Stochastic Perturbations of Periodic Orbits with Sliding

Abstract

Vector fields that are discontinuous on codimension-one surfaces can have attracting periodic orbits involving segments that are contained on a discontinuity surface of the vector field. In this paper, we consider the addition of small noise to a general piecewise-smooth vector field and study the resulting stochastic dynamics near such a periodic orbit. Since a straight-forward asymptotic expansion in terms of the noise amplitude is not possible due to the presence of discontinuity surfaces, in order to quantitatively determine the basic statistical properties of the dynamics, we treat different parts of the periodic orbit separately. Dynamics distant from discontinuity surfaces is analysed by the use of a series expansion of the transitional probability density function. Stochastically perturbed sliding motion is analysed through stochastic averaging methods. The influence of noise on points at which the periodic orbit escapes a discontinuity surface is determined by zooming into the transition point. We combine the results to quantitatively determine the effect of noise on the oscillation time for a three-dimensional canonical model of relay control. For some parameter values of this model, small noise induces a significantly large reduction in the average oscillation time. By interpreting our results geometrically, we are able to identify four features of the relay control system that contribute to this phenomenon.

Appendices

Appendix 1: Calculations for the Relay Control Example in the Absence of Noise

With $\varepsilon = 0$ (1.4) is the piecewise-linear ODE system

$$\begin{aligned} \dot{\mathbf{X}} = \left\{ \begin{array}{ll} A \mathbf{X}+ B, &{}\quad X_1 < 0 \\ A \mathbf{X}- B, &{}\quad X_1 > 0 \end{array} \right. , \end{aligned}$$

(8.1)

where $A$ (1.2) has eigenvalues $-\lambda$ and $-\omega \zeta \pm \mathrm{i} |\omega | \sqrt{1 - \zeta ^2}$. For the parameter values (1.3), $\lambda$ is relatively small, thus solutions to each linear half-system of (8.1) rapidly approach the eigenspace corresponding to the eigenvalue $-\lambda$. The eigenvector of $A$ for $-\lambda$ is

$$\begin{aligned} v_{-\lambda } = \left[ 1,~2 \zeta \omega ,~\omega ^2 \right] ^\mathsf{T}, \end{aligned}$$

(8.2)

and the equilibria of the left and right half-systems of (8.1) are, respectively,

$$\begin{aligned} \mathbf{X}^{*(L)} = \left[ \begin{array}{c} \frac{1}{\lambda \omega ^2} \\ -1 + \frac{2 \zeta }{\lambda \omega } + \frac{1}{\omega ^2} \\ 2 + \frac{2 \zeta }{\omega } + \frac{1}{\lambda } \end{array} \right] , \quad \mathbf{X}^{*(R)} = \left[ \begin{array}{c} -\frac{1}{\lambda \omega ^2} \\ 1 - \frac{2 \zeta }{\lambda \omega } - \frac{1}{\omega ^2} \\ -2 - \frac{2 \zeta }{\omega } - \frac{1}{\lambda } \end{array} \right] . \end{aligned}$$

(8.3)

$\mathbf{X}^{*(L)}$ and $\mathbf{X}^{*(R)}$ are both virtual equilibria of (8.1). The weak stable manifold for each equilibrium is the line that passes through the equilibrium in the direction $v_{-\lambda }$. These manifolds intersect $X_1 = 0$ at

$$\begin{aligned} \mathbf{X}_\mathrm{int}^{(L)} = \left[ 0,~-1+\frac{1}{\omega ^2},~2+\frac{2 \zeta }{\omega } \right] ^\mathsf{T}, \quad \mathbf{X}_\mathrm{int}^{(R)} = \left[ 0,~1-\frac{1}{\omega ^2},~-2-\frac{2 \zeta }{\omega } \right] ^\mathsf{T}. \end{aligned}$$

(8.4)

Consequently, $\Gamma$ arrives at $X_1 = 0$ at points extremely close to $\mathbf{X}_\mathrm{int}^{(L)}$ and $\mathbf{X}_\mathrm{int}^{(R)}$. For the purposes of applying the coordinate change described in Sect. 2.3, it is appropriate to approximate the point with $X_3 > 0$ at which $\Gamma$ returns to $X_1 = 0$ by $\mathbf{X}_\mathrm{int}^{(L)}$.

Stable sliding motion occurs on $X_1 = 0$ when $\dot{X_1} > 0$ for the left half-system of (8.1) and $\dot{X_1} < 0$ for the right half-system of (8.1). By (1.2), stable sliding motion occurs on the strip

$$\begin{aligned} \left\{ (0,X_2,X_3)^\mathsf{T} ~\big |~ -1 < X_2 < 1 \right\} . \end{aligned}$$

(8.5)

Sliding motion is specified by Filippov’s solution (Filippov 1988, 1964), which yields

$$\begin{aligned} \left[ \begin{array}{c} \dot{X}_2 \\ \dot{X}_3 \end{array} \right] = \left[ \begin{array}{cc} 2 &{} 1 \\ -1 &{} 0 \end{array} \right] \left[ \begin{array}{c} X_2 \\ X_3 \end{array} \right] . \end{aligned}$$

(8.6)

Equation (8.6) has the explicit solution

$$\begin{aligned} \left[ \begin{array}{c} X_{{d},2}(t;\mathbf{X}_0) \\ X_{{d},3}(t;\mathbf{X}_0) \end{array} \right] = \mathrm{e}^t \left[ \begin{array}{c} 1 \\ -1 \end{array} \right] X_{0,2} + \mathrm{e}^t \left( t \left[ \begin{array}{c} 1 \\ -1 \end{array} \right] + \left[ \begin{array}{c} 0 \\ 1 \end{array} \right] \right) (X_{0,2}+X_{0,3}), \end{aligned}$$

(8.7)

for any initial point $\mathbf{X}_0 = (0,X_{0,2},X_{0,3})$ in the stable sliding region.

