On the regularization of impact without collision: the Painlevé paradox and compliance

1. Introduction

In mechanics, in problems with unilateral constraints in the presence of friction, the rigid-body assumption can result in the governing equations having multiple solutions (the indeterminate case) or no solutions (the inconsistent case). The classical example of Painlevé [1–3], consisting of a slender rod slipping¹ along a rough surface (figure 1), is the simplest and most studied example of these phenomena, now known collectively as Painlevé paradoxes [5–8]. Such paradoxes can occur at physically realistic parameter values in many important engineering systems [9–15].

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When a system has no consistent solution, it cannot remain in that state. Lecornu [16] proposed a jump in vertical velocity to escape an inconsistent, horizontal velocity, state. This jump has been called impact without collision (IWC) [17], tangential impact [18] or dynamic jamming [13]. Experimental evidence of IWC is given in [15]. IWC can be incorporated into the rigid-body formulation [19,20] by considering the equations of motion in terms of the normal impulse, rather than time.

Génot & Brogliato [17] considered the dynamics around a critical point, corresponding to zero vertical acceleration of the end of the rod. They proved that, when starting in a consistent state, the rod must stop slipping before reaching the critical point. In particular, paradoxical situations cannot be reached after a period of slipping.

One way to address the Painlevé paradox is to regularize the rigid-body formalism. Physically, this often corresponds to assuming some sort of compliance at the contact point A, typically thought of as a spring, with stiffness (and sometimes damping) that tends to the rigid body model in a suitable limit. Mathematically, very little rigorous work has been done on how IWC and Painlevé paradoxes can be regularized. Dupont & Yamajako [21] treated the problem as a slow–fast system, as we will do. They explored the fast time-scale dynamics, which is unstable for the Painlevé paradoxes. Song et al. [22] established conditions under which these dynamics can be stabilized. Le Suan An [23] considered a system with bilateral constraints and showed qualitatively the presence of a regularized IWC as a jump in vertical velocity from a compliance model with diverging stiffness. Zhao et al. [24] considered the example in figure 1 and regularized the equations by assuming a compliance that consisted of an undamped spring. They estimated, as a function of the stiffness, the orders of magnitude of the time taken in each phase of the (regularized) IWC. Another type of regularization was considered by Neimark & Smirnova [25], who assumed that the normal and tangential reactions took (different) finite times to adjust.

In this paper, we present the first rigorous analysis of the regularized rigid-body formalism, in the presence of compliance with both stiffness and damping. We recover IWC in both the inconsistent and the indeterminate cases, and in the latter case, we present a formula for conditions that separate IWC and lift-off. We solve the problem for arbitrary values of the compliance damping and give explicit asymptotic expressions in the limiting cases of small and large damping. Our results apply directly to a general class of rigid bodies. Our approach is similar to that used in [26,27] to understand the forward problem in piecewise smooth (PWS) systems in the presence of a twofold.

The paper is organized as follows. In §2, we introduce the problem, outline some of the main results known to date and include compliance. In §3, we give a summary of our main results, theorems 3.1 and 3.2, before presenting their derivation in §§4 and 5. We discuss our results in §6 and outline our conclusion in §7.

2. Classical Painlevé problem

Consider a rigid rod AB, slipping on a rough horizontal surface, as depicted in figure 1.

The rod has mass m, length 2l, the moment of inertia of the rod about its centre of mass S is given by I and its centre of mass coincides with its centre of gravity. The point S has coordinates (X,Y) relative to an inertial frame of reference (x,y) fixed in the rough surface. The rod makes an angle θ with respect to the horizontal, with θ increasing in a clockwise direction. At A, the rod experiences a contact force (−F_T,F_N), which opposes the motion. The dynamics of the rod is then governed by the following equations:

The coordinates (X,Y) and (x,y) are related geometrically as follows:

We now adopt the scalings $(X,Y)=l(\tilde{X},\tilde{Y}),(x,y)=l(\tilde{x},\tilde{y}),(F_T,F_N)=mg({\tilde{F}}_T,{\tilde{F}}_N)$, $t=1/\omega \tilde{t},\alpha =m{l}^2/I$, where ω²=g/l. For a uniform rod, $I=\frac{1}{3}m{l}^2$, and so α=3 in this case.

