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Most probable transition paths in piecewise-smooth stochastic differential equations

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We develop a path integral framework for determining most probable paths in a class of systems of stochastic differential equations with piecewise-smooth drift and additive noise. This approach extends the Freidlin-Wentzell theory of large deviations to cases where the system is piecewise-smooth and may be non-autonomous. In particular, we consider an $n-$dimensional system with a switching manifold in the drift that forms an $(n-1)-$dimensional hyperplane and investigate noise-induced transitions between metastable states on either side of the switching manifold. To do this, we mollify the drift and use $\Gamma-$convergence to derive an appropriate rate functional for the system in the piecewise-smooth limit. The resulting functional consists of the standard Freidlin-Wentzell rate functional, with an additional contribution due to times when the most probable path slides in a crossing region of the switching manifold. We explore implications of the derived functional through two case studies, which exhibit notable phenomena such as non-unique most probable paths and noise-induced sliding in a crossing region.
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Most probable transition paths in piecewise-smooth stochastic
differential equations
Kaitlin Hill∗ †
, Jessica Zanetell
, and John A. Gemmer∗ ‡
Abstract
We develop a path integral framework for determining most probable paths in a class of systems
of stochastic differential equations with piecewise-smooth drift and additive noise. This approach
extends the Freidlin-Wentzell theory of large deviations to cases where the system is piecewise-
smooth and may be non-autonomous. In particular, we consider an ndimensional system with
a switching manifold in the drift that forms an (n1)dimensional hyperplane and investigate
noise-induced transitions between metastable states on either side of the switching manifold.
To do this, we mollify the drift and use Γconvergence to derive an appropriate rate functional
for the system in the piecewise-smooth limit. The resulting functional consists of the standard
Freidlin-Wentzell rate functional, with an additional contribution due to times when the most
probable path slides in a crossing region of the switching manifold. We explore implications
of the derived functional through two case studies, which exhibit notable phenomena such as
non-unique most probable paths and noise-induced sliding in a crossing region.
Keywords: Piecewise smooth dynamical systems, Filippov systems, Freidlin-Wentzell rate functional, Gamma-
convergence, noise induced tipping, rare events
2000 MSC: 37H10, 37J45
1 Introduction
In dynamical systems, a tipping event is loosely defined as occurring when a sudden or small change
to a variable or parameter induces a large change to the state of the system. While tipping is often
studied within the context of climate applications [1–7], it also has broad applications in ecology
[8, 9], ecosystems [10, 11] and epidemiology [12–15], to name a few. While there is no precise
mathematical definition of a tipping event, in [16] it was proposed that tipping events could be
classified according to whether the underlying mathematical mechanism involves, predominantly,
a bifurcation (B-tipping), noise-induced transitions (N-tipping), or transitions between basins of
attraction induced by fast changes in parameters (R-tipping), i.e. rate-induced tipping; see also
[17–20].
The term ‘noise-induced tipping’ encompasses a range of phenomena such as transitions between
metastable states, bursting, stochastic resonance, stochastic coherence and stochastic synchroniza-
tion [21] and can be analyzed from several points of view including the Freidlin-Wentzell (FW)
theory of large deviations, [22–24], the path integral framework [25, 26], transition path theory
[27, 28], and formal asymptotics [29, 30]. In particular, for smooth autonomous dynamical systems
additively perturbed by Gaussian white noise, the FW theory provides a framework for computing
Department of Mathematics and Statistics, Wake Forest University, Winston-Salem, NC 27109, USA
Corresponding author, Email : kaitlin.m.s.hill@gmail.com
Email : gemmerj@wfu.edu
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Most probable paths in piecewise-smooth SDEs
most probable transitions as minimizers of a rate functional in the asymptotic limit of vanishing
noise strength. The benefit of the FW framework is that most probable transition paths can be
numerically computed using iterative schemes such as the string method for gradient systems [31],
the minimum action method [23], the geometric minimum action method [32] and explicit gradient
descent [33]. Moreover, knowledge of the most probable transition path can then be coupled with
statistical techniques such as importance sampling to compute quantities of interest such as the
expected time of tipping for nonzero noise [24, 34].
In this paper we extend the path integral framework to a class of differential equations with
additive noise and piecewise-smooth drift. There has been considerable recent interest in under-
standing the dynamics of piecewise-smooth stochastic dynamical systems, particularly in relation
to rare events and tipping, or transitions between metastable states, but also in varying applications
[35–37]. In [38, 39], Chen et al. use the backward Fokker-Planck technique to derive the distribu-
tion of paths in a model of dry friction. Notably, in [38], Chen et al. use a similar method to that
of the present study, smoothing out the piecewise-smooth vector field to numerically analyze the
desingularized SDE. In [40, 41], Baule et al. use the path integral approach, as in the present study,
to investigate most probable paths in a piecewise-smooth model of stick-slip friction. Beyond me-
chanical systems, the study of noise-induced tipping in piecewise-smooth systems has applications
in biology [42, 43] and climate models [44–49]. To our knowledge, most probable paths in general
stochastic piecewise-smooth systems have not yet been addressed.
1.1 Most probable paths in smooth systems
During rare event transitions from one metastable state to another, a stochastic system will follow
paths according to some distribution, which in the limit of vanishing noise strength is generally
singly-peaked along a most probable transition path. Following [22], for a smooth vector field F,
the most probable transition path between x0and xfcan be defined as follows. We first define an
admissible set of transition paths A(t0,tf)by
A(t0,tf)={αH1([t0, tf]; Rn) : α(t0) = x0and α(tf) = xf},
where H1([t0, tf]; Rn) is the Hilbert space of weakly differentiable curves αsatisfying αH1if
and only if Rtf
t0|α|2+|˙
α|2dt < . The most probable transition path α∈ A(t0,tf)is then defined
to be the global minimizer of the Freidlin-Wentzell rate functional I(t0,tf):A(t0,tf)7→ R,given by
I(t0,tf)[α] = Ztf
t0k˙
α(t)F(α(t), t)k2dt, (1)
so that α= arg minα∈A(t0,tf)I(t0,tf)[α]. For this functional, the necessary condition satisfied by
second differentiable minimizers is the following Euler-Lagrange equations [24]:
¨
α=Ft+˙
α(F− ∇F|) + F|F.
Here, the subscript trefers to the partial derivative with respect to time.
1.2 Framing the piecewise-smooth problem
We determine the most probable transition path of a noise-induced tipping event when the drift
is piecewise-smooth. In general, for systems with a piecewise-smooth drift, the appropriate rate
functional to minimize is not known. Minimizers of the Freidlin-Wentzell rate functional may not
be well-defined when a region of the vector field is not continuous, or even Lipschitz continuous.
2
Most probable paths in piecewise-smooth SDEs
It is not clear how the most probable path may traverse across a switching manifold, which may
have an attracting or repelling sliding region that introduces more complex dynamics. Specifically,
letting F±:Rn7→ Rnbe smooth vector fields, we consider a system of stochastic differential
equations of the form
dxt=F(xt)dt +σdWt,(2)
where x= (x, y)Rnsuch that xRand yRn1,σR,W= (W1,...Wn) is an n-dimensional
Wiener process, and F:Rn\ {x= 0} 7→ Rnis defined by
F(x) = (F+(x), x > 0,
F(x), x < 0.(3)
We let S+={xRn:x > 0},S={xRn:x < 0}, and Σ = {xRn:x= 0}. The set Σ
is often called the switching manifold or discontinuity boundary [50]. In general Σ may be defined
as the zero level set of a smooth function H:Rn7→ R,but we assume for simplicity that Σ is a
hyperplane in Rn1.
We define the deterministic skeleton of Equation (2) as the dynamical system
˙
x=F(x).(4)
We assume that there exist asymptotically stable fixed points x0S+and xfSof Equation
(4), such that F+(x0) = 0 and F(xf) = 0. By noise-induced transitions, we mean realizations
of Equation (2) that transition from the basin of attraction of x0to that of xf. More precisely,
we define a noise-induced transition on the interval [t0, tf] as a solution to the stochastic boundary
value problem given by realizations of Equation (2) that satisfy x(t0) = x0and x(tf) = xf. Here
the boundary conditions mean that the process is conditioned to transition from x0to xf.
1.3 Background on piecewise-smooth systems
Dynamics that occur entirely within the smooth regions S±can be fully described by regular
(smooth) dynamical systems theory. On the other hand, dynamics on a switching manifold Σ may
not be defined and are typically imposed a posteriori, e.g. using Filippov’s convex combination
[50–52]. In general, the lack of smoothness requirements across the switching manifold allows for
more diverse phenomena than in smooth systems of similar dimension, since the vector field across
Σ may be discontinuous or even point in opposite directions. Regions of Σ where either occurs
are called sliding regions. A solution that reaches Σ at a sliding region may “slide” along it. On
the other hand, crossing regions occur on Σ where sliding is not possible. We differentiate these
regions using the following notation:
1. Attracting sliding regions: ΣA={xΣ : F+
1(0,y)0, F
1(0,y)0},
2. Repelling sliding regions: ΣR={xΣ : F+
1(0,y)0, F
1(0,y)0},
3. Positive crossing regions: Σ+={xΣ : F±
1(0,y)>0},
4. Negative crossing regions: Σ={xΣ : F±
1(0,y)<0},
where F±
1is the first component of F±.