The upper sliding segment of $\Gamma$ ends at $X_2 = 1$. With the approximation that the sliding segment starts at $\mathbf{X}_\mathrm{int}^{(L)}$, (8.4), the deterministic sliding time, here call it $T$, is therefore determined by

$$\begin{aligned} X_{{d},2} \left( T;\mathbf{X}_\mathrm{int}^{(L)} \right) = 1, \end{aligned}$$

(8.8)

and sliding ends at

$$\begin{aligned} (0,1,Z)^\mathsf{T}, \mathrm{~where~} Z \equiv X_{{d},3} \left( T;\mathbf{X}_\mathrm{int}^{(L)} \right) \approx 2.561. \end{aligned}$$

(8.9)

In the transformed coordinates (2.9), the initial point for the sliding phase and the end point for the excursion phase are, respectively,

$$\begin{aligned} \mathbf{x}_\mathrm{int}^{(L)} = P \mathbf{X}_\mathrm{int}^{(L)} + Q= & {} \left[ \begin{array}{c} 0 \\ -2+\frac{1}{\omega ^2} \\ 2+\frac{2 \zeta }{\omega } - Z - \left( 2-\frac{1}{\omega ^2} \right) \frac{1}{Z+2} \end{array} \right] , \end{aligned}$$

(8.10)

$$\begin{aligned} \mathbf{x}_\mathrm{int}^{(R)} = P \mathbf{X}_\mathrm{int}^{(R)} + Q= & {} \left[ \begin{array}{c} 0 \\ -\frac{1}{\omega ^2} \\ -2-\frac{2 \zeta }{\omega } - Z - \frac{1}{\omega ^2(Z+2)} \end{array} \right] . \end{aligned}$$

(8.11)

Appendix 2: Calculations of the Regular Phase for Relay Control

Here, we provide details of calculations for the relay control example that were outlined in Sect. 3.3.

The deterministic solution to (3.27) is given by

$$\begin{aligned} \mathbf{x}_{d}(t) = \mathrm{e}^{{\mathcal {A}} t} \left( \mathbf{x}_0 + {\mathcal {A}}^{-1} {\mathcal {B}}^{(R)} \right) - {\mathcal {A}}^{-1} {\mathcal {B}}^{(R)}, \end{aligned}$$

(8.12)

where $\mathbf{x}_0 = \mathbf{x}_{d}(0)$ denotes the initial point. Here, we take $\mathbf{x}_0 = \mathbf{x}_{\Gamma }^E$ (the deterministic end point of the previous escaping phase, refer to Fig. 3) with which first passage to the switching manifold occurs at $\mathbf{x}_{\Gamma }^R \approx \mathbf{x}_\mathrm{int}^{(R)}$ (8.10), see Sect. 2.3.

Through elementary use of (2.12)–(2.13), the coefficients in the PDE for ${\mathcal {P}}$ (3.24) are found to be

$$\begin{aligned} \phi _1^{(R)}\left( \mathbf{x}_{d}^R\right)= & {} x_{\Gamma ,2}^R, \end{aligned}$$

(8.13)

$$\begin{aligned} \phi _2^{(R)}\left( \mathbf{x}_{d}^R\right)= & {} \frac{-1}{Z+2} x_{\Gamma ,2}^R + x_{\Gamma ,3}^R + Z + 2, \end{aligned}$$

(8.14)

$$\begin{aligned} \phi _3^{(R)}\left( \mathbf{x}_{d}^R\right)= & {} \frac{-1}{(Z+2)^2} x_{\Gamma ,2}^R + \frac{1}{Z+2} x_{\Gamma ,3}^R, \end{aligned}$$

(8.15)

$$\begin{aligned} \frac{\partial \phi _1^{(R)}}{\partial x_2}\left( \mathbf{x}_{d}^R\right)= & {} 1 , \quad \frac{\partial \phi _1^{(R)}}{\partial x_3}\left( \mathbf{x}_{d}^R\right) = 0, \end{aligned}$$

(8.16)

$$\begin{aligned} \left( D D^\mathsf{T} \right) _{1,1}= & {} 1, \quad \left( D D^\mathsf{T} \right) _{2,1} = -2, \quad \left( D D^\mathsf{T} \right) _{3,1} = \frac{Z}{Z+2}. \end{aligned}$$

(8.17)

Then, by substituting

$$\begin{aligned} \mathbf{x}_{d} \left( \sqrt{\varepsilon } \tau + t_{\Gamma }^R \right) = \mathbf{x}_{\Gamma }^R + \sqrt{\varepsilon } \phi ^{(R)}\left( \mathbf{x}_{d}^R\right) \tau + O(\varepsilon ), \end{aligned}$$

(8.18)

with (8.13)–(8.15) into the expression for the free-space PDF (3.28), we obtain an expression for $f^{(0)}$ by the absorbing boundary condition (3.26). Specifically,

$$\begin{aligned}&f^{(0)}(u_2,u_3,\tau ) = \frac{1}{(2 \pi )^{\frac{3}{2}} \sqrt{\det \left( K\left( t_{\Gamma }^R\right) \right) }} \,\mathrm{exp} \left( -\frac{1}{2} \chi ^\mathsf{T} K\left( t_{\Gamma }^R\right) ^{-1} \chi \right) ,\nonumber \\&\mathrm{~where~} \chi = \left[ \begin{array}{c} -\phi _1^{(R)}\left( \mathbf{x}_{d}^R\right) \tau \\ u_2 - \phi _2^{(R)}\left( \mathbf{x}_{d}^R\right) \tau \\ u_3 - \phi _3^{(R)}\left( \mathbf{x}_{d}^R\right) \tau \end{array} \right] , \end{aligned}$$