Then for general α, (2.1) and (2.2) can be combined to become, on dropping the tildes,

Suppose F_N is known. Then system (2.5) is a Filippov system [4]. Hence, we obtain a well-defined forward flow when $\dot{x}=v=0$ and

Proposition 2.1.

The Filippov vector-field, within the subset of the switching manifold $\mathit{\Sigma }:\dot{x}=v=0$ where (2.7) holds, is given by

Remark 2.2.

Our results hold for mechanical systems with different q_±, p_± and c_± in (2.6) and even dependency on several angles $\theta \in {\mathbb{T}}^d$, e.g. the two-link mechanism of Zhao et al. [15]. As expected, S_w and S_ϕ in (2.9) are independent of μ, even for general q_±, p_± and c_±.

To solve (2.5) and (2.8), we need to determine F_N. The constraint-based method leads to the Painlevé paradox. The compliance-based method is the subject of this paper.

(a) Constraint-based method

In order that the constraint y=0 be maintained, $\ddot{y}(=\dot{w})$ and F_N form a complementarity pair given by

$$\dot{w}\ge 0,F_N\ge 0,F_N\cdot \dot{w}=0.$$2.10

Note that F_N≥0 since the rough surface can only push, not pull, the rod. Then for general motion of the rod, F_N and y satisfy the complementarity conditions

$$0\le F_N\perp y\ge 0.$$2.11

In other words, at most one of F_N and y can be positive.

For the system shown in figure 1, the Painlevé paradox occurs when v>0 and θ∈(0,π/2), provided p₊(θ)<0, as follows. From the fourth equation in (2.5), we can see that b is the free acceleration of the end of the rod. Therefore, if b>0, lift-off is always possible when y=0, w=0. But if b<0, in equilibrium we would expect a forcing term F_N to maintain the rod on y=0. From $\dot{w}=0$, we obtain

$$F_N=-\frac{b}{p_{+}}$$2.12

since v>0. If p₊>0, which is always true for θ∈(π/2,π), then F_N≥0, in line with (2.11). But if p₊<0, which can happen if θ∈(0,π/2), then F_N<0 in (2.12). Then, F_N is in an inconsistent (or non-existent) mode. On the other hand, if b>0 and p₊<0, then F_N>0 in (2.12). At the same time, lift-off is also possible from y=0 and hence F_N is in an indeterminate (or non-unique) mode. It is straightforward to show that p₊(θ)<0 requires

$$\mu >{\mu }_P(\alpha )\equiv \frac{2}{\alpha }\sqrt{1+\alpha }.$$2.13

Then, the Painlevé paradox can occur for θ∈(θ₁,θ₂), where

$$\begin{array}{rl}{\theta }_1(\mu ,\alpha )& =\arctan \hspace{0.17em}\frac{1}{2}(\mu \alpha -\sqrt{{\mu }^2{\alpha }^2-4(1+\alpha )})\\ \mathrm{and}{\theta }_2(\mu ,\alpha )& =\arctan \hspace{0.17em}\frac{1}{2}(\mu \alpha +\sqrt{{\mu }^2{\alpha }^2-4(1+\alpha )}).\end{array}\}$$2.14

For a uniform rod with α=3, we have μ_P(3)=4/3. For α=3 and μ=1.4, the dynamics can be summarized² in the (θ,ϕ)-plane, as in figure 2. Along θ=θ₁,θ₂, we have p₊(θ)=0. These lines intersect the curve b(θ,ϕ)=0 at four points: ${\varphi }_{1,2}^{\pm }=\pm \sqrt{\csc \hspace{0.17em}{\theta }_{1,2}}$. Génot & Brogliato [17] showed that the point $P:(\theta ,\varphi )=({\theta }_1,\sqrt{\csc \hspace{0.17em}{\theta }_1})$ is the most important and analysed the local dynamics around it. The rigid body equations (2.1) are unable to resolve the dynamics in the third and fourth quadrants. So, we regularize these equations using compliance.

