The ghosts of departed quantities in switches and transitions

Transitions between steady dynamical regimes in diverse applications are often modelled using discontinuities, but doing so introduces problems of uniqueness. No matter how quickly a transition occurs, its inner workings can affect the dynamics of the system significantly. Here we discuss the way transitions can be reduced to discontinuities without trivializing them, by preserving so-called hidden terms. We review the fundamental methodology, its motivations, and where their study seems to be heading. We derive a prototype for piecewise smooth models from the asymptotics of systems with rapid transitions, sharpening Filippov's convex combinations by encoding the tails of asymptotic series into nonlinear dependence on a switching parameter. We present a few examples that illustrate the impact of these on our standard picture of smooth or only piecewise smooth dynamics.

the ambiguity that accompanies a discontinuity. 26 Many dynamic systems involve intervals of smooth steady change punctuated by 27 sharp transitions. Some we take for granted, such as light refraction or reflection, elec-28 tronic switches, and physical collisions. In elementary mechanics, for example, when 29 two objects collide, a switch is made between 'before' and 'after' collision regimes, 30 which are each themselves well understood. The patching of the two regimes leaves 31 certain artefacts, such as the choice between a physical rebound solution, and an un-32 physical (or virtual) solution in which the objects pass through each other without 33 deviating. More exotic switches arise in climate models, for instance as a jump in the 34 Earth's surface albedo at the edge of an ice shelf [14,22], in superconductivity as a 35 jump in conductivity at the critical temperature [3], in models of cellular mitosis [10], 36 in dynamics of socio-economic and ecological decision implementation [27, 6, 28], and 37 so on. 38 In the case of the collision model, we do not find the discontinuity or virtual 39 solutions too disturbing when first encountered, and move on to apply such insights 40 to the dynamics of nonlinear mechanical systems, and thereafter to electronics, the 41 climate, living processes, etc., perhaps becoming too comfortable with patching over so it seems futile to look deeper. Fortunately, the mathematics of matching such 44 'piecewise-defined' systems turn out to be richer than might be expected. 45 Consider a system whose behaviour is modelled by a system of ordinary differential 46 equationsẋ = f (x; y), where y ≈ sign(h(x)) for some smooth function h. Our first 47 aim here is to show that, for many classes of behaviour, such approximations take the where y(h) = sign(h) + O (ε/h) , 50 for arbitrarily small ε. Our second aim is to show why the tails of these expansions 51 matter, and how they can be retained in a piecewise-smooth model as ε → 0. This 52 information seems not to be part of established piecewise dynamical systems theory, 53 but their omission is easily remedied.

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The modern era of piecewise-smooth systems begins with Filippov and contem-55 poraries, who showed that differential equations with "discontinuous righthand sides" 56 can at least be solved (e.g. in [2,11,12]). What those solutions look like remains an 57 active and flourishing field of enquiry.

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As an example, take an oscillator given byẋ 2 = x 1 andẋ 1 = −0.01x 1 − x 2 − 59 sin(ωt), where the forcing sin(ωt) overcomes the damping −0.01x 1 to produce sus-60 tained oscillations. Say the frequency ω switches between two values, ω = π/2 for 61 x 1 < 0 and ω = 3π/2 for x 1 > 0. The method usually used to study such switching 62 is due to Filippov [12,33,24,7], and handles the discontinuity at   We may then ask whether the behaviour in figure 1(b.i) is an aberration of the 90 simulation method, or the true behaviour of (1b) as a discontinuous system. We may . What we will show is that nonlinear dependence on λ introduces 94 fine structure to the switching process, which is captured in (b.i), but can be missed 95 in a less precise simulation as in (b.ii) or by neglecting nonlinearity outright as in (a).