When solutions may slide along the switching manifold Σ,we impose a flow using the Filippov
convex method [50, 52]. That is, we define the sliding flow as a convex combination of F±,
Fs(0,y) = λF+(0,y) + (1 λ)F(0,y) (5)
3
Most probable paths in piecewise-smooth SDEs
with
λ(y) = F
1(0,y)
F
1(0,y)F+
1(0,y)[0,1].(6)
Notice that Equation (6) fixes λ(y). This flow naturally arises in the context of our main result
without a priori imposing it; see Theorem 3.8. However, the most probable paths predicted by
our result does not precisely match observed most probable paths, which may be in part due
to nonlinear contributions. There has been recent work on generalizing sliding flow to nonlinear
combinations [53, 54], but we will not consider such possible extensions in this study. A point
xs= (0,ys)Σ is a pseudoequilibrium if it is an equilibrium of the sliding flow; i.e., for some
λ(0,1),Fs(0,ys) = 0.
Filippov’s convex combination interprets solutions of System (3),(4) as a continuous curve x(t)
satisfying the following conditions:
(˙
x(t) = F(x),
x(0) = x0,(7)
where F:Rn7→ Rnis defined as
F(x) =
F+(x),xS+Σ+,
F(x),xSΣ,
Fs(x),xΣAΣR.
(8)
That is, for points in time in which a solution curve x(t) intersects Σ it will either cross Σ,tracking
the flow of either F+or Fdepending on whether the curve is crossing into S+or Sor it will
track Σ in the sliding regions until entering a crossing region. Note, implicit in this definition is
that solution curves are differentiable everywhere except on Σ.
To provide a brief illustration, Figure 1(a)-(c) shows some possible dynamics across Σ in a two-
dimensional system, with (a) Σ+, (b) Σ±and ΣA, and (c) Σ±and ΣR. In (a) the flow traverses
across Σ continuously but not smoothly; in (b) and (c) the flow may also remain on Σ,in intervals
indicated by the blue line (for ΣA) and the green line (for ΣR). Note that the flow leaving a
repelling sliding region is non-unique in forward time, while the flow leaving an attracting sliding
region is non-unique in backward time. In addition, complex dynamics may appear on Σ itself,
depending on the manner in which the imposed vector field on Σ transitions between S+and S.
Figure 1(d) shows one example of a path traversing from a stable equilibrium in S+, through the
switching manifold in an attracting sliding region using dynamics imposed using Filippov’s convex
combination [52], then traversing to a stable equilibrium in S. The system in this illustration is
a piecewise-smooth version of the Lorenz 63 model,
˙x
˙y
˙z
=
σ(yy0x+x0)
(xx0)(ρz+z0)y+y0
(xx0)(yy0)β(zz0)
, x < 0,and
˙x
˙y
˙z
=
σ+(yx)
x(ρ+z)y
xy β+z
, x > 0,
(9)
originally studied in [55]. Here, σ±, ρ±, β±>0 are parameters and x0, y0, z0R. For this example,
S±and Σ are defined as previously, with n= 3.
1.4 Outline of analysis
For our problem, noise-induced transitions must cross Σ and the Freidlin-Wentzell rate functional,
Equation (1), must be modified to account for the discontinuity in Fand possible discontinuity-
induced dynamics. For example, a most probable path that reaches a repelling sliding region ΣR
4
Most probable paths in piecewise-smooth SDEs
(a) (b) (c) (d)
Figure 1: (a)-(c) Phase plane generic Fwith n= 2, with (a) a crossing region, (b) crossing and
attracting sliding, and (c) crossing and repelling sliding. (d) Phase diagram of a piecewise-smooth
version of the Lorenz 63 model, System (9). Red points are equilibria of S±,and the black point is
a pseudoequilibrium of the imposed flow on Σ. The sliding region, ΣA,is shaded blue. The black
curve connects the two equilibria, going through the peudoequilibrium at the origin and sliding
in ΣA. The blue curve is the sliding trajectory. Parameters used are σ+= 10, β+= 2, ρ±= 2,
σ= 11, β= 3, x0= 1, y0=1, and z0= 0.
leaves it non-uniquely due to the non-uniqueness of the flow, as illustrated in the case studies in
Sections 4 and 5. Additionally, if one were to naively piece together the most probable paths of
the smooth regions and neglect possible contribution from the manner in which the path crosses
the switching manifold, this may lead to paths that do not reflect a global minimum of the rate
functional.
We resolve these issues by smoothing out Fvia mollification with a compactly supported,
smooth, radially symmetric kernel of characteristic width ε, and consider the sequence of minimizers
to the rate functional Equation (1) as ε0. We show for any piecewise-smooth system of the
form given by System (2),(3) that the most probable path minimizes Equation (1) in the smooth
regions S±, plus an additional functional whose contribution represents time spent sliding in a
crossing region of the switching manifold. For F= (F1,G) such that F1:Rn\Σ7→ Rand
G:Rn\Σ7→ Rn1,and α= (α, β)∈ A(t0,tf)and 0 < ε 1, the appropriate rate functional with
both contributions is
I(t0,tf)[α] = Z{tS+S}k˙
α(t)F(α(t))k2dt
+Z{tΣ±}
min
λ[0,1] nλF +
1(0,β) + (1 λ)F
1(0,β)2
+
˙
βλG+(0,β)(1 λ)G(0,β)
2dt.
We make the following observations about this result:
1. The form of the above rate functional is independent of the chosen mollifier.
2. When there is a sliding region, the most probable path may track the Filippov dynamics;
that is, no additional contribution is made during times where the most probable path slides
via the imposed flow on the switching manifold.
We rigorously prove the above result using a technique from the calculus of variations called
Γ-convergence to show that I(t0,tf)is the Γ-limit of the sequence of rate functionals for the mollified
5
Most probable paths in piecewise-smooth SDEs
system [56, 57]. The Γ-limit is the natural notion of a limiting functional in this context in the
sense that minimizers of the rate functional with the mollified drift converge (weakly) to a minimizer
of I(t0,tf)in the limit as ε0. While Γ-convergence has been used to study the convergence of
minimizers of the the Onsager-Machlup functional in the small noise limit [58–62], to our knowledge
we are the first to use Γ-convergence to compute an appropriate extension of the Freidlin-Wentzell
rate functional for piecewise smooth systems.
The utility of the derived rate functional is demonstrated through two case studies. Each case
study also presents distinct phenomena in their most probable transition paths that are not possible
in systems with smooth drift. The first case study analyzes the most probable transition path in
a two-dimensional piecewise-linear system, a simple demonstrative case in which the typical rate
functional formulation of the most probable path breaks down. Notably, in this case the most
probable path follows the switching manifold along an attracting sliding region, but it does not
following the switching manifold along a repelling sliding region.
The second case study analyzes the most probable path in a simple one-dimensional periodically-
forced piecewise-smooth system, constructed similarly to some conceptual models for Arctic energy
balance [4, 63]. A significant portion of the analysis of the deterministic dynamics and the Monte
Carlo simulations were first derived in Zanetell’s Master’s thesis [64]. Here we again observe the
breakdown of the typical formulation of the most probable path. Additionally, we observe an
emergent dynamical phenomenon, which we describe as noise-induced sliding, in which the most
probable path follows the switching manifold in a crossing region, despite the energy cost of the
associated rate functional.
The paper is organized as follows: in Section 2 we briefly highlight notation we use and
some technical assumptions on F. Then in Section 3 we derive the mollified vector field and
use Γconvergence to determine the appropriate limiting functional for calculating the most prob-
able path for stochastic differential equations of the form given by System (2),(3). Next, in Section
4 we provide a simple case study of a planar linear piecewise-smooth system as an illustration
of the most probable path calculated in Section 3. Finally, in Section 5 we provide a case study
that extends the most probable path derivation to the case of a one-dimensional non-autonomous
system.
2 Notation and Assumptions
In this paper we use the following conventions for notation:
1. We use the convention that k · k represents the L2norm for functions and | · | represents the
`2norm for vectors in Rn. Furthermore, we use ,·i to denote the inner product in `2.
2. Subscripts refer to the index of a vector unless otherwise indicated. E.g., for F= (F1, F2, . . . , Fn),
we write the components as Fj, j = 1, . . . , n.
3. Unless noted otherwise, Fmay be non-autonomous. Although our analysis applies to non-
autonomous systems, for simplicity of presentation we suppress the time dependence.
4. Since we assume that the underlying drift is only piecewise-smooth in the first component,
we use a convenient shorthand that separates this first component from the remaining ones.
Namely, we write x= (x, y)Rn,where xRand yRn1,and F= (F1,G),where
F1:Rn\ {x= 0} 7→ Rand G:Rn\ {x= 0} 7→ Rn1.
6
Most probable paths in piecewise-smooth SDEs
5. For the path α∈ A(t0,tf),we separate the first component from the remaining ones as
α= (α, β),so that αH1([t0, tf]; R), α(t0) = x0, and α(tf) = xf,where x0,f = (x0,f ,y0,f ).
Similarly, βH1[t0, tf]; Rn1,so that β(t0) = y0and β(tf) = yf.
6. We denote the Jacobian of a function F(x, y) as F(x, y). For the Jacobian comprised of
only the partial derivatives with respect to the components of y,we write yF(x, y).
7. We indicate that a variable is being treated as fixed or pre-determined by separating it with
a semicolon. E.g., if the value of yhas been set, then F(x;y) is only a function of x.