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3. Main results

We now present the main results of our paper, theorems 3.1 and 3.2. Theorem 3.1 shows that, if the rod starts in the fourth quadrant of figure 2, it undergoes (regularized) IWC for a time of $\mathcal{O}(ϵ\hspace{0.17em}\ln \hspace{0.17em}{ϵ}^{-1})$. The same theorem also gives expressions for the resulting vertical velocity of the rod in terms of the compliance damping and initial horizontal velocity and orientation of the rod.

Theorem 3.1.

Consider an initial condition

Theorem 3.2 is similar to theorem 3.1, but now the rod starts in the first quadrant of figure 2. This theorem also gives an exact formula for initial conditions that separate (regularized) IWC and lift-off.

Theorem 3.2.

Consider an initial condition

Remark 3.3.

These two theorems have not appeared before in the literature. In the rigid-body limit ($ϵ\to 0$), we recover IWC in both cases. Previous authors have not carried out the ‘very difficult’ calculation [28], performed numerical calculations [6,21] or given a range of estimates for the time of (regularized) IWC in the absence of damping [24]. We give exact and asymptotic expressions for key quantities as well as providing a geometric interpretation of our results, for a large class of rigid bodies, in the presence of a large class of normal forces, as well as giving a precise estimate for the time of (regularized) IWC, all in the presence of both stiffness and damping. Note that we are not attempting to describe all the dynamics around P. There is a canard connecting the third quadrant with the first, and the analysis of it is exceedingly complicated [29] due to fast oscillatory terms. Instead, we follow [24] and consider that the rod dynamics starts in a configuration with p₊(θ₀)<0.

4. Proof of theorem 3.1: impact without collision in the inconsistent case

The proof of theorem 3.1 is divided into three phases, illustrated in figure 3. These phases are a generalization of the phases of IWC in its rigid-body formulation [15].

• — Slipping compression (§4b): During this phase, y, w and v all decrease. The dynamics follow an unstable manifold γ^u of a set of critical points C, given in (4.4) below, as $ϵ\to 0$. Along γ^u the normal force $F_N=\mathcal{O}({ϵ}^{-1})$ and v will therefore quickly decrease to 0. Mathematically, this part is complicated by the fact that the initial condition (3.1) belongs to the critical set C as $ϵ\to 0$.

• — Sticking (§4c): Since $F_N=\mathcal{O}({ϵ}^{-1})$ and q₊q₋<0, the rod will stick with v≡0. During this phase, $\ddot{y}=\dot{w}>0$ and eventually sticking ends with F_N=0 as $ϵ\to 0$.

• — Lift-off (§4d): In the final phase F_N=0, lift-off occurs and the system eventually returns to y=0.

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(b) Slipping compression

The first phase of the regularized IWC: slipping compression ends on the switching manifold

$$\mathit{\Sigma }=\{(\hat{y},w,\theta ,\varphi ,v)|v=0\},$$4.8

shown in figure 3a. Proposition 4.4 describes the intersection of the forward flow of initial conditions (3.1) with Σ.

Proposition 4.4.

The forward flow of the initial conditions (3.1) under (4.3) intersects Σ in

$$\begin{array}{rl}{\gamma }^{\mathrm{u}}\cap \mathit{\Sigma }+o(1)& \equiv \{\phantom{\frac{1}{2}}(\hat{y},w,\theta ,\varphi ,v)\in \mathit{\Sigma }|\hspace{0.17em}\hat{y}=-\frac{p_{+}({\theta }_0)}{q_{+}({\theta }_0){\mathrm{\lambda }}_{+}({\theta }_0)}{v}_0+o(1),\hspace{0.17em}w=-\frac{p_{+}({\theta }_0)}{q_{+}({\theta }_0)}{v}_0+o(1),\\ & \theta ={\theta }_0+o(1),\hspace{0.17em}\varphi ={\varphi }_0-\frac{c_{+}({\theta }_0)}{q_{+}({\theta }_0)}{v}_0+o(1)\},\end{array}$$4.9

as $ϵ\to 0$.