(8.19)

which is used in (3.31) to obtain ${\mathcal {P}}^{(0)}$. The function $g^{(1)}$ [which appears in the second term of the expression for ${\mathcal {P}}^{(1)}$ (3.33)] is determined from (3.32) and is given by

$$\begin{aligned} g^{(1)}(u_2,u_3,\tau )= & {} - \frac{2}{\left( D D^\mathsf{T} \right) _{1,1}} \left( \frac{\partial \phi _1^{(R)}}{\partial x_2}\left( \mathbf{x}_{d}^R\right) u_2 + \frac{\partial \phi _1^{(R)}}{\partial x_3}\left( \mathbf{x}_{d}^R\right) u_3 \right) f^{(0)} \nonumber \\&\quad -\, \frac{1}{\phi _1^{(R)}\left( \mathbf{x}_{d}^R\right) } f^{(0)}_{\tau }+\,\left( \frac{2 \left( D D^\mathsf{T} \right) _{2,1}}{\left( D D^\mathsf{T} \right) _{1,1}} - \frac{\mu _1}{\phi _1^{(R)}\left( \mathbf{x}_{d}^R\right) } \right) f^{(0)}_{u_2}\nonumber \\&\quad +\, \left( \frac{2 \left( D D^\mathsf{T} \right) _{3,1}}{\left( D D^\mathsf{T} \right) _{1,1}} - \frac{\mu _2}{\phi _1^{(R)}\left( \mathbf{x}_{d}^R\right) } \right) f^{(0)}_{u_3}. \end{aligned}$$

(8.20)

1.1 Calculation of ${\mathbb {E}}[t^R]$

From (3.12), we can write

$$\begin{aligned} {\mathbb {E}}[t^R]= & {} \int _0^\infty \int _0^\infty \int _{-\infty }^\infty \int _{-\infty }^\infty p_f(\mathbf{x},t) \,\mathrm{d}x_3 \,\mathrm{d}x_2 \,\mathrm{d}x_1 \,\mathrm{d}t \nonumber \\&\quad +\,\int _0^\infty \int _0^\infty \int _{-\infty }^\infty \int _{-\infty }^\infty {\mathcal {P}}(z,u_2,u_3,\tau ) \,\mathrm{d}x_3 \,\mathrm{d}x_2 \,\mathrm{d}x_1 \,\mathrm{d}t. \end{aligned}$$

(8.21)

Using $\Psi (s) \!\equiv \! -\frac{x_{{d},1} \left( s+t_{\Gamma }^R \right) }{\sqrt{2 \kappa _{11} \left( s+t_{\Gamma }^R \right) }}, \xi \!=\! \frac{x_1 - x_{{d},1}(t)}{\sqrt{2 \kappa _{11}(t)}}$ and $s \!=\! t - t_{\Gamma }^R$, the first integral in (8.21) is

$$\begin{aligned}&\int _0^\infty \int _0^\infty \int _{-\infty }^\infty \int _{-\infty }^\infty p_f(\mathbf{x},t) \,\mathrm{d}x_3 \,\mathrm{d}x_2 \,\mathrm{d}x_1 \,\mathrm{d}t \nonumber \\&\quad = \int _0^\infty \int _0^\infty \int _{-\infty }^\infty \int _{-\infty }^\infty \frac{1}{(2 \pi \varepsilon )^{\frac{3}{2}} \sqrt{\det (K(t))}} \mathrm{e}^{-\frac{1}{2 \varepsilon } (\mathbf{x}- \mathbf{x}_{d}(t))^\mathsf{T} K(t)^{-1} (\mathbf{x}- \mathbf{x}_{d}(t))}\nonumber \\&\qquad \mathrm{d}x_3 \,\mathrm{d}x_2 \,\mathrm{d}x_1 \,\mathrm{d}t \nonumber \\&\quad = \int _0^\infty \int _0^\infty \frac{1}{\sqrt{2 \pi \varepsilon \kappa _{11}(t)}} \mathrm{e}^{-\frac{(x_1 - x_{{d},1}(t))^2}{2 \varepsilon \kappa _{11}(t)}} \,\mathrm{d}x_1 \,\mathrm{d}t \nonumber \\&\quad = \frac{1}{\sqrt{\pi \varepsilon }} \int _{-t_{\Gamma }^R}^\infty \int _{\Psi (s)}^\infty \mathrm{e}^{-\frac{\xi ^2}{\varepsilon }} \,\mathrm{d}\xi \,ds. \end{aligned}$$

(8.22)

Then, reversing the order of integration and expanding $s = \Psi ^{-1}(\xi )$ as a Taylor series centred at $\xi = 0$ produce

$$\begin{aligned}&\frac{1}{\sqrt{\pi \varepsilon }} \int _{-t_{\Gamma }^R}^\infty \int _{\Psi (s)}^\infty \mathrm{e}^{-\frac{\xi ^2}{\varepsilon }} \,\mathrm{d}\xi \,ds \nonumber \\&\quad = \frac{1}{\sqrt{\pi \varepsilon }} \int _{-\infty }^\infty \left( t - \frac{\sqrt{2 \kappa _{11}}}{\dot{x}_{{d},1}} \xi + \left( \frac{\dot{\kappa }_{11}}{\dot{x}_{{d},1}^2} - \frac{\kappa _{11} \ddot{x}_{{d},1}}{\dot{x}_{{d},1}^3} \right) \xi ^2 + O(\xi ^3) \right) \Bigg |_{t = t_{\Gamma }^R} \mathrm{e}^{-\frac{\xi ^2}{\varepsilon }} \,\mathrm{d}\xi \nonumber \\&\quad = t_{\Gamma }^R + \frac{1}{2} \left( \frac{\dot{\kappa }_{11}}{\left( \phi _1^{(R)}\left( \mathbf{x}_{d}^R\right) \right) ^2} + \frac{\kappa _{11} \ddot{x}_{{d},1}}{\left( \phi _1^{(R)}\left( \mathbf{x}_{d}^R\right) \right) ^3} \right) \Bigg |_{t = t_{\Gamma }^R} \varepsilon + O(\varepsilon ^2). \end{aligned}$$

(8.23)

The second integral in (8.21) is

$$\begin{aligned}&\int _0^\infty \int _0^\infty \int _{-\infty }^\infty \int _{-\infty }^\infty {\mathcal {P}}(z,u_2,u_3,\tau ) \,\mathrm{d}x_3 \,\mathrm{d}x_2 \,\mathrm{d}x_1 \,\mathrm{d}t \nonumber \\&\quad = -\varepsilon \int _{-\frac{t_{\Gamma }^R}{\sqrt{\varepsilon }}}^\infty \int _0^\infty \int _{-\infty }^\infty \int _{-\infty }^\infty f^{(0)}(u_2,u_3,\tau ) \mathrm{e}^{2 x_{\Gamma ,2}^R z} \,\mathrm{d}u_3 \,\mathrm{d}u_2 \,\mathrm{d}z \,\mathrm{d}\tau + O \left( \varepsilon ^{\frac{3}{2}} \right) \nonumber \\&\quad = -\frac{\varepsilon }{2 \left( \phi _1^{(R)}\left( \mathbf{x}_{d}^R\right) \right) ^2} + O \left( \varepsilon ^{\frac{3}{2}} \right) , \end{aligned}$$

(8.24)

and the sum of (8.23) and (8.24) produces (3.34).