We also make the following assumptions for each smooth vector field F±.We will use these as-
sumptions to prove two lemmas for the mollified vector field Fεin Section 3 which are necessary
to establish existence of minimizers to the FW functional. Note, these assumptions also apply in
the non-autonomous case but the explicit dependence on time in the argument of Fis suppressed.
Assumptions. Let F±:Rn\ {x= 0} 7→ Rnbe smooth vector fields. We assume the following
properties about F±.
1. (Growth conditions) There exist R1, c1, c3, c5>0and c2, c4, c6Rsuch that c5< c1and for
some p > 1,|x|> R1implies
(|F(x)| ≥ c1|x|p+c2,if x < 0,
|F+(x)| ≥ c1|x|p+c2,if x > 0,(10)
and
F
∂x (x)c3|x|p+c4,if x < 0,
F+
∂x (x)c3|x|p+c4,if x > 0,(11)
and
|F+(0,y)F(0,y)| ≤ c5|y|p+c6.(12)
2. (Asymptotically inward-flowing) There exist R2, c7>0such that |x|> R2implies F±(x)6= 0
and (hF(x),r(x)i<c7|F(x)|,if x0,
hF+(x),r(x)i<c7|F+(x)|,if x0,(13)
where r(x) = x/|x|is the normalized outward-pointing radial vector at x.
3 Mollified Freidlin-Wentzell rate functional
As discussed in Section 1, the standard form of the Freidlin-Wentzell rate functional cannot be
directly used to calculate most probable paths in piecewise-smooth systems. In this section, we
seek to ameliorate this deficiency by studying the convergence of the minimizer of the Freidlin-
Wentzell functional, where we approximate Equation (3) as a smooth vector field obtained by
mollification in the xdirection by a smooth, compactly supported, symmetric kernel. Specifically,
by smoothing Fvia mollification in xover a region of width 2ε, we obtain a sequence of functionals
I(t0,tf)
εand a corresponding sequence of minimizers αε∈ A(t0,tf), which we prove converge — up
to a subsequence — to a minimizer of a limiting functional I(t0,tf)in the limit as ε0.
7
Most probable paths in piecewise-smooth SDEs
3.1 Mollified functional and existence of a minimum
We begin this section by recalling the definition of a (Friedrichs) mollifier, following the presentation
in [65]. Let ζ:R7→ Rbe a smooth even bump function, defined by
ζ(x) = (Ch(x),|x|<1,
0,|x| ≥ 1,(14)
where h:RRis a smooth function satisfying h(x)0, h(1) = h(1) = 0 and all of its
derivatives vanish at x=1 and x= 1, e.g., h(x) = exp 1/|x|21. Additionally, C > 0 is
determined so that R
−∞ ζ(x)dx = 1; i.e., C= 1/R
−∞ h(x)dx. For each ε > 0,we define a sequence
of functions ζε(x) = ζ(x/ε)which satisfy Rε
εζε(x)dx = 1 and are compactly supported on
[ε, ε]. The mollification Fεof Fin xis then defined by convolution with ζεin the xdirection:
Fε(x, y) = ζε(x)F(x, y) = Z0
−∞
ζε(xs)F(s, y)ds +Z
0
ζε(xs)F+(s, y)ds,
=
Zε
ε
ζε(u)F(xu, y)du, x ≤ −ε,
Zx
ε
ζε(u)F+(xu, y)du +Zε
x
ζε(u)F(xu, y)du, |x|< ε,
Zε
ε
ζε(u)F+(xu, y)du, x ε,
=Zε
ε
ζε(u)F(xu, y){ux}(u) + F+(xu, y){ux}(u)du,
(15)
where Ais the indicator function on the set A. The mollified Freidlin-Wentzell rate functional
I(t0,tf)
ε:A(t0,tf)7→ Ris then defined by:
I(t0,tf)
ε[α] = Ztf
t0|˙
αFε(α)|2dt. (16)
Note, by construction, Fε(x) is a smooth function that converges pointwise to F(x) as ε0
except at x= 0 [65]. However, since F(x) is not continuous this convergence is necessarily not
uniform and thus it is not a priori clear which properties of Fare preserved under mollification.
The following lemmas ensure that Fεalso has p-growth and is asymptotically inward-flowing. These
properties are crucial to establishing the existence of a minimum for I(t0,tf)
ε,but to streamline our
main results, the proofs of these lemmas are in Appendix A.
Lemma 3.1 (pgrowth).There exists εsuch that for all ε > 0satisfying ε< ε, there exist
Rε
1, cε
1>0and cε
2Rsuch that |x|> Rε
1implies
|Fε(x)| ≥ cε
1|x|p+cε
2
for some p > 1.
Lemma 3.2 (Asymptotically inward-flowing).For all ε > 0,there exist Rε
2, cε
7>0such that
|x|> Rεimplies Fε(x)6= 0 and
hFε(x),r(x)i<cε
7|Fε(x)|,
where r(x) = x/|x|is the normalized outward-pointing radial vector at x.
8
Most probable paths in piecewise-smooth SDEs
We now prove the existence of a minimizer of I(t0,tf)
εusing Tonelli’s direct method of the calculus
of variations; see, e.g., [57, 66]. This procedure consists of first proving that a minimizing sequence
of I(t0,tf)
εis bounded with respect to the H1norm and thus has a weakly convergent subsequence
in the H1topology. Note, it is necessary to work in the H1topology since closed and bounded sets
are not necessarily compact in the strong topology. Nevertheless, upon passing to a subsequence,
we can establish a limit point for the minimizing sequence. Finally, by proving that I(t0,tf)
εis lower
semi-continuous with respect to weak convergence in the H1topology, it follows that limit points
of the minimizing sequence are indeed minimizers.
To begin the procedure for the direct method of the calculus of variations outlined in the
previous paragraph, we first recall the definition of weak convergence in the H1topology in the
context of our problem.
Definition 3.3 (Weak convergence in H1[67]).A sequence {αm}
m=1 H1([t0, tf]; Rn)converges
weakly to αH1([t0, tf]; Rn), written
αm*αin H1([t0, tf]; Rn)or αm
H1
*α,
provided αm*αin L2([t0, tf]; Rn)and ˙
αm*˙
αin L2([t0, tf]; Rn). That is, for all ω
L2([t0, tf]; Rn),
lim
m→∞ Ztf
t0hαm,ωidt =Ztf
t0hα,ωidt and lim
m→∞ Ztf
t0h˙
αm,ωidt =Ztf
t0h˙
α,ωidt.
The following lemma establishes that boundedness of I(t0,tf)
ε[αm] implies boundedness of αm
with respect to the H1norm.
Lemma 3.4. There exists ε>0such that if ε<εand if αm∈ A satisfies I(t0,tf)
ε[αm]< M1for
some M1>0, then there exists M2>0such that kαmkH1< M2.
Proof. For contradiction, suppose αmis unbounded with respect to the H1norm. Therefore, there
exists a subsequence αmksuch that kαmkkH1→ ∞.
1. Suppose there exists M > 0 such that kαmkk< M. Consequently, since Fεis smooth
there exists C1>0 such that |Fε(αmk)|< C1. By the Cauchy-Schwarz inequality,
I(t0,tf)
ε=Ztf
t0|˙
αmk|22hαmk,Fε(αmk)i+|Fε(αmk)|2dt
Ztf
t0|˙
αmk|22hαmk,Fε(αmk)idt
≥ k ˙
αmkk22C1(tft0)1
2k˙
αmkk.
Therefore, since kαmkkH1→ ∞, it follows from Poincare’s inequality that k˙
αmkk → ∞. Thus,
I(t0,tf)
ε→ ∞, which is a contradiction.
2. Suppose kαmkkis unbounded. Upon passing to another subsequence which we also label
αmk, it follows that kαmkk→ ∞. Applying the elementary inequality a2+b22ab, it follows
that
I(t0,tf)
ε[αmk]2Ztf
t0
[|˙
αmk||Fε(αmk)|−hαm,Fε(αmk)i]dt
= 2 Ztf
t0|˙
αmk||Fε(αmk)|sin2θ
2dt,
9
Most probable paths in piecewise-smooth SDEs
where θ(t) is the angle between ˙
αmk(t) and Fε(αmk(t)).
We now choose ε, Rε
1as in Lemma 3.1 and Rε
2as in Lemma 3.2, set Rε= max{Rε
1, Rε
2}, and
let Ak={t:|αmk(t)|> Rεand d
dt |αmk(t)| ≥ 0}. It follows from Lemmas 3.1 and 3.2 that there
exists a constant C2>0 such that
I(t0,tf)
ε[αmk]C2ZAk|˙
αmk||Fε(αmk)|dt
C2ZAk|˙
αmk||αmk|pdt
=C2ZAk|˙
αmk||αmk||αmk|p1dt.
Applying the Cauchy-Schwarz inequality and the fundamental theorem of calculus, we obtain
I(t0,tf)
ε[αmk]C2ZAkh˙
αmk,αmki|α|p1dt
=C2ZAk
d
dt|α|p+1dt
=C2max
t[t0,tf]|αmk|p+1 (Rε)p+1.
Since kαmkk→ ∞ implies maxt[t0,tf]|αmk(t)|→∞, it follows that I(t0,tf)
ε[αmk]→ ∞, which is a
contradiction. It follows from items 1 and 2 that αmis bounded with respect to the H1norm.
Finally, we use the direct method of the calculus of variations to prove the existence of a
minimum.