Remark 4.5.

The o(1)-term in (4.9) is $\mathcal{O}({ϵ}^c)$ for any c∈(0,1) (see also lemma 4.8).

(i) Proof of proposition 4.4

We prove proposition 4.4 using Fenichel’s normal form theory [33]. But since (4.3)${\hspace{0.17em}}_{{\hat{F}}_N=[-\hat{y}-\delta w]}$ with v>0 is PWS, care must be taken. There are at least two ways to proceed. One way is to consider the smooth system (4.3)${\hspace{0.17em}}_{{\hat{F}}_N=-\hat{y}-\delta w}$, then rectify C_ϵ by straightening out its stable and unstable manifolds. Then, (4.3)${\hspace{0.17em}}_{{\hat{F}}_N=-\hat{y}-\delta w}$ will be a standard slow–fast system to which Fenichel’s normal form theory applies. Subsequently, one would then have to ensure that conclusions based on the smooth (4.3)${\hspace{0.17em}}_{{\hat{F}}_N=-\hat{y}-\delta w}$ also extend to the PWS system (4.3)${\hspace{0.17em}}_{{\hat{F}}_N=[-\hat{y}-\delta w]}$. One way to do this is to consider the following scaling

$${\kappa }_1:\hspace{0.17em}\hat{y}=r_1{\hat{y}}_1,w=r_1{w}_1,ϵ=r_1,$$4.10

zooming in on C at $\hat{y}=0,w=0$. In terms of the original variables, $y={ϵ}^2{\hat{y}}_1$, w=ϵw₁. Both the scalings $(\hat{y},w)$ and $({\hat{y}}_1,w_1)$ have appeared in the literature [6,21,28].

In this paper, we follow another approach (basically reversing the process described above) which works more directly with the PWS system. Therefore, in §4b(ii), we study the scaling (4.10) first. We will show that the $({\hat{y}}_1,w_1)$-system contains important geometry of the PWS system (significant, for example, for the separation of initial conditions in theorem 3.2). Then in §4b(iii), we connect the ‘small’ $(\hat{y}=\mathcal{O}(ϵ),w=\mathcal{O}(ϵ))$ described by (4.10) with the ‘large’ ($\hat{y}=\mathcal{O}(1),w=\mathcal{O}(1)$) in (4.3) by considering coordinates described by the following transformation:

$${\kappa }_2:\hspace{0.17em}\hat{y}=-r_2,w=r_2{w}_2,ϵ=r_2{ϵ}_2.$$4.11

For y₁<0, we have the following coordinate change κ₂₁ between κ₁ and κ₂:

$${\kappa }_{21}:\hspace{0.17em}{r}_2=-r_1{y}_1,w_2=-w_1{y}_1^{-1},{ϵ}_2=-y_1^{-1}.$$4.12

The coordinates in κ₂ (4.11) appear as a directional chart obtained by setting $\overline{y}=-1$ in the blowup transformation $(r,\overline{\hat{y}},\overline{w},\overline{ϵ})\mapsto (\hat{y},w,ϵ)$ given by³

$$\hat{y}=r\overline{\hat{y}},w=r\overline{w},ϵ=r\overline{ϵ},r\ge 0,(\overline{\hat{y}},\overline{w},\overline{ϵ})\in S^2=\{(\overline{\hat{y}},\overline{w},\overline{ϵ})|{\overline{\hat{y}}}^2+{\overline{w}}^2+{\overline{ϵ}}^2=1\}.$$4.13