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The consequences of overlooking such nonlinearities of switching can be far more  The starting point to a more general approach to piecewise-smooth systems is 101 quite simple. If a quantity f switches between states f + and f − as a threshold Σ is 102 crossed, f can be expressed as where a step function λ switches between ±1 across Σ and in [−1, +1] on Σ. The first 105 two terms have an obvious interpretation, namely the linear interpolation across the 106 jump in f . The last term is less obvious, but the role and origins of each term in (2) 107 are what we seek to understand here.

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Strictly speaking (2) may be treated as a differential inclusion. When x lies on heuristic [17] and some rigorous [26] justification. Here we provide a more substantive 121 derivation based on asymptotic transition models. 122 We begin in section 2 by deriving (2) as a uniform model of switching. The 123 argument begins with a general asymptotic expression of a switch, representative 124 of various differential, integral, or sigmoid models that exhibit abrupt transitions. 125 We delay exploring the motivations for this model to section 5, as it is somewhat 126 discursive, since discontinuities arise in so many contexts and yet in similar form.

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The mathematics itself in these sections is quite standard, but the universal oc- where f + and f − are independent vector fields (but each is itself smooth in x). Some 146 kind of abrupt switch occurs across |h(x)| < ε for small ε. The behaviour inside 147 |h(x)| < ε may be of unknown nature, or of such complexity that our state of knowl-148 edge is well represented by the approximation The question in either (3) or (4) is how to model the system at and around h = 0.

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For motivation we may consider systems whose full definition we do know, and in terms of smooth functions p 0 (x), p 1 (x), q(h/ε), r n (x), and a sigmoid function which tends to a discontinuous function, Λ(h/ε) → sign(h), as ε → 0. The term 160 p 1 Λ in (5) encapsulates the switching in the system, the summation term contains 161 behaviour that is asymptotically vanishing away from the switch, and the term p 0 is 162 switch independent.

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The expression (5) is the starting point for the analysis which follows, hereon 164 until section 4. 165 We begin by re-writing (5) in a form that behaves uniformly as ε → 0. Since Since this is now a function of x and Λ, assume that the righthand side of (7) can be 169 expressed as a formal series in Λ, We can relate the c n 's to the r n 's, but more useful is to relate them directly to the 172 large h/ε behaviour ofẋ in (3)-(4), giving f (x, ±1) ≡ f ± (x). Taking the sum and 173 difference of these gives , 176 177 so we can eliminate the first two coefficients c 0 (x) and c 1 (x) in (8) to give dynamical models for smooth systems that may be nonlinear in a state x).

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The result is that, given an asymptotic expression (5) for a switch across an 194 ε-width boundary in a dynamical system, we obtain an ε-independent form 196 as promised in (2). This remains valid as ε → 0, and the switch, whether smooth or 197 discontinuous, is hidden implicitly inside λ. If we let ε → 0, then by (6) the switching 198 multiplier λ obeys The essential point is that we are left with the ghosts, in the term λ 2 − 1 g(x which 201 vanishes away from the switch where λ = ±1, but does not vanish on h = 0. The 202 next two sections show how to handle them, and why their existence is non-trivial.

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In section 5 we return to how and when such switches arise in various contexts,

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including sigmoid-like transitions, higher dimensional ordinary or partial differential 205 equations, and oscillatory integrals. We now turn to the methods used to solve the 206 piecewise-smooth system (12)-(13).

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3. The switching layer. We have derived in (12) an expression for the vector 208 field in our systemẋ = f (x; λ) which, with (13), remains valid in the discontinuous 209 limit ε → 0. One last thing is needed to complete the description of the piecewise-