Theorem 3.5 (Existence of minimizer of the mollified functional).There exists an ε>0such
that if ε<ε, then there exists an α
ε∈ A(t0,tf)such that for all α∈ A(t0,tf),
I(t0,tf)
ε[α
ε]I(t0,tf)
ε[α].
Proof. Let ε > 0,and let αmbe a minimizing sequence of I(t0,tf)
ε[α]; i.e.,
lim
m→∞ I(t0,tf)
ε[αm] = inf
α∈A(t0,tf)I(t0,tf)
ε[α].
Consequently, there exists M1>0 such that I(t0,tf)
ε[αm]< M1and thus by Lemma 3.4 there
exists M2>0 such that kαmkH1< M2. Thus, by the Banach–Alaoglu theorem and the Rellich-
Kondrachov theorem there exists α
ε∈ A(t0,tf)and a subsequence αmksuch that ˙
αmk
L2
*˙
α
εand
αmk
L
α
εas k→ ∞ [68]. Since the integrand appearing in I(t0,tf)
εis convex with respect to ˙
α,it
follows that I(t0,tf)
εis lower semicontinuous with respect to weak convergence [67], and therefore
I(t0,tf)
ε[α
ε]lim inf
k→∞ I(t0,tf)
ε[αmk] = lim
m→∞ I(t0,tf)
ε[αm] = inf
α∈A(t0,tf)I(t0,tf)
ε[α].
10
Most probable paths in piecewise-smooth SDEs
3.2 Euler-Lagrange equations
Now that the existence of minimizers is established for the Freidlin-Wentzell rate I(t0,tf)
ε[αε] cor-
responding to the mollified system ˙
x=Fε, we can establish a necessary condition for minimizers,
which is that α∈ A(t0,tm)
Σis a C2integral curve of the flow generated by the following system of
Euler-Lagrange (EL) equations:
¨
x=Fε
t+ (Fε− ∇Fε|)˙
x+Fε|Fε,(17)
where the subscript tdenotes the partial derivative with respect to time. It is convenient to
introduce the scaling z=x/ε, z [1,1],so the EL Equations (17) become the fast-slow system
ε¨z=Fε
1,t + (yFε
1Gε
z)˙y +Gε|
zGε,
¨
y=Gε
t+ (Gε
z− ∇yFε
1)ε˙z+ (yGε− ∇yGε|)˙
y+yGε|Gε.
Introducing the conjugate momenta ϕ=ε˙zFε
1and ψ=˙
yGεleads to the Hamiltonian form,
ε˙z=Fε
1+ϕ,
˙
y=Gε+ψ,
˙ϕ=ϕF ε
1,z − hψ,Gε
zi,
˙
ψ=ϕyFε
1− hψ,yGεi.
(18)
The most probable path of System (2),(3) corresponds to solutions of the EL Equations (18) subject
to boundary conditions for appropriate values of t. In the case studies of Sections 4 and 5, we use
the gradient flow to numerically solve System (18) when an explicit solution is not available.
3.3 Limiting functional
In this section, we consider the problem of computing an appropriate limiting functional I(t0,tf):
A(t0,tf)7→ R. Theorem 3.5 guarantees the existence of a minimizer of I(t0,tf)
ε[α] for fixed εand is
given by α
ε= arg minα∈A(t0,tf)I(t0,tf)
ε[α(t)].If we now consider the sequence of minimizers α
εand
suppose there exists α∈ A(t0,tf)such that α
ε
H1
*α, an appropriate limiting functional I(t0,tf)
should have the property that αminimizes I(t0,tf). Introduced by de Giorgi, the Γ-limit is the
precise notion of a limiting functional that ensures this property [56, 57]. The following definition
of a Γ-limit is adapted from [57] for our specific problem.
Definition 3.6 Convergence [57]).We say that I(t0,tf)
εΓconverges to I(t0,tf):A(t0,tf)7→ R
with respect to H1weak convergence if for all α∈ A(t0,tf), we have
1. (liminf inequality) for every sequence αε∈ A(t0,tf)satisfying αε
H1
*α,
I(t0,tf)[α]lim inf
ε0I(t0,tf)
ε[αε];
2. (recovery sequence) there exists a sequence αε
H1
*αsuch that
lim
ε0I(t0,tf)
ε[αε] = I(t0,tf)[α].
11
Most probable paths in piecewise-smooth SDEs
If I(t0,tf)
εΓ-converges to I(t0,tf)we write Γlim
ε0I(t0,tf)
ε=I(t0,tf).
Showing both the lim inf inequality and the existence of a recovery sequence establish that both
the lim sup and the lim inf of I(t0,tf)
εover all weakly converergent sequences exists and is equal
to I(t0,tf). Indeed, these two conditions ensure that I(t0,tf)
εhas a common lower bound, the lower
bound is optimal, and the limiting functional is precisely this lower bound. Moreover, the definition
of Γ-convergence is constructed in such a manner that ensures the convergence of minimizers of
I(t0,tf)
εto the minimizer of I(t0,tf)if an additional equicoecervity condition is satisfied. The following
theorem adapted from [57] ensures this convergence for our specfic problem.
Theorem 3.7. Suppose I(t0,tf)
εΓ-converges to I(t0,tf):A(t0,tf)7→ Ras ε0with respect to H1
weak convergence and suppose for all α
εthat minimize I(t0,tf)
εthere exists a constant M > 0such
that kα
εkH1< M. Then there exists an α∈ A(t0,tf)such that
inf
α∈A(t0,tf)I(t0,tf)[α] = I(t0,tf)[α].
Moreover, every limit of a subsequence of α
εas ε0is a minimizer of I(t0,tf).
One of the challenges with proving a Γ-limit is that the target limiting functional must be formed
from an educated guess based on analysis of minimizers. In the system of interest, it is natural
to expect that the appropriate Γ-limit will correspond to the standard FW functional except for
intervals of time on which the curve tracks Σ. However, for α∈ A(t0,tf)satisfying {tR:α(t)=0}
has nonzero measure, the recovery sequence must be constructed to minimize the functional for
times in which α(t)Σε={xRn:|x|< ε}. To compute the appropriate Γ-limit for I(t0,tf)
ε,we
introduce the following intervals of time:
I[α] = {t(t0, tf) : α(t)6∈ Σ}={t(t0, tf) : α1(t)6= 0},
Iε[α] = {t(t0, tf) : α(t)6∈ Σε}={t(t0, tf) : |α1(t)| ≥ ε},
IΣ[α] = {t(t0, tf) : α(t)Σ}={t(t0, tf) : α1(t)=0},
IΣ+[α] = {t(t0, tf) : α(t)Σ+}={t(t0, tf) : α1(t) = 0 and F±
1(0,β)>0},
IΣ[α] = {t(t0, tf) : α(t)Σ}={t(t0, tf) : α1(t) = 0 and F±
1(0,β)<0},
Iε
Σ[α] = {t(t0, tf) : α(t)Σε}={t(t0, tf) : |α1(t)|< ε},
I±[α] = {t(t0, tf) : α(t)S±},
Iε
±[α] = {t(t0, tf) : α(t)Sε
±}.
as well as introduce the function Λ : R7→ R, defined by
Λ(z) = 1
2+Zz
0
ζ(u)du. (19)
Note that Λ(z) = 1 if z > 1,Λ(z) = 0 if z < 1,and Λ(z) smoothly transitions from 0 to 1 on the
interval [1,1]. We now state the following Γlimit result.
Theorem 3.8. There exists I(t0,tf):A(t0,tf)7→ Rsuch that
Γlim
ε0I(t0,tf)
ε=I(t0,tf),
12
Most probable paths in piecewise-smooth SDEs
where for α= (α, β)∈ A(t0,tf),
I(t0,tf)[α] = ZI[α]k˙
α(t)F(α(t))k2dt
+ZIΣ[α]
min
λ[0,1] nλF +
1(0,β) + (1 λ)F
1(0,β)2
+
˙
βλG+(0,β)(1 λ)G(0,β)
2dt.
(20)
Proof. For fixed α= (α, β)∈ A(t0,tf), we first show the existence of a recovery sequence, αε. We
will construct αεover three intervals, Iε
+[α],Iε
[α],and Iε
Σ[α],by
αε(t) = α(t){t∈Iε
+[α]}+α(t){t∈Iε
[α]}+ (εz(t),β(t)) {t∈I ε
Σ[α]}
=α(t){αSε
+}+α(t){αSε
}+ (εz(t),β(t)) {(εz,β)Σε},
where z(t) is given by
z= arg min
z0[1,1] nΛ(z0)F+
1(0,β) + 1Λ(z0)F
1(0,β)2
+
˙
βΛ(z0)G+(0,β)1Λ(z0)G(0,β)
2.
(21)
To prove convergence of I(t0,tf)
ε[αε] to I(t0,tf)[α] and αε
H1
*α,we first consider the interval of
time Iε
+[α]. Since αε(t){αεSε
+}=α(t){αSε
+}it follows that αε(t){αεSε
+}
H1
*α(t){αSε
+}.
Moreover, we have
ZIε
+[αε]k˙
αεFε(αε)k2dt ZI+[α]k˙
αF+(α)k2dt
=ZIε
+[α]∩I+[α]2h˙
α,F+(α)Fε(α)i+|Fε(α)|2− |F+(α)|2dt
+ZIε
+[α]\I+[α]k˙
αFε(α)k2dt ZI+[α]\Iε
+[α]k˙
αF+(α)k2dt.