1.2 Calculation of ${\mathbb {E}}[\mathbf{x}^R]$

Here, we briefly describe the manner by which we evaluate ${\mathbb {E}}[\mathbf{x}^R]$ numerically.

Equation (3.17) gives

$$\begin{aligned} {\mathbb {E}}\left[ x_j^R\right]= & {} \frac{\varepsilon }{2} (D D^\mathsf{T})_{11} \int _0^\infty \int _{-\infty }^\infty \int _{-\infty }^\infty x_j^R \left( \frac{\partial p_f}{\partial x_1}(0,x_2,x_3,t) + \frac{1}{\varepsilon } \frac{\partial {\mathcal {P}}}{\partial z}(0,u_2,u_3,\tau ) \right) \,\nonumber \\&\quad \times \, \mathrm{d}x_2 \,\mathrm{d}x_3 \,\mathrm{d}t, \end{aligned}$$

(8.25)

for $j = 2,3$, and changing to the local variables (3.22) yields

$$\begin{aligned} {\mathbb {E}}\left[ x_j^R\right]= & {} \frac{\varepsilon ^{\frac{5}{2}}}{2} (D D^\mathsf{T})_{11} \int _{\frac{t_{\Gamma }^R}{\sqrt{\varepsilon }}}^\infty \int _{-\infty }^\infty \int _{-\infty }^\infty \left( \sqrt{\varepsilon } u_j \!+\! x_{{\Gamma },j}^R \right) \bigg ( \frac{\partial p_f}{\partial x_1} (0,\sqrt{\varepsilon } u_2 \!+\! x_{{\Gamma },2}^R,\sqrt{\varepsilon } u_3 \nonumber \\&\quad +\,x_{{\Gamma },3}^R, \sqrt{\varepsilon } \tau + t_{\Gamma }^R) +\frac{1}{\varepsilon } \frac{\partial {\mathcal {P}}}{\partial z} (0,u_2,u_3,\tau ) \bigg ) \,\mathrm{d}u_2 \,\mathrm{d}u_3 \,\mathrm{d}\tau . \end{aligned}$$

(8.26)

Since $p_f$ is Gaussian with covariance matrix, $K(t)$, it is straightforward to derive

$$\begin{aligned}&\frac{\partial p_f}{\partial x_1}(0,x_2,x_3,t)\nonumber \\ {}&\quad = -\frac{1}{\varepsilon \det (K)} \left( -\left( \kappa _{22} \kappa _{33} - \kappa _{23}^2\right) x_{\Gamma ,1}^R + \left( \kappa _{13} \kappa _{23} - \kappa _{12} \kappa _{33}\right) \left( x_2 - x_{\Gamma ,2}^R\right) \right. \nonumber \\&\qquad \left. +\,\left( \kappa _{12} \kappa _{23} - \kappa _{13} \kappa _{22}\right) \left( x_3 - x_{\Gamma ,3}^R\right) \right) p_f(0,x_2,x_3,t). \end{aligned}$$

(8.27)

We also have from (3.26)

$$\begin{aligned} \frac{\partial {\mathcal {P}}}{\partial z}(0,u_2,u_3,\tau )= & {} \frac{1}{\varepsilon ^{\frac{3}{2}}} \left( -\frac{2 \phi _1^{(R)}\left( \mathbf{x}_{d}^R\right) }{(D D^\mathsf{T})_{11}} \left( f^{(0)} + \sqrt{\varepsilon } f^{(1)} \right) + \sqrt{\varepsilon } g^{(1)} + O(\varepsilon ) \right) \nonumber \\= & {} -2 x_{\Gamma ,2}^R p_f|_{x_1 = 0} + \frac{1}{\varepsilon } g^{(1)} + O \left( \frac{1}{\sqrt{\varepsilon }} \right) . \end{aligned}$$

(8.28)

Finally, we obtain

$$\begin{aligned} {\mathbb {E}}\left[ x_j^R\right]= & {} \frac{\varepsilon ^{\frac{3}{2}}}{2} \int _{\frac{t_{\Gamma }^R}{\sqrt{\varepsilon }}}^\infty \int _{-\infty }^\infty \int _{-\infty }^\infty \left( \sqrt{\varepsilon } u_j + x_{{\Gamma },j}^R \right) \bigg ( -\frac{1}{\det (K)} \Big ( -\left( \kappa _{22} \kappa _{33} - \kappa _{23}^2\right) x_{\Gamma ,1}^R \nonumber \\&\quad +\,\left( \kappa _{13} \kappa _{23} - \kappa _{12} \kappa _{33}\right) \left( x_2 - x_{\Gamma ,2}^R\right) + \left( \kappa _{12} \kappa _{23} - \kappa _{13} \kappa _{22}\right) \left( x_3 - x_{\Gamma ,3}^R\right) \Big ) \nonumber \\&\quad \times ~p_f \left( 0,\sqrt{\varepsilon } u_2 + x_{{\Gamma },2}^R,\sqrt{\varepsilon } u_3 + x_{{\Gamma },3}^R, \sqrt{\varepsilon } \tau + t_{\Gamma }^R \right) \nonumber \\&\quad +\,\frac{2 \kappa }{\alpha } p_f \big |_{x_1 = 0} + \frac{1}{\varepsilon } \,g^{(1)}(u_2,u_3,\tau ) \bigg ) \,\mathrm{d}u_2 \,\mathrm{d}u_3 \,\mathrm{d}\tau + O \left( \varepsilon ^{\frac{3}{2}} \right) . \end{aligned}$$

(8.29)

To produce the black lines in panels b and c of Fig. 4, we have numerically evaluated the leading-order component of (8.29), which is $O(\varepsilon )$.