The blowup is chosen so that the zoom in (4.10) coincides with the scaling chart obtained by setting $\overline{ϵ}=1$. The blowup transformation blows up C to $\overline{C}:\hspace{0.17em}r=0,\hspace{0.17em}(\overline{\hat{y}},\overline{w},\overline{ϵ})\in S^2$ a space $(\theta ,\varphi ,v)\in {\mathbb{R}}^3$ of spheres.⁴

The main advantage of our approach is that in chart κ₂ we can focus on $\overline{C}\cap \{{\overline{\hat{y}}}^{-1}\overline{w}>-{\delta }^{-1},\overline{\hat{y}}<0\}$ (or simply w₂<δ⁻¹ in (4.11)) of $\overline{C}$, the grey area in figure 3, where

$$r^{-1}{\hat{F}}_N(\hat{y},w)=[-\overline{\hat{y}}-\delta \overline{w}]=-\overline{\hat{y}}(1+\delta {\overline{\hat{y}}}^{-1}\overline{w})>0,$$4.14

and the system will be smooth. This enables us to apply Fenichel’s normal form theory [33] there. All the necessary patching for the PWS system is done independently in the scaling chart κ₁.

(ii) Chart κ₁

Let ${\hat{F}}_{N,1}({\hat{y}}_1,w_1)={ϵ}^{-1}{\hat{F}}_N(ϵ{\hat{y}}_1,ϵw_1)=[-{\hat{y}}_1-\delta w_1].$ Then applying chart κ₁ in (4.10) to the non-standard slow–fast system (4.3) gives the following equations:

$$\begin{array}{rl}{\hat{y}}_1^{\prime }& =w_1,w_1^{\prime }=b(\theta ,\varphi )+p_{+}(\theta ){\hat{F}}_{N,1}({\hat{y}}_1,w_1),\\ {\theta }^{\prime }& =ϵ\varphi ,{\varphi }^{\prime }=ϵc_{+}(\theta ){\hat{F}}_{N,1}({\hat{y}}_1,w_1)\\ \mathrm{and}{v}^{\prime }& =ϵ(a(\theta ,\varphi )+q_{+}(\theta ){\hat{F}}_{N,1}({\hat{y}}_1,w_1)).\end{array}\}$$4.15

The above equation is a slow–fast system in standard form: $({\hat{y}}_1,w_1)$ are fast variables, whereas (θ,ϕ,v) are slow variables. By assumption (3.2) of theorem 3.1, b<0, p₊<0 and so, since ${\hat{F}}_{N,1}({\hat{y}}_1,w_1)\ge 0$, we have w₁′<0 in (4.15). Hence, there exists no critical set for the PWS system (4.15)_ϵ=0. (The critical set $C_1=\{({\hat{y}}_1,w_1,\theta ,\varphi ,v)|{\hat{y}}_1=b(\theta ,\varphi )/p_{+}(\theta ),w_1=0\}$ of the smooth system (4.15)${\hspace{0.17em}}_{{\hat{F}}_{N,1}=-{\hat{y}}_1-\delta w_1,ϵ=0}$ lies within ${\hat{y}}_1>0$. It is therefore an artefact of the PWS system, as illustrated in figure 4 (recall also remark 4.2).)

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The unstable manifold ${\gamma }_1^{\mathrm{u}}$ of C₁ in the smooth system (4.15)${\hspace{0.17em}}_{{\hat{F}}_{N,1}=-{\hat{y}}_1-\delta w_1}$ is given by $w_1={\mathrm{\lambda }}_{+}({\theta }_0)({\hat{y}}_1-b({\theta }_0,{\varphi }_0)/p_{+}({\theta }_0))$ and its restriction

$${\gamma }_1^{\mathrm{u}}({\theta }_0,{\varphi }_0,v_0)=\{({\hat{y}}_1,w_1,{\theta }_0,{\varphi }_0,v_0)|\hspace{0.17em}{w}_1={\mathrm{\lambda }}_{+}({\theta }_0)({\hat{y}}_1-\frac{b({\theta }_0,{\varphi }_0)}{p_{+}({\theta }_0)}),\hspace{0.17em}{\hat{y}}_1\le 0\},$$4.16