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This manuscript is for review purposes only.
Rescaling time in (15) to τ = t/ε, then settingε = 0, gives the fast critical 231 subsystem (denoting the derivative with respect to τ by a prime) which gives the fast dynamics of transition through the switching layer. The equilibria 234 of this one-dimensional system form the so-called sliding manifold When M exists, it forms an invariant manifold of (15) in theε = 0 limit, at least 237 everywhere that M is normally hyperbolic, which excludes the set where ∂f1 ∂λ = 0, 240 Isolating the slow system in (15), and settingε = 0 in (15), gives the slow critical  In the context of piecewise-smooth systems, we are concerned with these results 249 only in the limitε → 0, not the perturbation toε > 0 that is typically of interest in 250 singular perturbation studies. However, for more general interest it is worth relating 251 these to singular perturbation theory. The system (15) is the singular limit of which is a more commonly seen expression in recent singular perturbation studies of 256 piecewise-smooth systems (see e.g. [32]). In [20] it is shown than (15) has equivalent 257 slow-fast dynamics to (20) on the discontinuity set x 1 = 0 in the critical limit ε = 0.

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With this we depart the smooth world. In section 2 we showed how a prototype 259 asymptotic expansion (5) could be represented as a discontinuous system in the small   and possibly also by g.  If (12) depends linearly on λ, i.e. if g ≡ 0, then the sliding mode given by (19) 285 is unique (and is exactly that described by Filippov [12]). If g is nonzero then there 286 may be multiple sliding modes, and the precise dynamics must be found using (15).

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Example 1. A simple example of hidden dynamics is given by comparing the two systems with λ = sign(x 1 ), shown in figure 2. These appear to be identical for x 1 = 0, where (ẋ 1 ,ẋ 2 ) = (− sign(x 1 ), 1). It is only on x 1 = 0 that their behaviour may differ. To find this we blow up x 1 = 0 into the switching layer λ ∈ [−1, +1], given by applying respectively, for ε → 0. We seek sliding modes by solvingλ = 0. Both have sliding manifolds M at λ = 0, and therefore sliding modes with, however, contradictory vector fields (a) (ελ,ẋ 2 ) = (0, −1) , Hence systems that appear the same outside the switching surface can have dis-288 tinct, and even directly opposing, sliding dynamics on the surface, due to nonlinear 289 dependence on λ.

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This manuscript is for review purposes only. to cross it. If g ≡ 0, in fact, the flow will cross the surface, because f ± (x) · ∇h have 293 the same sign as each other, so the linear interpolation (12) (as λ varies between ±1) 294 cannot pass through zero and there can be no sliding modes (no solutions of (19)).

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If g is nonzero then the flow may stick to the surface, and solutions sliding along the 296 surface are found using (19).

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A more coarse simulation may miss the hidden oscillation, for example with a fixed 335 discretization time step s ≥ 4ε the state x 2 seems to instead reach the equilibrium 336 x 1 = x 2 = 0 of the linear theory (simulations not shown). We will comment more on 337 the general principles behind such sensitivity at the end of section 4.3. where some lengthy algebra yields   where a heuristic case was made that 'unmodelled errors' could kill off hidden dy-366 namics, i.e. mask (or essentially eliminate) the nonlinear dependence on λ in (12).

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Unmodelled errors might include the discretization step of a simulation, time delay or 368 hysteresis of a switching process, or external noise. Essentially, large perturbations by 369 unmodelled errors can kick a system far enough that nonlinear features are missed. 370 We saw in figure 1 for a function r(ε) whose form depends on g, e.g. r(ε) = −ε/ log ε for a friction 399 example in [21].