Appendix 3: Calculation of $\sigma$

Here, we derive the formula (4.32):

$$\begin{aligned} \sigma \sigma ^\mathsf{T} = \frac{(b_L-b_R) (b_L-b_R)^\mathsf{T}}{(a_L+a_R)^2}, \end{aligned}$$

(8.30)

where $\sigma$ appears in (4.31). This is achieved by employing a linear diffusion approximation to reduce the drift term, $\big ( F_0(z(t),\mathbf{y}_{d}(t)) - \Omega (\mathbf{y}_{d}(t)) \big ) \,\mathrm{d}t$, of (4.30), to a diffusion term that approximates this drift term, and in the limit, $\varepsilon \rightarrow 0$ has an equivalent distribution. This is possible because the evolution of $z(t)$ is fast relative to that of $\mathbf{y}_{d}(t)$.

Since we are taking the limit $\varepsilon \rightarrow 0$, we may neglect higher-order terms in the stochastic differential equation for $z(t)$, (4.23). Furthermore, the vector noise term in (4.23) is equivalent to a scalar noise term $\sqrt{\alpha } \,\mathrm{d}W(t)$, where $\alpha = (D D^\mathsf{T})_{11}$. It is convenient to further replace this term with simply $\mathrm{d}W(t)$, as the noise amplitude $\sqrt{\alpha }$ appears as only a multiplicative factor in the final result. We let

$$\begin{aligned} r = \frac{t}{\varepsilon }, \end{aligned}$$

(8.31)

represent the fast timescale. Then (4.23) becomes

$$\begin{aligned} \mathrm{d}q(r;\mathbf{y}) = \left\{ \begin{array}{lc} a_L(\mathbf{y}), &{}\quad q < 0 \\ -a_R(\mathbf{y}), &{}\quad q > 0 \end{array} \right\} \,\mathrm{d}r + \,\mathrm{d}W(r), \end{aligned}$$

(8.32)

where we have replaced $z$ with the symbol $q$ to indicate that changes mentioned above have been made. In (8.32), $\mathbf{y}$ is treated as a constant, so (8.32) represents Brownian motion with two-valued drift (Karatzas and Shreve 1991).

In order to approximate the behaviour of the drift term, $\big ( F_0(z(t),\mathbf{y}_{d}(t)) - \Omega (\mathbf{y}_{d}(t)) \big ) \,\mathrm{d}t$, in distribution, we let

$$\begin{aligned} R(r,\mathbf{y}) = {\mathbb {E}} \left[ \left( F_0(q(\tilde{r}+r;\mathbf{y}),\mathbf{y}) - \Omega (\mathbf{y}) \right) \left( F_0(q(\tilde{r};\mathbf{y}),\mathbf{y}) - \Omega (\mathbf{y}) \right) ^\mathsf{T} \right] . \end{aligned}$$

(8.33)

For $r \ge 0, R(r,\mathbf{y})$ denotes the autocovariance of the function $F_0$ (4.25) with (8.32). In (8.33), we take $q(\tilde{r};\mathbf{y})$ to be at steady state and thus $R(r,\mathbf{y})$ is independent of the value of $\tilde{r}$. By stochastic averaging theory (Freidlin and Wentzell 2012; Pavliotis and Stuart 2008; Monahan and Culina 2011; Khas’minskii 1966), in the limit $\varepsilon \rightarrow 0$, the drift term may be replaced by the diffusion term $\sigma (\mathbf{y}_{d}(t)) \sqrt{\alpha \varepsilon } \,\mathrm{d}W(t)$, where

$$\begin{aligned} \sigma (\mathbf{y}) \sigma (\mathbf{y})^\mathsf{T} = 2 \int _0^\infty R(r,\mathbf{y}) \,\mathrm{d}r. \end{aligned}$$

(8.34)

Below, we derive (8.30) by evaluating (8.34).

Let $p(q,r|q_0)$ denote the transitional PDF of (8.32) with $q(0) = q_0$. When $a_L, a_R > 0$, (8.32) has the steady-state PDF

$$\begin{aligned} p_\mathrm{ss}(q) = \frac{2 a_L a_R}{a_L+a_R} \left\{ \begin{array}{lc} \mathrm{e}^{2 a_L q}, &{}\quad q \le 0 \\ \mathrm{e}^{-2 a_R q}, &{}\quad q \ge 0 \end{array} \right. . \end{aligned}$$

(8.35)

Then, by (8.34), we can write

$$\begin{aligned} \sigma (\mathbf{y}) \sigma (\mathbf{y})^\mathsf{T}= & {} 2 \int _0^\infty \int _{-\infty }^\infty \int _{-\infty }^\infty \big ( F_0(q,\mathbf{y}) - \Omega (\mathbf{y}) \big ) \big ( F_0(q_0,\mathbf{y})\nonumber \\&\quad -\, \Omega (\mathbf{y}) \big )^\mathsf{T} p(q,r|q_0) p_\mathrm{ss}(q_0) \,\mathrm{d}q \,\mathrm{d}q_0 \,\mathrm{d}r, \end{aligned}$$

(8.36)

where $F_0$ is given by (4.25). Since

$$\begin{aligned} {\mathbb {E}} \left[ F_0(q,\mathbf{y}) - \Omega (\mathbf{y}) \right] \equiv 0, \end{aligned}$$

(8.37)

it follows that

$$\begin{aligned} \sigma (\mathbf{y}) \sigma (\mathbf{y})^\mathsf{T}= & {} 2 \int _0^\infty \int _{-\infty }^\infty \int _{-\infty }^\infty \big ( F_0(q,\mathbf{y}) - \Omega (\mathbf{y}) \big ) \big ( F_0(q_0,\mathbf{y}) - \Omega (\mathbf{y}) \big )^\mathsf{T} \big ( p(q,r|q_0)\nonumber \\&- p_\mathrm{ss}(q) \big ) p_\mathrm{ss}(q_0) \,\mathrm{d}q \,\mathrm{d}q_0 \,\mathrm{d}r. \end{aligned}$$