to the subset ${\hat{y}}_1\le 0$, w₁≤0 where ${\hat{F}}_{N,1}\ge 0$, is locally invariant for the PWS system (4.15)${\hspace{0.17em}}_{{\hat{F}}_{N,1}=[-{\hat{y}}_1-\delta w_1]}$. In chart κ₁, initial conditions (3.1) now become

$$({\hat{y}}_1,w_1,\theta ,\varphi ,v)=(0,\mathcal{O}(1),{\theta }_0,{\varphi }_0,v_0).$$4.17

Lemma 4.6.

Consider ${\mathit{\Lambda }}_1=\{({\hat{y}}_1,w_1,\theta ,\varphi ,v)|{\hat{y}}_1=-{\nu }^{-1}\}$ with ν>0 small. Then, the forward flow of (4.17) under (4.15) intersects Λ₁ in

$$z_1(ϵ)\equiv (-{\nu }^{-1},w_{1c}(\nu )+\mathcal{O}(ϵ),{\theta }_0+\mathcal{O}(ϵ),{\varphi }_0+\mathcal{O}(ϵ),v_0+\mathcal{O}(ϵ)),$$4.18

where $w_{1c}(\nu )=-{\mathrm{\lambda }}_{+}{\nu }^{-1}(1+o(1)),\hspace{0.17em}\nu \to 0$.

Proof.

Consider the layer problem (4.15)_ϵ=0. Since b<0, initial conditions (4.17) with w₁>0 return to ${\hat{y}}_1=0$ with w₁<0, see figure 4. Therefore, we consider w₁(0)≤0 subsequently. From λ₋<0<λ₊, it then follows that the solution remains within ${\hat{F}}_{N,1}>0$ for τ>0 for ϵ=0. The problem is therefore linear. The remaining details of the proof are straightforward and hence omitted. ▪

For ϵ>0, the variables (ϕ,v) will vary by $\mathcal{O}(1)$-amount as ${\hat{y}}_1,w_1\to -\mathrm{\infty }$. But the variables (ϕ,v) are fast in (4.3) and slow in (4.15). To describe this transition, we change to chart κ₂.

(iii) Chart κ₂

Writing the non-standard slow–fast PWS system (4.3)${\hspace{0.17em}}_{{\hat{F}}_N=[-\hat{y}-\delta w]}$ in chart κ₂, given by (4.11), gives the following smooth (as anticipated by (4.14)) system:

$$\begin{array}{rl}{ϵ}_2^{\prime }& ={ϵ}_2{w}_2,w_2^{\prime }={ϵ}_{2}b(\theta ,\varphi )+p_{+}(\theta )(1-\delta w_2)+w_2^2,\\ {\theta }^{\prime }& =ϵ\varphi ,{\varphi }^{\prime }=c_{+}(\theta )r_2(1-\delta w_2),\\ v^{\prime }& =ϵa(\theta ,\varphi )+q_{+}(\theta )r_2(1-\delta w_2)\text{and}{r}_2^{\prime }=-r_2{w}_2,\end{array}\}$$4.19

on the box U₂={(ϵ₂,w₂,θ,ϕ,v,r₂)|ϵ₂∈[0,ν],w₂∈[−λ₊−ρ,−λ₊+ρ],r₂∈[0,ν]}, for ρ>0 sufficiently small (so that w₂<δ⁻¹) and ν as above. Notice that z₁(ϵ) from (4.18) in chart κ₂ becomes

$$z_2(ϵ)\equiv {\kappa }_{21}(z_1(ϵ)):\hspace{0.17em}{r}_2=ϵ{\nu }^{-1},w_2=w_{1c}\nu +\mathcal{O}(ϵ),{ϵ}_2=\nu ,$$4.20

using (4.12). Clearly, z₂(ϵ)∈κ₂₁(Λ₁)⊂U₂, κ₂₁(Λ₁) being the face of the box U₂ with ϵ₂=ν. For simplicity, we will write subsets such as {(ϵ₂,w₂,θ,ϕ,v,r₂)∈U₂|⋯ } by {U₂|⋯ }.