Since Fε(x, y){xε}=R1
1ζ(v)F+(xεv, y)dv {xε}and R1
1ζ(v)F+(xεv, y)dv converges
uniformly to F+(x, y) as ε0,the integral RIε
+[α]∩I+[α]h˙
α,F+(α)Fε(α)idt vanishes as ε0.
By Minkowski’s inequality and the triangle inequality,
ZIε
+[α]∩I+[α]|Fε(α)|2− |F+(α)|2dtZIε
+[α]∩I+[α]kFε(α)k2− kF+(α)k2dt
ZIε
+[α]∩I+[α]kFε(α)F+(α)k2dt,
and thus by the same uniform convergence argument as above vanishes as ε0. Since the measures
of Iε
+[α]\ I+[α] and I+[α]\ Iε
+[α] vanish as ε0 it follows that the remaining integrals also
13
Most probable paths in piecewise-smooth SDEs
vanish. The analogous case for I[α] is identical. Hence,
lim
ε0I(t0,tf)
ε[αε{αεSε
±}] = lim
ε0ZIε[α]k˙
αFε(α)k2dt
=ZI[α]k˙
αF(α)k2dt
=I(t0,tf)[α{α6∈Σ}].
We now consider the interval of time Iε
Σ[α]. Note that using the substitution v=u/ε, we have
lim
ε0Fε(εz, y){|εz|}= lim
ε0Zz
1
ζ(v)F+(εz εv, y)dv +Z1
z
ζ(v)F(εz εv, y)dv
= Λ(z)F+(0,y) + (1 Λ(z)) F(0,y).
Let
H(y;z) = Λ(z)F+(0,y) + (1 Λ(z)) F(0,y) = λF+(0,y) + (1 λ)F(0,y),
where we set z(t) as in Equation (21), so that λ= Λ(z)[0,1] is a constant. Thus, in general
limε0Fε(εz, y) = H(z, y), where zvaries in [1,1],but we evaluate Hat a particular zvalue to
correspond with the Γlimit. Since αε(t){αεΣε}= (εz, β)∈ A(t0,tf)it follows that
ZIε
Σ[α]k˙
αεFε(αε)k2dt ZIΣ[α]k˙
αH(β;z)k2dt
=ZIε
Σ∩IΣk˙
αεFε(αε)k2− k ˙
αH(β;z)k2dt
+ZIε
Σ\IΣk˙
αεFεk2dt ZIΣ\Iε
Σk˙
αHk2dt.
As above, the last two integrals vanish as ε0,since the measure of Iε
Σ\ IΣconverges to zero
and IΣ\ Iε
Σ=,and the integrands are bounded. Since Iε
Σ∩ IΣ=IΣ,
ZIε
Σ∩IΣk˙
αεFε(αε)k2− k ˙
αH(β;z)k2dt
=ZIΣ|˙
αε|22h˙
αε,Fε(αε)i+|Fε(αε)|2− | ˙
α|2+ 2h˙
α,H(β;z)i−|H(β;z)|2dt.
Since ˙
αε{αεΣε}
H1
*˙
α{αΣ}and Fε(αε){αεΣε}
H1
*H(β;z){αΣ},
ZIΣ|˙
αε|2− | ˙
α|2dt =Ztf
t0|˙
αε{αεΣε}|2− | ˙
α{αΣ}|2dtZtf
t0k(˙
αε˙
α){αεΣε}k2dt 0
as ε0, and
lim
ε0ZIΣ|Fε(αε)|2− |H(β;z)|2dt = lim
ε0Ztf
t0|Fε{αεΣε}|2− |H{αΣ}|2dt
lim
ε0Ztf
t0k(FεH){αεΣε}k2dt
= 0,
14
Most probable paths in piecewise-smooth SDEs
where we have used Minkowski’s inequality and the triangle inequality, as above. Also, as a conse-
quence of the uniform boundedness of sequences in H1,the product of a weakly convergent sequence
and a strongly convergent sequence is weakly convergent [67], and thus
ZIΣh˙
α,H(β;z)i−h˙
αε,Fε(αε)idt =Ztf
t0h˙
α,H(β;z)i{αΣ}− h ˙
αε,Fε(αε)i{αεΣε}dt 0
as ε0. Hence,
lim
ε0I(t0,tf)
ε[αε{αεΣε}] = lim
ε0ZIε
Σk˙
αεFε(αε)k2dt
=ZIΣk˙
αH(β;z)k2dt
=ZIΣ
min
λ[0,1] nλF +
1(0,β) + (1 λ)F
1(0,β)
2
+
˙
βλG+(0,β)(1 λ)G(0,β)
2dt
=I(t0,tf)[α{αΣ}].
We now prove the liminf inequality. For the portion of αin S±,the liminf inequality follows
directly from the recovery sequence calculation, since we chose αε=αfor any α∈ A(t0,tf). For
the portion of αin Σ,we have that for every αε,α∈ A(t0,tf)satisfying αε{αεΣε}
H1
*α{αΣ},
lim
ε0I(t0,tf)
ε[αε{αεΣε}] = lim
ε0Ztf
t0|˙
αε{αεΣε}Fεαε{αεΣε}|2dt
=ZIΣ[α]|˙
α{αΣ}H(α{αΣ})|2dt
ZIΣ[α]
min
λ[0,1] nλF +
1(0,β) + (1 λ)F
1(0,β)
2
+
˙
βλG+(0,β)(1 λ)G(0,β)
2dt.
Then by the lower semicontinuity of the rate functional with respect to weak convergence,
I(t0,tf)[α{αΣ}] = ZIΣ
min
λ[0,1] nλF +
1(0,β) + (1 λ)F
1(0,β)
2
+
˙
βλG+(0,β)(1 λ)G(0,β)
2dt
lim inf
ε0I(t0,tf)
ε[αε{αεΣε}].
In other words, this result shows that in the smooth regions S±,the minimum of the rate
functional I(t0,tf)by α(t),where αis the solution to the EL Equations (18). Then in the switching
manifold Σ, the rate functional is minimized by some α(t;z) = (0,β(t;z)) ∈ A(t0,tf),where zis
defined using Equation (21). Note that the critical manifold of the fast-slow System (18) is defined
implicitly as solutions in Rnto following system of equations
M0(0 = Λ(z)F+
1(0,y) + (1 Λ(z)) F
1(0,y) + ϕ
0 = ϕF+
1(0,y) + F
1(0,y)+hψ,G+(0,y) + G(0,y)i),
15
Most probable paths in piecewise-smooth SDEs
so when the conjugate momentum has ϕ= 0,Λ(z)F+
1(0,y) + (1 Λ(z)) F
1(0,y) = 0. Thus, the
path that minimizes the rate functional in Σ is given by α= (0,β),where
˙
β= Λ(z)G+(0,β) + (1 Λ(z)) G(0,β).(22)
Practically, to obtain the most probable path in the switching manifold, we first determine the
value of zso that Λ(z)F+
1(0,y) + (1 Λ(z)) F
1(0,y) = 0 for each yRn1. This fixes z, so we
may calculate the path using Equation (22).
To make explicit the connection of this result to the dynamics imposed on the switching manifold
by the Filippov convex combination, recall the definition of sliding flow given by Equations (5),(6).
Here, we may fix λ= Λ(z) so that the first component of the sliding flow in Equation (5) becomes
zero. The remaining components describing the sliding flow in Equation (5) then either coincide
with Equation (22) or its time-reversed flow, with t→ −t. That is, Theorem 3.8 indicates that the
intersection of the most probable path with the switching manifold, α(t){αΣ},may correspond
to the solution of the flow given by ˙
y=Gs(0,y;z),where Gsis the convex combination
Gs(0,y;z) = λG+(0,y;z) + (1 λ)G(0,y;z),
and λ[0,1]. Thus, the most probable path in the switching manifold precisely recovers the
Filippov dynamics with this particular form of λ(or its time-reversed dynamics). This result is
independent of the mollifier imposed, as long as it fits the form defined in Equation (14), though
the path does depend on the mollifier via Λ(z).
Furthermore, this indicates that the functional given by Equation (20) has no contribution
during times when the path slides in a sliding region, but it does when the path slides in a crossing
region.
4 Case Study 1: Linear system in 2D
In this section, we explore the application of the most probable path derived in Section 3 to a
simple piecewise-linear system in R2. Consider System (2),(3), where F±:R27→ R2are defined as
F+(x, y) = f+
g+=a(x1) + b(yη)
c(x1) + yη,
F(x, y) = f
g=p(x+ 1) + q(y+η)
r(x+ 1) + y+η.
(23)
Here, a, b, c, p, q, r, η Rare parameters. This splits up the plane into two smooth regions separated
by the switching manifold Σ = {(x, y)|x= 0}. This system has been nondimensionalized to scale
out a parameter.
We begin by investigating the dynamics of the deterministic skeleton, System (4),(23), as the
parameter ηvaries. For all values of η, there are two stable fixed points separated by the switching
manifold Σ. For most values of η, dynamics fall under two broad categories: (1) the two fixed
points are the only asymptotic states possible for x0R2\Σ, and (2) there is a sliding-cycle
surrounding each fixed point, with a crossing-cycle surrounding all sliding-cycles (see Figure 2(f)).
For intermediate values of ηwe observe (3) both standard and nonstandard discontinuity-induced
bifurcations.