(8.38)

By (4.5), (4.11) and (4.25), we have

$$\begin{aligned} F_0(q,\mathbf{y}) - \Omega (\mathbf{y}) = \left\{ \begin{array}{lc} \frac{a_L (b_L-b_R)}{a_L+a_R}, &{} \quad q < 0 \\ \frac{-a_R (b_L-b_R)}{a_L+a_R}, &{} \quad q > 0 \end{array} \right. . \end{aligned}$$

(8.39)

Therefore, we can write

$$\begin{aligned} \sigma (\mathbf{y}) \sigma (\mathbf{y})^\mathsf{T}= & {} 2 \frac{(b_L-b_R)(b_L-b_R)^\mathsf{T}}{(a_L+a_R)^2} \Bigg ( a_L^2 \int _0^\infty \int _{-\infty }^0 \int _{-\infty }^0 \big ( p(q,r|q_0)\nonumber \\&\quad -\,p_\mathrm{ss}(q) \big ) p_\mathrm{ss}(q_0) \,\mathrm{d}q \,\mathrm{d}q_0 \,\mathrm{d}r \nonumber \\&\quad -\,a_L a_R \int _0^\infty \int _{-\infty }^0 \int _0^\infty \big ( p(q,r|q_0) - p_\mathrm{ss}(q) \big ) p_\mathrm{ss}(q_0) \,\mathrm{d}q \,\mathrm{d}q_0 \,\mathrm{d}r \nonumber \\&\quad -\,a_L a_R \int _0^\infty \int _0^\infty \int _{-\infty }^0 \big ( p(q,r|q_0) - p_\mathrm{ss}(q) \big ) p_\mathrm{ss}(q_0) \,\mathrm{d}q \,\mathrm{d}q_0 \,\mathrm{d}r \nonumber \\&\quad +\,a_R^2 \int _0^\infty \int _0^\infty \int _0^\infty \big ( p(q,r|q_0) - p_\mathrm{ss}(q) \big ) p_\mathrm{ss}(q_0) \,\mathrm{d}q \,\mathrm{d}q_0 \,\mathrm{d}r \Bigg ).\nonumber \\ \end{aligned}$$

(8.40)

Next, we show that

$$\begin{aligned}&\int _0^\infty p(q,r|q_0) - p_\mathrm{ss}(q) \,\mathrm{d}r \nonumber \\&\quad = \left\{ \begin{array}{lc} \left( \frac{a_L^3+a_R^3}{a_L a_R (a_L+a_R)^2} + \frac{2 a_R}{a_L+a_R} (q+q_0) \right) \mathrm{e}^{2 a_L q} \\ \quad +~\frac{1}{a_R} \left( \mathrm{e}^{-a_R(q-q_0)-a_R|q-q_0|} - \mathrm{e}^{-2 a_R q} \right) , &{} q_0 \le 0,\, q \le 0 \\ \left( \frac{a_L^3+a_R^3}{a_L a_R (a_L+a_R)^2} - \frac{2 a_L}{a_L+a_R} q + \frac{2 a_R}{a_L+a_R} q_0 \right) \mathrm{e}^{-2 a_R q}, &{} q_0 \le 0,\, q \ge 0 \\ \left( \frac{a_L^3+a_R^3}{a_L a_R (a_L+a_R)^2} + \frac{2 a_R}{a_L+a_R} q - \frac{2 a_L}{a_L+a_R} q_0 \right) \mathrm{e}^{2 a_L q}, &{} q_0 \ge 0,\, q \le 0 \\ \left( \frac{a_L^3+a_R^3}{a_L a_R (a_L+a_R)^2} - \frac{2 a_L}{a_L+a_R} (q+q_0) \right) \mathrm{e}^{-2 a_R q} \\ \quad +~\frac{1}{a_L} \left( \mathrm{e}^{a_L(q-q_0)-a_L|q-q_0|} - \mathrm{e}^{2 a_L q} \right) , &{} q_0 \ge 0,\, q \ge 0 \end{array} \right. ,\qquad \qquad \end{aligned}$$

(8.41)

and from (8.35) and (8.41) straight-forward integration reveals that the integrals that appear in (8.40) are given simply by

$$\begin{aligned} \int _0^\infty \int _{-\infty }^0 \int _{-\infty }^0 \big ( p(q,r|q_0) - p_\mathrm{ss}(q) \big ) p_\mathrm{ss}(q_0) \,\mathrm{d}q \,\mathrm{d}q_0 \,\mathrm{d}r&= \frac{1}{2 (a_L+a_R)^2}, \nonumber \\ \int _0^\infty \int _{-\infty }^0 \int _0^\infty \big ( p(q,r|q_0) - p_\mathrm{ss}(q) \big ) p_\mathrm{ss}(q_0) \,\mathrm{d}q \,\mathrm{d}q_0 \,\mathrm{d}r&= \frac{-1}{2 (a_L+a_R)^2}, \nonumber \\ \int _0^\infty \int _0^\infty \int _{-\infty }^0 \big ( p(q,r|q_0) - p_\mathrm{ss}(q) \big ) p_\mathrm{ss}(q_0) \,\mathrm{d}q \,\mathrm{d}q_0 \,\mathrm{d}r&= \frac{-1}{2 (a_L+a_R)^2}, \nonumber \\ \int _0^\infty \int _0^\infty \int _0^\infty \big ( p(q,r|q_0) - p_\mathrm{ss}(q) \big ) p_\mathrm{ss}(q_0) \,\mathrm{d}q \,\mathrm{d}q_0 \,\mathrm{d}r&= \frac{1}{2 (a_L+a_R)^2}, \end{aligned}$$

(8.42)

with which we immediately arrive at the desired result (8.30).