Lemma 4.7.

The set M₂={U₂| r₂=0,ϵ₂=0,w₂=−λ₊} is a set of critical points of (4.19). Linearization around M₂ gives only three non-zero eigenvalues −λ₊<0, λ₋−λ₊<0, λ₊>0, and so M₂ is of saddle-type. The stable manifold is W^s(M₂)={U₂|r₂=0} while the unstable manifold is W^u(M₂)={U₂|ϵ₂=0,w₂=−λ₊}. In particular, the one-dimensional unstable manifold γ^u₂(θ₀,ϕ₀,v₀)⊂W^u(M₂) of the base point (ϵ₂,w₂,θ,ϕ,v,r₂)=(0,0,θ₀,ϕ₀,v₀,0)∈M₂ is given by

$$\begin{array}{rl}{\gamma }_2^{\mathrm{u}}({\theta }_0,{\varphi }_0,v_0)& =\{\phantom{\frac{1}{2}}{U}_2|\hspace{0.17em}{w}_2=-{\mathrm{\lambda }}_{+}({\theta }_0),\hspace{0.17em}\theta ={\theta }_0,\hspace{0.17em}\varphi ={\varphi }_0-\frac{c_{+}({\theta }_0)}{p_{+}({\theta }_0)}{\mathrm{\lambda }}_{+}({\theta }_0)r_2,\\ & v=v_0-\frac{q_{+}({\theta }_0)}{p_{+}({\theta }_0)}{\mathrm{\lambda }}_{+}({\theta }_0)r_2,\hspace{0.17em}{r}_2\ge 0,\hspace{0.17em}{ϵ}_2=0\}.\end{array}$$4.21

Proof.

The first two statements follow from straightforward calculation. For ${\gamma }_2^{\mathrm{u}}({\theta }_0,{\varphi }_0,v_0)$, we restrict to the invariant set ϵ₂=0, w₂=−λ₊ and solve the resulting reduced system. ▪

Notice that the set γ^u₂(θ₀,ϕ₀,v₀) is just γ^u(θ₀,ϕ₀,v₀) in (4.5) written in chart κ₂ for ϵ₂=0. Furthermore, note that z₂(0)⊂W^s(M₂). In the subsequent lemma, we follow z₂(ϵ)⊂{ϵ₂=ν} up until r₂=ν, with ν sufficiently small, by applying Fenichel’s normal form theory.

Lemma 4.8.

Let c∈(0,1) and set Λ₂={U₂|r₂=ν}. Then as $ϵ\to 0$, for ν and ρ sufficiently small, the forward flow of z₂(ϵ) in (4.20) intersects Λ₂ in

$$\begin{array}{rl}& \{{\mathit{\Lambda }}_2|\hspace{0.17em}{w}_2=-{\mathrm{\lambda }}_{+}+\mathcal{O}({ϵ}^c),\hspace{0.17em}\theta ={\theta }_0+\mathcal{O}(ϵ\hspace{0.17em}\ln \hspace{0.17em}{ϵ}^{-1}),\hspace{0.17em}\varphi ={\varphi }_0-\frac{c_{+}({\theta }_0)}{p_{+}({\theta }_0)}{\mathrm{\lambda }}_{+}({\theta }_0)\nu +\mathcal{O}({ϵ}^c),\\ & v=v_0-\frac{q_{+}({\theta }_0)}{p_{+}({\theta }_0)}{\mathrm{\lambda }}_{+}({\theta }_0)\nu +\mathcal{O}({ϵ}^c)\}.\end{array}$$4.22

Proof.

By Fenichel’s normal form theory, we can straighten out stable and unstable fibres.

Lemma 4.9.