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The counterintuitive outcome is that errors like noise can cause a system to be-   One particular sigmoid function introduced by Hill [16] has become prevalent in biological models, and that is Z(z) = z r z r +θ r for z, θ > 0, r ∈ N. The function Z(z) often represents the switching on/off of ligand binding or gene production in a larger modelẋ = f (x; y) of biological regulation. If we let z = θe h and r = 1/ε, for large argument the Hill function has an expansion In computation, commonly used sigmoids are the inverse or hyperbolic tangents, with expansions  Differentiable but non-analytic sigmoid functions are often used in theoretical approaches to smoothing discontinuities. An example is where r(h) = e 2ε/(h−ε) . Its asymptotic form is rather messier than the examples above, but it is better behaved since its convergence to sign(h) is even faster, being given for |h| < ε by y(h) = sign(h) 1 − 2e 2/(|h/ε|−1) |h/ε| + 1 + O e 4/(|h/ε|−1) − e 2e −1 {1+O(|h/ε|−1)}/(|h/ε|−1) .
In all cases the leading order term is made discontinuous by the presence of a Transitioning between steady states y ∼ ±1 for |h| ≫ ε and relaxing asẏ ∼ −y 448 for |h| ≪ ε, for small ε, is consistent with (1 − y 2 )h = ε(y +ẏ), and results in the The quantity ε is small (the two ε's that appear here need not be the same, but for 452 simplicity let us assume they are). Treating the y system in (31) as infinitely fast (for 453 ε → 0 so x is pseudo-static), its solution is easily found to be 454 (32) y(t, h) = −(ε/2h) + α tanh (αth/ε + k 0 ) ,

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This manuscript is for review purposes only.

A PDE:
Large-scale bistability, small scale dissipation. If instead 463 y represents a physical property like temperature, it might have both spatial and 464 temporal variation that become significant during transition.

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For |h| ≪ ε assume that y satisfies the heat equation y hh ∼ εẏ for some small 466 positive ε, where y h denotes ∂y/∂h. For |h| ≫ ε assume asymptotes y ∼ ±1, implying 467 y h ∼ 0. This character is satisfied for example by the system h ε y h + y hh − εẏ = 0, 468 giving overall The y system evolves on a fast timescale t/ε 2 to the slow subsystem h(x)y h +εy hh = 0, 471 which has solutions y = y * (h) given by where Erf denotes the standard error function [1]. The asymptotes y → ±1 for large 474 h imply y * (0) = 0 and y h * (0) = 2/πε. Solutions of the full system can be found 475 in the form y(t, h) = y * (h) + e −t/ε Y (h). Substituting this into the partial differential 476 equation for y in (34) yields

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(again treating x as pseudo-static for small ε). The first bracket vanishes by the definition of y * , the second gives an ordinary differential equation for Y with solution where 1 F 1 is the Kummer confluent hypergeometric function [1]. The exact functions 479 are less interesting to us than their large variable asymptotics, given by

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(and for completeness, The function Y (h) deviates from the sigmoid of y * (h) by an amount greatest near  495 We can obviously now redefineȳ = e 1 2 ρ 2 y − 1 so thatȳ = sign h + ... as in previous 496 sections. Here y is a simple sigmoid for ρ = 0, but otherwise has peaks of height 497 |ȳ| ≈ 1 + 2 π 4ρ 3 π 2 e −π 2 /8ρ 2 at h ≈ ±επ/2ρ, illustrated in figure 7. As we take the limit 498 ε → 0 for h = 0, however, all graphs limit toȳ(h) = sign(h) regardless of ρ, any peaks The graphs ofȳ(h) for different values of ρ, which all limit to a sign function as ε → 0. For ρ > 0 the graph has peaks (multiple peaks for larger ρ), whose height is εindependent and therefore do not disappear as we shrink ε, but merely get squashed into the region |h| = O (ε).

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The term a(k) is taken to be slow (polynomially) varying, while the term e ψ(k) is 508 fast (exponentially) varying. This is typical when solving differential equations using

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≈ a(k s )e ψ(ks) to leading order. This is an incredibly simple, but also accurate, result, when properly 530 used. In line 2 we just assume such an expansion is valid along a path P (we will 531 come back to this), and line 3 is just the leading order term. The clever bit is the 532 simple substitution u = (k − k s ) −ψ ′′ (k s ) to obtain line 4, and this actually defines 533 P by demanding that P transforms back to the real line.