To prove (8.41), we first note that, as shown in Karatzas and Shreve (1984), $p(q,r|q_0)$ is given by

$$\begin{aligned}&\!\! p(q,r|q_0)\nonumber \\&\!\!\quad = \left\{ \begin{array}{lc} 2 \mathrm{e}^{2 a_L q} \int _0^\infty h(r,b,a_R) * h(r,b-q-q_0,a_L) \,db + G(q,r,a_L|q_0), &{} q_0 \le 0,\, q \le 0 \\ 2 \mathrm{e}^{-2 a_R q} \int _0^\infty h(r,b+q,a_R) * h(r,b-q_0,a_L) \,db, &{} q_0 \le 0,\, q \ge 0 \\ 2 \mathrm{e}^{2 a_L q} \int _0^\infty h(r,b+q_0,a_R) * h(r,b-q,a_L) \,db, &{} q_0 \ge 0,\, q \le 0 \\ 2 \mathrm{e}^{-2 a_R q} \int _0^\infty h(r,b+q+q_0,a_R) * h(r,b,a_L) \,db + G(q,r,-a_R|q_0), &{} q_0 \ge 0,\, q \ge 0 \end{array} \right. ,\nonumber \\ \end{aligned}$$

(8.43)

where

$$\begin{aligned} h(r,q_0,\omega )= & {} \frac{|q_0|}{\sqrt{2 \pi r^3}} \mathrm{e}^{-\frac{(q_0 - \omega r)^2}{2 r}}, \end{aligned}$$

(8.44)

$$\begin{aligned} G(q,r,\omega |q_0)= & {} \frac{1}{\sqrt{2 \pi r}} \mathrm{e}^{-\frac{(q-q_0-\omega r)^2}{2 r}} - \mathrm{e}^{-2 \omega q_0} \frac{1}{\sqrt{2 \pi r}} \mathrm{e}^{-\frac{(q+q_0-\omega r)^2}{2 r}}, \end{aligned}$$

(8.45)

and $*$ denotes convolution with respect to $r$. Here, we derive (8.41) from (8.43)–(8.45) for $q_0, q \ge 0$. The case $q_0 \ge 0, q \le 0$ is similar, and the remaining two cases follow by symmetry.

For $q_0, q \ge 0$, direct integration yields

$$\begin{aligned}&\int _0^\infty p(q,r|q_0) - p_\mathrm{ss}(q) \,\mathrm{d}r = \frac{1}{a_R} \left( \mathrm{e}^{-a_R(q-q_0)-a_R|q-q_0|} - \mathrm{e}^{-2 a_R q} \right) \nonumber \\&\quad +\lim _{\nu \rightarrow 0^+} {\mathcal {L}} \left( 2 \mathrm{e}^{-2 a_R q} \int _0^\infty h(r,b+q+q_0,a_R) * h(r,b,a_L) \,db - p_\mathrm{ss}(q) \right) , \end{aligned}$$

(8.46)

where

$$\begin{aligned} {\mathcal {L}}[f(r)] = \int _0^\infty \mathrm{e}^{-\nu r} f(r) \,\mathrm{d}r. \end{aligned}$$

(8.47)

denotes a Laplace transform in $r$. Next, we recall (8.35) and note that

$$\begin{aligned} {\mathcal {L}}[h(r,q_0,\omega )] = \mathrm{e}^{\omega q_0 - \sqrt{\omega ^2 + 2 \nu } |q_0|}, \end{aligned}$$

(8.48)

to obtain

$$\begin{aligned}&\int _0^\infty p(q,r|q_0) - p_\mathrm{ss}(q) \,\mathrm{d}r = \frac{1}{a_R} \left( \mathrm{e}^{-a_R(q-q_0)-a_R|q-q_0|} - \mathrm{e}^{-2 a_R q} \right) \nonumber \\&\quad +\,2 \mathrm{e}^{-2 a_R q} \lim _{\nu \rightarrow 0^+} \left( \frac{\mathrm{e}^{\left( a_R - \sqrt{a_R^2 + 2 \nu } \right) (q+q_0)}}{-a_R + \sqrt{a_R^2 + 2 \nu } - a_L + \sqrt{a_L^2 + 2 \nu }} - \frac{a_L a_R}{\nu (a_L+a_R)} \right) . \end{aligned}$$

(8.49)

Finally, by substituting $-a + \sqrt{a^2 + 2 \nu } = \frac{\nu }{a} - \frac{\nu ^2}{2 a^3} + O(\nu ^3)$, with $a = a_L,a_R$ in the above equation, terms involving $\frac{1}{\nu }$ vanish and we arrive at (8.41) for $q_0, q \ge 0$.

Appendix 4: Derivations of Formulas for $\mathrm{Diff} \left( t^S \right)$ and $\mathrm{Var}(t^S)$

In this section, we derive (6.7) and (6.16):

$$\begin{aligned} \mathrm{Diff} \left( t^S \right)&= \mathrm{Diff} \left( t^S \big | \mathbf{x}_{\Gamma }^M \right) + \mathrm{D}_{\mathbf{x}} t_{d}^S \left( \mathbf{x}_{\Gamma }^M \right) ^\mathsf{T} \mathrm{Diff} \left( \mathbf{x}^M \right) \nonumber \\&\quad + \sum _{i=1}^N \sum _{j=1}^N \mathrm{D}_{\mathbf{x}}^2 t_{d}^S \left( \mathbf{x}_{\Gamma }^M \right) _{i,j} \mathrm{Cov} \left( x_{\Gamma }^M \right) _{i,j} + O \left( \varepsilon ^{\frac{3}{2}} \right) , \end{aligned}$$

(8.50)

$$\begin{aligned} \mathrm{Var} \left( t^S \right)&= \mathrm{Var} \left( t^S \big | \mathbf{x}_{\Gamma }^M \right) + \mathrm{D}_{\mathbf{x}} t_{d}^S \left( \mathbf{x}_{\Gamma }^M \right) ^\mathsf{T} \mathrm{Cov} \left( \mathbf{x}^M \right) \mathrm{D}_{\mathbf{x}} t_{d}^S \left( \mathbf{x}_{\Gamma }^M \right) + O \left( \varepsilon ^{\frac{3}{2}} \right) , \end{aligned}$$

(8.51)

which express the leading-order terms for $\mathrm{Diff} \left( t^S \right)$ and $\mathrm{Var}(t^S)$ in terms of conditioned quantities and may be evaluated using the results of Sect. 4. Analogous formulas in Sect. 6 relating to other components of the stochastic dynamics may be derived in the same fashion.