For ν and ρ sufficiently small, then within U₂ there exists a smooth transformation $({ϵ}_2,w_2,\varphi ,v,r_2)\mapsto (\tilde{\varphi },\tilde{v})$ satisfying

$$\begin{array}{rl}\tilde{\varphi }& =\varphi +\frac{c_{+}(\theta )}{p_{+}(\theta )}{\mathrm{\lambda }}_{+}(\theta )r_2+\mathcal{O}(r_2(w_2+{\mathrm{\lambda }}_{+}))\\ \mathit{and}\tilde{v}& =v+\frac{q_{+}(\theta )}{p_{+}(\theta )}{\mathrm{\lambda }}_{+}(\theta )r_2+\mathcal{O}(r_2(w_2+{\mathrm{\lambda }}_{+})+ϵ),\end{array}\}$$4.23

which transforms (4.19) into

$$\begin{array}{rl}{ϵ}_2^{\prime }& ={ϵ}_2{w}_2,w_2^{\prime }={ϵ}_{2}b(\theta ,\tilde{\varphi })+p_{+}(\theta )(1-\delta w_2)+w_2^2+\mathcal{O}(ϵ),\\ {\theta }^{\prime }& =ϵ\tilde{\varphi },{\tilde{\varphi }}^{\prime }=0,\\ {\tilde{v}}^{\prime }& =0\mathit{and}{r}_2^{\prime }=-r_2{w}_2.\end{array}\}$$4.24

Proof.

Replace r₂ by νr₂ in (4.19) and consider ν small. Using ϵ=ϵ₂r₂, this brings the system into a classical slow–fast system for 0<ν≪1, where (ϵ₂,w₂,r₂) are fast variables while (θ,ϕ,v) are slow. In particular, ϵ₂=r₂=0, w₂=−λ₊ is a saddle-type slow manifold for ν small. The system is therefore amenable to Fenichel’s normal form theory [33]. The result then follows by returning to the original r₂ and using $\varphi =\tilde{\varphi }+\mathcal{O}(r_2)$ together with r₂ϵ₂=ϵ in the w₂ equation. ▪

To prove lemma 4.8, we then integrate the normal form (4.24) with initial conditions z₂(ϵ) from (4.20) from (a reset) time τ=0 up to τ=T, defined implicitly by r₂(T)=ν. Clearly, $\theta (T)={\theta }_0+\mathcal{O}(ϵT)$, $\tilde{\varphi }(T)={\tilde{\varphi }}_0$ and $\tilde{v}(T)={\tilde{v}}_0$. Then, from (4.12), Gronwall’s inequality and the fact that 1−λ₋λ⁻¹₊>1, we find

$$\begin{array}{rl}T& ={\mathrm{\lambda }}_{+}^{-1}\hspace{0.17em}\ln \hspace{0.17em}{ϵ}^{-1}(1+o(1))\\ \mathrm{and}{w}_2(T)& =-{\mathrm{\lambda }}_{+}(1+\mathcal{O}(e^{-{\mathrm{\lambda }}_{+}T}+e^{({\mathrm{\lambda }}_{-}-{\mathrm{\lambda }}_{+})T}+ϵ))\\ & =-{\mathrm{\lambda }}_{+}+\mathcal{O}({ϵ}^{c(1-{\mathrm{\lambda }}_{-}{\mathrm{\lambda }}_{+}^{-1})}+{ϵ}^c)=-{\mathrm{\lambda }}_{+}+\mathcal{O}({ϵ}^c),\end{array}\}$$4.25

for c∈(0,1). Then, we obtain the expressions for θ=θ(T), ϕ=ϕ(T) and v=v(T) in (4.22) from (4.23) in terms of the original variables. ▪

(iv) Completing the proof of proposition 4.4

To complete the proof of proposition 4.4, we then return to (4.3) using (4.11) and integrate initial conditions (4.22) within $\{\hat{y}=-r_2=-\nu \}$, up to the switching manifold Σ={v=0}, using regular perturbation theory and the implicit function theorem. This gives (4.9), which completes the proof of proposition 4.4.