6.27) in ss in xss = ψ (xss, xwu, xuu), (6.28) in u in xwu = ψ (xss, xwu, xuu), (6.29) uu in xuu = ψ (xss, xwu, xuu). (6.30) out out out out Similarly one relates the coordinates xss , xws , xuu on Σ to coordinates on Σ by a local diffeomorphism χout. The asymptotics are now implied by Proposition 6.4.

21 6.4 Families of vector ﬁelds

Consider a familyx ˙ = fµ(x) depending on a parameter µ.

Notation 6.6. Let (x, y) 7→ fµ(x, y) with x ∈ R, be a function that depends smoothly on y, α on the parameter µ, and on x for x > 0. When writing fµ(x, y) = O(x ), we will assume k l α−l that kDy,µDxfµ(x, y)k ≤ Ck+lx for positive constants Ck+l.

Proposition 6.7. Let Ψµ : Σ → Σ be the ﬁrst return map given by the ﬂow of fµ, and let points x(j), x(j + 1) in Σ be such that x(j +1)=Ψµ(x(j)). If (6.15), (6.16) holds, then a parameter dependent coordinate system x =(xss, xwu, xuu) on Σ can be chosen so that

ss,ws W (p) ∩ Σ = {xwu,uu =0}, wu,uu W (p) ∩ Σ = {xss,wu =0}, ss,ws,wu W (p) ∩ Σ ∩W ss,ws(p)∩Σ {xuu =0}, ws,wu,uu W (p) ∩ Σ ∩W wu,uu(p)∩Σ {xss =0}.

In these coordinates, x(j + 1) and Ψµ(x(j)) are related by

(xss(j + 1), xwu(j + 1), xuu(j)) = Tµ(xss(j), xwu(j), xuu(j + 1)),

for some map Tµ, with the following asymptotics:

β+ω xss(j +1) = O(xwu(j) ), β β+ω xwu(j +1) = a(µ)+ ϕ(µ)xwu(j) + O(xwu(j) ), (6.31) 1+ω xuu(j) = O(xwu(j + 1) ). Here a and ϕ are smooth functions of µ.

Proof. The normal form given in Lemma 6.1 also holds for parameter dependent vector ﬁelds, where the linear and higher order terms depend smoothly on the parameter. The corresponding coordinate system depends smoothly on the parameter. The parameter dependent version of Proposition 6.4 is proven in much the same way as the proposition itself; one considers in addition derivatives with respect to the parameter, see also [8, 9]. The choice of coordinates in Proposition 6.5 can be made parameter dependent. An additional translation results from breaking the homoclinic connection, yielding the term a(µ).

6.5 Bifurcation equations

The equations for a periodic orbit intersecting the cross section Σ once, are

β+ω xss = O(xwu ), β β+ω xwu = a(µ)+ ϕ(µ, xss, xuu)xwu + O(xwu ), (6.32) 1+ω xuu = O(xwu ).

22 We can solve xss, xuu as function of xwu and the parameter µ by the implicit function theorem. Note that the equations are not smooth in xwu, see Remark 2.4 and [5, 6]. This leads to a β β+ω reduced bifurcation equation xwu = a(µ)+ ϕ(µ)xwu + O(xwu ), from which the bifurcation theorem is readily proved.

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