First, by definition,

$$\begin{aligned} \mathrm{Diff} \left( t^S \right) \equiv {\mathbb {E}} \left[ t^S \right] - t_{\Gamma }^S = \int t^S p \left( t^S \right) \,\mathrm{d}t^S - t_{\Gamma }^S, \end{aligned}$$

(8.52)

where throughout this exposition $p(\cdot )$ denotes the PDF of the indicated variable. Conditioning over the starting point $\mathbf{x}^M$ gives

$$\begin{aligned} \mathrm{Diff} \left( t^S \right) = \int t^S \int p \left( t^S \big | \mathbf{x}^M \right) p \left( \mathbf{x}^M \right) \,\mathrm{d}\mathbf{x}^M \,\mathrm{d}t^S - t_{\Gamma }^S. \end{aligned}$$

(8.53)

By then reversing the order of integration and using $\mathrm{Diff} \left( t^S \big | \mathbf{x}^M \right) \equiv {\mathbb {E}} \left[ t^S \big | \mathbf{x}^M \right] - t_{d}^S \left( \mathbf{x}^M \right)$, we obtain

$$\begin{aligned} \mathrm{Diff} \left( t^S \right) = \int \left( t_{d}^S \left( \mathbf{x}^M \right) + \mathrm{Diff} \left( t^S \big | \mathbf{x}^M \right) \right) p \left( \mathbf{x}^M \right) \,\mathrm{d}\mathbf{x}^M - t_{\Gamma }^S. \end{aligned}$$

(8.54)

By replacing $t_{d}^S \left( \mathbf{x}^M \right)$ in (8.54) with its Taylor series centred at the deterministic value $\mathbf{x}^M = \mathbf{x}_{\Gamma }^M$:

$$\begin{aligned} t_{d}^S \left( \mathbf{x}^M \right)= & {} t_{d}^S \left( \mathbf{x}_{\Gamma }^M \right) + \mathrm{D}_{\mathbf{x}} t_{d}^S \left( \mathbf{x}_{\Gamma }^M \right) ^\mathsf{T} \left( \mathbf{x}^M - \mathbf{x}_{\Gamma }^M \right) \nonumber \\&\quad +\,\left( \mathbf{x}^M - \mathbf{x}_{\Gamma }^M \right) ^\mathsf{T} \mathrm{D}_{\mathbf{x}}^2 t_{d}^S \left( \mathbf{x}_{\Gamma }^M \right) \left( \mathbf{x}^M - \mathbf{x}_{\Gamma }^M \right) + O \left( \varepsilon ^{\frac{3}{2}} \right) , \end{aligned}$$

(8.55)

and evaluating the integral in (8.54), we arrive at (8.50). The error term in (8.55) is $O \left( \varepsilon ^{\frac{3}{2}} \right)$ because $\mathbf{x}^M - \mathbf{x}_{\Gamma }^M = O \left( \sqrt{\varepsilon } \right)$.

Second, to derive (8.51) we begin by writing

$$\begin{aligned} \mathrm{Var} \left( t^S \right) = \int \left( t^S - {\mathbb {E}} \left[ t^S \right] \right) ^2 p \left( t^S \right) \,\mathrm{d}t^S. \end{aligned}$$

(8.56)

Conditioning over $\mathbf{x}^M$ gives

$$\begin{aligned} \mathrm{Var} \left( t^S \right) = \int \left( t^S - {\mathbb {E}} \left[ t^S \right] \right) ^2 \int p\left( t^S \big | \mathbf{x}^M \right) p \left( \mathbf{x}^M \right) \,\mathrm{d}\mathbf{x}^M \,\mathrm{d}t^S. \end{aligned}$$

(8.57)

Reversing the order of integration and adding and subtracting ${\mathbb {E}} \left[ t^S \big | \mathbf{x}^M \right]$ produce

$$\begin{aligned} \mathrm{Var} \left( t^S \right)= & {} \int \int \left( t^S - {\mathbb {E}} \left[ t^S \big | \mathbf{x}^M \right] + {\mathbb {E}} \left[ t^S \big | \mathbf{x}^M \right] \right. \nonumber \\&\quad \left. -\,{\mathbb {E}} \left[ t^S \right] \right) ^2 p \left( t^S \big | \mathbf{x}^M \right) \,\mathrm{d}t^S p \left( \mathbf{x}^M \right) \,\mathrm{d}\mathbf{x}^M. \end{aligned}$$

(8.58)

Since the mean values differ from their deterministic values by $O(\varepsilon )$, we have

$$\begin{aligned} {\mathbb {E}} \left[ t^S \big | \mathbf{x}^M \right] - {\mathbb {E}} \left[ t^S \right] = t_{\Gamma }^S \left( \mathbf{x}^M \right) - t_{\Gamma }^S + O(\varepsilon ). \end{aligned}$$

(8.59)

By substituting (8.55) and (8.59) into (8.58), and noting $t_{d}^S \left( \mathbf{x}_{\Gamma }^M \right) = t_{\Gamma }^S$, we obtain

$$\begin{aligned} \mathrm{Var} \left( t^S \right)= & {} \int \int \left( t^S - {\mathbb {E}} \left[ t^S \big | \mathbf{x}^M \right] + \mathrm{D}_{\mathbf{x}} t_{d}^S \left( \mathbf{x}_{\Gamma }^M \right) ^\mathsf{T} \left( \mathbf{x}^M - \mathbf{x}_{\Gamma }^M \right) \right. \nonumber \\&\quad \left. +\,O(\varepsilon ) \right) ^2 p \left( t^S \big | \mathbf{x}^M \right) \,\mathrm{d}t^S p \left( \mathbf{x}^M \right) \,\mathrm{d}\mathbf{x}^M. \end{aligned}$$

(8.60)

Finally, by expanding the square in (8.60) and evaluating the double integral, we arrive at (8.51).