Taking the two most prevalent cases (1) and (2), we perform Monte Carlo simulations of System
(2),(23). In the case of attracting sliding, Monte Carlo simulations match the most probable path
derived by minimizing the functional Equation (20), which slides along Σ. However, in the case of
repelling sliding, Monte Carlo simulations match a most probable path that does not slide, instead
crossing Σ at the onset of sliding.
16
Most probable paths in piecewise-smooth SDEs
(a) (b)
-10 0 10
-2
0
2
(c)
-10 0 10
-2
0
2
(d)
-10 0 10
-2
0
2
(e)
-10 0 10
-2
0
2
(f)
Figure 2: (a) Stability plane for x+. Shaded regions have stable nodes (blue) or stable spirals
(yellow). (b) A bifurcation diagram for System (4),(23). Red lines are stable foci and the black
line is a pseudosaddle. Brown curves are the crossing points of the large crossing limit cycle. The
orange and purple dashed curves are the upper and lower limits of the sliding portion of the two
(unstable) sliding limit cycles. The grey shaded region indicates ηvalues where the two sliding
limit cycles may have both period-1 and period-2. Vertical dashed lines indicate bifurcations
identified: a visible tangency bifurcation (V V2), a crossing cycle-tangency bifurcation (I),two
sliding cycle-tangency bifurcations (II) and (III),and a sliding cycle-nonunique period bifurcation
(IV ). (c)-(f) Phase portraits for illustrative values of η. Colors correspond to (b), with the period-
2 unstable sliding limit cycle indicated as a dashed gray curve. Parameters used in all plots are
a=p=2, b =q=7,and c=r= 1.
4.1 Deterministic dynamics
In System (4),(23), there is one equilibrium in each smooth region, at x±= (±1,±η). Figure
2(a) summarizes the linear stability analysis of x+; due to the symmetry of the system, the linear
stability of xis identical if we consider the vertical axis as qr and the horizontal axis as p. The
solid line separates saddle (a < bc) from non-saddle equilibria. For non-saddle equilibria, the dashed
line further separates stable equilibria (a < 1) from unstable equilibria. Stable spirals occur in
the yellow region and stable nodes occur in the blue region, which are separated by the curve
bc =(a1)2/4.
Figure 2(b) shows the bifurcations in the system as ηvaries, in the case where x±are stable
spirals (ηdoes not affect the linear stability of x±). The remainder of this section discusses
17
Most probable paths in piecewise-smooth SDEs
dynamics shown in this bifurcation diagram. We identify bifurcations using existing terminology
where possible, though some bifurcations do not appear to have existing classification. Note,
although we discuss the time-reversed dynamics in this section, in general most probable paths are
not time-reversible unless there is a gradient structure [69].
For large η, in addition to the two stable foci x±,the system has three limit cycles of the
piecewise-smooth flow: one stable crossing limit cycle with large amplitude and two unstable sliding
limit cycles around each focus. See Figure 2(f) for a visualization of the phase portrait with all
three limit cycles. In (b) the points at which the crossing limit cycle crosses Σ are shown as brown
curves, and the maximum and minimum of the intersection of each sliding limit cycle with the
(repelling) sliding region are dashed orange and purple curves, respectively. Although dynamics on
Σ are not defined a priori, if we impose a sliding flow using the Filippov convex method, with
λ(y) = a+b(yη)
a+b(yη)pq(y+η),
then there are at most two pseudoequilibria xs= (xs, ys), with xs= 0 and ysgiven by the root(s)
of the quadratic [λ(ys)(r+ys+η) + (1 λ(ys))(c+ysη)]. For a=p, b =q, and c=rthere
is one pseudoequilibrium, at the origin.
As we decrease ηfrom 0.8, the two sliding cycles collide (though not at a point of tangency). Due
to the repelling sliding region and nonuniqueness of solution curves leaving the sliding region, this
collision leads to higher period behavior for both limit cycles while preserving the period-1 behavior,
in that on the limit cycle a solution may either cycle around one focus, or both, or switch between
the two. Although similar to a period-doubling bifurcation, in this case the limit cycle may have any
number of periods, and the stability is unchanged by the bifurcation. This bifurcation is labeled as
(IV ) in Figure 2(b); for brevity, we refer to it as a ‘sliding cycle-nonunique period’ bifurcation. We
indicate ηvalues with potential higher-period limit cycles using gray shading. In reverse time, in
which the unstable sliding cycles become stable and the repelling sliding region becomes attracting,
this behaves like a period-doubling bifurcation of two sliding limit cycles. Neither the repelling nor
attracting sliding cases appear to fit within recent classifications of piecewise-smooth bifurcations;
see [70–72].
At (II I ) in Figure 2(b), the point of tangency for the right sliding limit cycle collides with the
maximum of the left sliding limit cycle on Σ,annihilating the left sliding cycle and thereby the
higher-period limit cycles as well. However, the trajectory that formed the left sliding limit cycle
persists, though it can no longer slide. Then at (II) a similar bifurcation occurs when the minimum
of the right sliding cycle collides with the tangent point of the left formerly-sliding cycle trajectory,
breaking the right limit cycle.
At (I) in Figure 2(b), the crossing limit cycle collides with both the left and right formerly-
sliding cycle trajectories, breaking the cycle. Although the collision does not occur at the tangency
points of the formerly-sliding trajectories, for brevity we refer to this as a ‘crossing cycle-tangency’
bifurcation. As with (IV ),bifurcations (I)(III) do not appear to have been previously classified.
Finally, at η= 0 the two points of tangency at the switching manifold collide and slide past each
other, causing the repelling sliding region to become an attracting sliding region. In the context
of Kuznetsov, Rinaldi, and Gragnani’s classification of bifurcations in planar piecewise-smooth
systems, this appears to be a V V2type bifurcation [70].
4.2 Most probable paths
We determine the most probable path through the switching manifold by smoothing out the bound-
ary, then taking the limit of the resulting most probable path as our smoothing parameter goes to
18
Most probable paths in piecewise-smooth SDEs
zero to return to the original piecewise-smooth system. We define the mollification of the vector
field as in Equation (15), so that Fε(x, y)=(fε(x, y), gε(x, y)) = (fε(x), gε(x)), where
fε(x) =
f(x), x ≤ −ε,
f+(xx
ε+f(x)1Λx
εaΦ(ε, x)pΦ(x, ε),|x|< ε,
f+(x), x ε,
gε(x) =
g(x), x ≤ −ε,
g+(xx
ε+g(x)1Λx
εcΦ(ε, x)rΦ(x, ε),|x|< ε,
g+(x), x ε.
(24)
Here, Λ(x/ε) is defined as in Equation (19), and we define Φ(x1, x2) as
Φ(x1, x2) = Zx2
x1
ε(u)du.
Note that for |x|< ε, if a=pthe second two terms of fεvanish, and if c=rthe second two terms
of gεvanish. There are now two switching manifolds in the system, Σε
±={xR2:x=±ε},but
the flow is smooth across them. Note that due to the linearity of the system, mollification does not
affect the fixed points x±as long as ε < 1.
The persistence of the pseudoequilibrium xsthrough mollification is not as straightforward as
that of x±. Although a regular equilibrium may limit to a pseudoequilibrium as a smooth system
becomes piecewise-smooth, nonlinear dynamics on the switching manifold may lead to multiple
possible phenomena in smoothing out a piecewise-smooth field [53]. However, for the particular
choice of Fin this case study, after mollification there is a saddle equilibrium at the origin, which
was the pre-mollification location of the pseudosaddle.
We introduce the scaling z=x/ε, z [1,1], to simplify the limit as ε0. With this change
of variables, System (4),(24) becomes a fast-slow system,
ε˙z=fε(εz, y),
˙y=gε(εz, y).
Minimizers of the rate functional I(t0,tf)(Equation (1)) in this rescaled flow are solutions of the EL
equations
ε¨z= ˙yfε
ygε
z+fεfε
z+gεgε
z,
¨y=ε˙zgε
zfε
y+fεfε
y+gεgε
y,
or in Hamiltonian form, with the conjugate momenta ϕ=ε˙zfεand ψ= ˙ygε,
ε˙z=fε(εz, y) + ϕ,
˙y=gε(εz, y) + ψ,
˙ϕ=fε
z(εz, y)ϕgε
z(εz, y)ψ,
˙
ψ=fε
y(εz, y)ϕψ.
(25)
This ensures that the stable fixed points x±persist through the rescaling, extended by two ad-
ditional components due to ϕand ψ. Fixed points in this extended system are given by z±=
(z±, y±,0,0) = (±1/ε, ±η, 0,0).
The projection onto the xyplane of the solution to System (25), which is the most probable
path of System (4),(23), is shown in Figure 3. Note that we have used xinstead of zto plot the
19
Most probable paths in piecewise-smooth SDEs
-1 0 1 2
-2
0
2
(a)
-1 0 1 2
-2
0
2
(b)
-0.4 -0.2 0 0.2 0.4
-2
0
2
(c)
Figure 3: (a) Most probable path of System (4),(23) from x+to x. (b) Most probable path as
in (a), with varying ε. (c) Zoomed region of (b). All parameters given by a=p=2, b =q=
7, c =r= 1,and η=2. In (a) ε= 0.5, and in (b),(c) ε= 0.5,0.1,0.05,and ε0 (light grey
to black curves).
most probable path for several values of εtogether. Figure 3(a) shows the most probable path for
the mollified rescaled system, System (25) with ε= 0.5,including the deterministic flow from xs
to x. Figure 3(b) shows the most probable path for decreasing values of ε, and Figure 3(c) shows
a zoomed-in region of (b). As illustrated in (b) and (c), this is consistent with the minimizer of the
rate functional for the piecewise-smooth system as ε0. Here and in the remaining figures for
this case study, we have used the function h(x) = exp 1/x21to construct the mollifier. See
Appendix B for the derivation of the most probable path as the solution of System (25).
We can numerically validate the most probable path using Monte Carlo simulations of System
(2),(23) with additive noise using the Euler-Maruyama scheme. Note that this system has a dis-
continuous and non-monotonic drift coefficient. Euler-Maruyama has been shown to converge in
probability for such systems [73, 74], but results for stronger convergence remain elusive; see [75]
and references therein. Since we are concerned with convergence of the mean of the distribution of
paths, Euler-Maruyama is sufficient for this investigation.
We performed Monte Carlo simulations for two values of η:η= 1,where there is repelling
sliding along Σ, and η=2,where there is attracting sliding. See Figure 4 for histograms of the
solutions to System (2),(23) that ‘tip’ from x+to a neighborhood of x. For (a) η=2, transition
paths tip over the basin boundary Σ by tracking the attracting sliding region until it ends. Recall
that in this parameter regime, there are only two limit points in the deterministic system. For (b)
η= 1,transition paths tip to a neighborhood of xin a sequence of basin-crossings, first over the
x > 0 unstable sliding cycle, then primarily following the deterministic flow toward the crossing
limit cycle, but ultimately crossing over the x < 0 unstable sliding cycle and approaching x. For
all figures, we have overlaid the most probable path calculated as the path αthat minimizes the
rate functional, Equation (20).
Note that in Figure 4(b), we indicate both the most probable path followed by the Monte Carlo
simulations (black curve) and the predicted family of nonunique most probable paths that slide (red
curves). These paths coincide for x > 0 but are distinct for x0.The observed most probable path
in the left-half plane corresponds to the solutions to the EL equations with boundary conditions
x(t1) = (0,maxyΣRy) and x(tf) = x.We calculate such solutions using the gradient flow; that
20
Most probable paths in piecewise-smooth SDEs
-0.1 0 0.1
0.4
0.8
1.2
1.6
104
-1 0 1
-2
-1
0
1
2
(a)
-2 0 2
-4
-2
0
2
4
2
50
100
150
200
250
300
-10 -5 0 5 10
-4
-2
0
2
4
2
50
100
150
200
250
300
(b)
Figure 4: Distribution of tipping events from Monte Carlo simulations and most probable paths (red
and black curves) from x+to x, with (a) attracting sliding (η=2; 6,487 tips with N= 1.056×107
simulations) and (b) repelling sliding (η= 1; 1,960 tips with N= 1.676 ×107simulations).
Parameters used are a=p=2, b =q=7, c =r= 1,and σ= 0.3. All other curves are as in
Figure 2.
is, we consider solutions α: [t1, tf]7→ R2to
α
∂s =δI(t0,tf)
δα=¨
α− ∇F|F+ (F− ∇F|)˙
α,
α(s, t1) = (0,max
yΣR
y),
α(s, tf) = x,
α(0, t) = xg(t),
(26)
where xg(t) is a smooth function satisfying xg(t1) = (0,maxyΣRy) and xg(tf) = x,and s > 0 is
an artificial time. Along solution curves α(s, t),
d
dsI(t0,tf)[α(s, t)] 0.
Therefore I(t0,tf)is a Lyapunov functional and thus lims→∞ α(s, t) = α(t), where α(t) is a
21
Most probable paths in piecewise-smooth SDEs
Figure 5: Values of the derived rate functional, Equation (20) for η= 1 and F=Fε, as εvaries.
The black line corresponds to the rate functional value for the crossing most probable path and
the red lines correspond to the functional values for the non-unique family of sliding most probable
paths.
solution to the EL equations, System (25). That is, steady state solutions of Equation (26) solve
the Euler-Lagrange equations. The converged solution to the IBVP in Equation (26), approximated
using the Forward Time Centered Space finite difference scheme with sas the time variable and t
as the space variable, is shown as the black curve in Figure 4(b).
From Equation (20) we might expect the predicted most probable paths in Figure 4(b) to
minimize I(t0,tf)because I(t1,tf)= 0. However, if we consider the predicted most probable path of
the piecewise-smooth system in the context of the mollified system and calculate I(t0,tf)
εfor several
values of ε > 0,the contribution of the rate functional as ε0 is substantial: the minimum rate
functional values for the predicted family of most probable paths are two magnitudes greater than
those of the observed most probable path; see Figure 5. This shows anecdotally that although
the minimizers of the Freidlin-Wentzell rate functional converge weakly to minimizers of Equation
(20), they may not converge strongly. Furthermore, since the repelling sliding region has Lebesgue
measure zero, the probability that the Euler-Maruyama simulations lie on Σ at time tis zero for
all t.
5 Case Study 2: Non-autonomous system in 1D
In this section, we extend the most probable path framework to a linear one-dimensional non-
autonomous system with periodic forcing. Consider Equation (2), where F±:R×R7→ Rare
defined by
F+=f+(x, t) = r+(x1) + A+cos(2πt),
F=f(x, t) = r(xa) + Acos(2π(tp)).(27)
Here, r±, A±, p 0 and a0 are parameters. Since Fis a scalar here, we will use the notation
F=fthroughout this case study. Since we are considering periodic forcing, we view the flow
generated by Equations (4) and (27) as lying on the cylindrical extended phase space Π = R×S1,
where S1denotes a circle. More precisely, we make the change of variable τ=t, augmenting
22
Most probable paths in piecewise-smooth SDEs
Equation (4) by the additional equation /dt = 1, and apply periodic boundary conditions at
τ= 0 and τ= 1. However, for ease of presentation, we will continue to use Equation (4) to
represent the dynamics on Π. With this convention, Σ and S±are again defined as in Section 1
and the underlying SDE is given by
dx =f(x)dt +σdW, (28)
where, following Filippov’s convex combination method, the deterministic skeleton is given by
˙x=f(x) =
f+(x, t),(x, t)S+Σ+
f(x, t),(x, t)SΣ
0,(x, t)ΣAΣR
.(29)
5.1 Deterministic dynamics
Since f(x, t) is linear in xfor (x, t)/Σ, the general solutions to Equation (7) on S±are given by
x±(t) = C±er±t+h±(t),
respectively. Here C±Rare integration constants and
h+(t) = 1 + r+A+
r2
++ 4π2cos(2πt) + 2A+π
r2
++ 4π2sin(2πt),
h(t) = a+rA
r2
+ 4π2cos(2π(tp)) + 2Aπ
r2
+ 4π2sin(2π(tp)).
A solution with initial conditions in either S+or Sis then constructed by selecting C±to satisfy
the initial condition as well as the continuity assumption on Σ. Note, in this construction solutions
to Equation (7) may be non-unique, since solutions can intersect in forward time when tracking
ΣAand in backwards time on ΣR.
We now consider the various parameter regimes of interest. First, if the following inequalities
are satisfied:
A+<qr2
++ 4π2and A<aqr2
+ 4π2,(30)
then h±(t) do not intersect Σ and thus are stable limit cycle solutions to Equation (7). Since we
are ultimately interested in noise-induced transitions from one stable limit cycle to another, we will
assume these inequalities hold throughout the rest of the case study. Second, the geometry of the
basins of attraction B±for h±(t), respectively, and the existence of ΣA,ΣR,and Σ±depend on
whether the nullclines of fintersect Σ. The nullclines are given by curves along which f±(x, t) = 0;
for this system, this occurs along n±(t),where
n+(t) = 1 + A+
r+
cos(2πt) and n(t) = a+A
r
cos(2π(tp).
There are thus four cases to consider which are illustrated in Figure 6:
1. If A+< r+and A< r|a|,then neither nullcline intersects Σ and thus ΣR= Σ and
Σ±= ΣA=; see Figure 6(a). In this case B±correspond to S+and S.
2. If A+> r+and A< r|a|,then n+intersects Σ and thus Σ6=, ΣR=RΣ, and
Σ+= ΣA=; see Figure 6(b). In this case Bhas a nontrivial intersection with S+and Σ.
23
Most probable paths in piecewise-smooth SDEs
(a) Neither n±intersect Σ (b) n+intersects Σ (c) nintersects Σ
(d) n±both intersect Σ (e) n±both intersect Σ (f) n±both intersect Σ
Figure 6: Example phase portraits for Equations (7)-(8). The red curves show the stable limit
cycles h±, the black curves show the nullclines n±, the green line is ΣR, the blue line is ΣA, and the
dashed black lines are Σ±. The dark and light gray regions correspond to the basins of attraction
B±for h±respectively.
3. If A+< r+and A> r|a|,then nintersects Σ and thus Σ+6=, ΣR= Σ Σ+, and
Σ= ΣA=; see Figure 6(c). In this case B+has a nontrivial intersection with Sand Σ.
4. If A+> r+and A> r|x0|,then both n±intersect Σ. In this case Σ±6=but, depending
on the phase shift p, ΣAcan be either empty or nonempty; see Figure 6(d)-(f). Moreover,
B±can both have a nontrivial intersection with S+and S.
5.2 Most probable paths
In contrast to Case Study 1, this system does not contain stable fixed points but instead stable limit
cycles. Consequently, for t0< tf, we consider the problem of determining the most probable path
from h(t0) to h+(tf). That is, we consider the family of optimization problems parameterized by
t0and tfand define the most probable transition path from hto h+as the minimum over this
family. To do so, we redefine the admissible set of transition paths A(t0,tf)by
A(t0,tf)={αH1([t0, tf]; R) : α(t0) = h(t0) and α(tf) = h+(tf)},(31)
24
Most probable paths in piecewise-smooth SDEs
and consider the optimization problem
inf
t0<tf
inf
α∈A(t0,tf)I(t0,tf)[α],(32)
where it follows from Theorem 3.8 that
I(t0,tf)[α] = ZI[α]|˙α(t)f(α(t), t)|2dt +ZIΣ[α]
min
λ[0,1] nλf+(0, t) + (1 λ)f(0, t)2odt.
In this case study, the minimum over λ[0,1] in the second integral of I(t0,tf)can be explicitly
calculated. First, for times in which αlies in ΣAΣR, the minimum is obtained by setting
λ=f(0, t)/(f(0, t)f+(0, t)) .Second, for times in which αlies in Σ±,f+(0, t) and f(0, t)
have the same sign and thus the minimum is obtained when λ= 0 or λ= 1. Therefore, it follows
that
I(t0,tf)[α] = ZI[α]|˙α(t)f(α, t)|2dt +ZIΣ±[α]
min{|f+(0, t)|,|f(0, t)|}2dt. (33)
To study the optimization problem defined by Equation (33) we first present the following
lemma which simplifies the analysis.
Lemma 5.1. For the vector field defined by Equation (27) with parameters satisfying (30):
inf
t0<tf
inf
α∈A(t0,tf)I(t0,tf)[α] = inf
α∈A(−∞,)I(−∞,)[α].(34)
Proof. Since f±(x, t) have linear growth in xand are asymptotically inward-flowing, it follows from
Theorem 3.7 that for t0< tfsatisfying t0>−∞ and tf<there exists α∈ A(t0,tf)such that
I(t0,tf)[α] = inf
α∈A(t0,tf)I(t0,tf)[α].
Define ¯α∈ A(−∞,)by
¯α(t) =
h(t), t t0,
α(t), t0< t < tf,
h+(t), t > tf.
By construction, ˙
¯α(t) = fα(t), t) for tt0and ttf,and thus
inf
α∈A(−∞,)I(−∞,)[α]I(−∞,)[¯α] = I(t0,tf)[α] = inf
α∈A(t0,tf)I(t0,tf)[α].
Since this inequality is true for all values of t0< tfsatisfying t06=−∞ and tf6=,the result
follows.
Interestingly, it follows from Lemma 5.1 that minimizers of Equation (5.2) are not unique.
Specifically, since f±(x, t) = f±(x, t +T) for TZ, it follows that if α∈ A(−∞,)is a minimum
then α
T∈ A(−∞,)defined by α
T(t) = α(t+T) is a minimum as well. Nevertheless, when
represented as curves in the cylndrical phase space Π these curves are identical. However, I(−∞,)
can possibly contain an uncountable number of non-unique minimizers even on Π. This possibility
arises when ΣRand the closure of B+,which we denote as B+,have a non-trivial intersection,
25
Most probable paths in piecewise-smooth SDEs
(a) (b)
Figure 7: (a) Non-unique minimizers of I(−∞,)(blue curves) overlaid on the deterministic dynam-
ics in the case when Σ = ΣR. (b) Most probable path overlaid on the probability density generated
by Monte-Carlo simulations of Equation (28) in a parameter regime in which (tmax,0) B+. The
solid black line corresponds to the most probable path constructed using Theorem 5.2 and the
dashed magenta curve is the mean of the probability density. All other curves in both (a) and
(b) follow the same convention as in Figure 6. The parameter values used in both (a) and (b) are
r+= 2, r= 3, T = 0, A+= 1, A= 1,and a=0.5. We set σ= 0.2 for the Monte-Carlo
simulations.
and a global minimizer of I(−∞,)intersects this region of ΣR. Each member of the uncountable
family of global minimizers can then be constructed by joining three curves as follows: the first
minimizes the functional to ΣR, the second tracks ΣRuntil reaching an arbitrary point t=sin the
intersection of ΣRwith B+, and the third curve leaves ΣRand tracks the drift; see Figure 7(a).
Key to this construction is that for Filippov systems there is a fundamental non-uniqueness in how
solution curves exit a repelling sliding region, and minimizers of I(−∞,)can track the drift at no
cost.
The following theorem summarizes when explicit minimizers of I(−∞,)can be calculated for
this case study. The key to the proof is the fact that since the system is piecewise linear, a change
of variables converts the problem of determining the most probable path of the non-autonomous
system to Σ to that of an autonomous system. Moreover, the autonomous system is a gradient
system and thus by Kramers’ law, the most probable transition through Σ occurs when the potential
difference is a minimum [22, 76]. In this case, this corresponds precisely to when the separation
between hand Σ is minimal, i.e. times when hhas a maximum.
Theorem 5.2. If (tmax ,0) B+, where tmax =T+1
2πtan12π
r, and α∈ A(−∞,)is defined
piecewise by
(α(t) = h(tmax)er(ttmax )+h(t), t tmax,
˙α(t)satisfies (29), t > tmax,
then
2h(tmax)2=I(−∞,)[α] = inf
α∈A(−∞,)I(−∞,)[α].
26
Most probable paths in piecewise-smooth SDEs
Proof. We first compute the most probable transition paths to Σ by determining the most probable
transition paths for the SDE
dx =f(x, t)dt +σdW
from h(t) to Σ. To do so, it is convenient to introduce the change of variables y(t) = x(t)h(t),
yielding
dy =rydt +σdW,
which corresponds to a gradient system with potential V(y) = ry2/2. With this change of
variables, the relevant admissible set A(t0,tm)
Σand functional I(t0,tm)
Σ:A(t0,tm)
Σ7→ Rare defined by
A(t0,tm)
Σ={γH1([t0, tm]; R) : γ(t0)=0, γ (tm) = h(tm),and γ(t)<h(t))},
I(t0,tm)
Σ[γ] = Ztm
t0|˙γ(t) + rγ(t)|2dt =Ztm
t0˙γ(t) + V0(γ(t))
2dt.
Expanding and integrating, it follows that for all γ∈ A(t0,tm)
Σ:
I(t0,tm)
Σ[γ] = Ztm
t0˙γ(t)2+ 2 ˙γ(t)V0(γ(t)) + V0(γ(t))2dt
=Ztm
t0˙γV0(γ(t))2+ 4 d
dtV(γ(t))dt
=Ztm
t0˙γV0(γ(t))2dt + 4 [V(γ(tm)) V(γ(t0))]
4V(γ(tm))
= 2h(tm)2.
This lower bound is an equality if t0=−∞ and ˙γ=V0(y), i.e. γtracks the time-reversed dynamics.
Consequently, setting tm=tmax it implies that γ(t) = h(tmax)er(ttmax )minimizes I(−∞,tm)
Σ
over all tmR.
Finally, for all α∈ A(−∞,),if we let tc= min{t:α(t)=0}denote the first time αcrosses Σ,
then by the above inequality,
I(−∞,)[α]I(−∞,tc)
Σ[αh]=2h(tm)22h(tmax )2.
Therefore, since the integrand of I(−∞,)[α] vanishes for times in which ˙αsatisfies Equation (29),
it follows that if (tmax,0) B+then αas constructed in the hypothesis of the theorem achieves
this lower bound.
5.3 Comparison of most probable paths to Monte-Carlo simulations
As in Section 4, we compare the most probable paths with Monte-Carlo simulations of tipping
events using the Euler-Maruyama scheme to numerically approximate realizations of Equation
(28). Specifically, for a realization χof Equation (28) we define the following tipping time
τ+(χ) = min{t:χ(t)> h+(t)},
and given t0< tfwe say χis a tipping event on the interval [t0, tf] if τ+(χ)< tf. The Monte-Carlo
simulations are then conducted until the distribution of tipping events on [t0, tf] converges.
27
Most probable paths in piecewise-smooth SDEs
In the case when (tmax,0) B+, Theorem 5.2 provides an explicit construction for the most
probable paths which can be directly compared with the distribution of tipping events. In Figure
7(b) we plot the most probable path overlaid on the probability density of the tipping events
on the interval [0,3] for the same parameter values used to generate Figure 7(a). In this case,
B+={x:x0}and thus (tmax,0) B+. Figure 7(b) illustrates that there is excellent agreement
between the predicted most probable path and the mean and mode of the probability density
generated by the Monte-Carlo simulations. Note, however, that the most probable path plotted
is one that crosses and does not slide as in the other potential most probable paths illustrated in
Figure 7(a). Clearly, while the minimizers of I(−∞,)are not unique, the tipping events concentrate
about the most probable path that does not slide.
When (tmax,0) /B+the most probable paths cannot be directly computed and thus we again
approximate the most probable paths by numerically computing the stationary curves of the gradi-
ent flow. In this case, to make the problem numerically tractable, we use the mollified vector field
computed using a Gaussian kernel
ζε(x) = 1
ε2πexp x2
2ε2.(35)
While ζεdoes not have compact support, ζ1(x)<1016 for |x| ≥ 9 and thus these kernels serve
as an accurate approximation of a function with compact support. With this kernel the mollified
vector field for this problem becomes
fε(x, t) = ε(rr+)