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Some basic complex geometry makes all this work. Complex function theory tells 535 us that the path P we seek can indeed be found. By virtue of the Cauchy-Riemann 536 equations, a path along which Im ψ = constant is also a steepest descent path of 537 Re ψ, so along such a path the function is non-oscillating (because the phase Im ψ 538 is constant), and its magnitude |e ψ | = e Re ψ is exponentially fast varying (where 539 |e ψ | is therefore exponentially). This only breaks down if the path encounters a 540 maximum or minimum k s , where ψ ′ (k s ) = 0. That is exactly the point k s which 541 (40) approximates about, integrating along the steepest descent path P, and the 542 approximation is 'exponentially good' because the integrand decays exponentially 543 away from k s . 544 We have neglected the endpoint α. Because the integral is exponentially fast 545 varying, the cutoff at the endpoint creates another exponentially strong maximum 546 (or minimum, in which case we discard it), where typically ψ ′ (α) is non-vanishing.

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Approximating to leading order about k = α as above gives So the endpoint, k = α, contributes to the integral if (41) converges. The contri-550 bution of stationary point k s is conditional, since it may or may not lie on the contour 551 P, so we have The factor (1 + sign h) /2 is a switch that turns on the stationary point contribution 554 for h > 0 if k s ∈ P, and turns it off for h < 0 if k s / ∈ P. The transition between 555 cases is a bifurcation in P when the path connects k = α directly to k = k s , i.e. when 556 Im ψ(0) = Im ψ(k s ) (since the path is a stationary phase contour). Typically we find 557 up to a sign that

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In general there may be many stationary phase points k s1 , k s2 , ..., turned on and off at switching surfaces (Stokes lines) of the form Finding the correct expansion (42) requires inspection of the phase contours in the 560 complex k plane, to find a path through the stationary points k si and the endpoints 561 of k ∈ (−∞, α], with P permitted to pass through the 'point at infinity' |k| → ∞, 562 such that the integral converges. One may also calculate the higher order corrections, 563 and a wealth of theory exists to assist, a good starting point is [8].

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The point of all this is simply that, just as with our previous examples in this 568 section, the discontinuity (the sign term) has again appeared as an inescapable part 569 of the leading order behaviour (42), and remains there as we add higher orders in the 570 tail of the series. The reader must pick apart the details to gain a fuller picture, but 571 we have laid out the basics to illustrate how the sign function arises.

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They describe a switch in some unknown variable y, whose effect we then seek to 588 understand on the bulk systemẋ = f (x; y), using the methods of sections 2-4. In 589 practice, the origins of discontinuity explored in section 5 are often unknown, but we 590 found them all to take a universal form, and we have shown how to express it in a 591 manner that retains the asymptotic tails -the ghosts of switching -in the limit of 592 a piecewise-smooth model. 593 We have barely begun discovering the consequences of nonlinear switching for 594 piecewise-smooth systems. The interaction of multiple switches, for example, opens 595 up a vast world of attractors and bifurcations to be discovered. We have tried only 596 to revisit the foundations of piecewise-smooth dynamics in a way that enables future 597 study to embrace the ambiguity of the discontinuity, not to present a theory ready 598 accomplished, and so many avenues are left to be explored in more rigour.

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Discontinuities seem to be a symptom of interaction between incongruent objects 600 or media, and the nature of such interactions is often difficult to model precisely.

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Whereas in some areas of physics we have a governing law, a wave or heat equa-602 tion perhaps, to guide the transition or permit asymptotic matching, in many of the 603 engineering and life science where discontinuous models are becoming increasingly 604 prominent, we rely on much less perfect information.

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Piecewise-smooth dynamical theory attempts to address this, but we have seen or in friction such efforts are in progress.

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In addition we understand something of how sensitive a piecewise-smooth system 614 is to its idealization of the switching as a discontinuity at a simple threshold. We and in simulations the discretization step provides another κ (as in figure 1(b.ii)).

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The stiffness ε and unmodelled errors κ compete, and in a κ dominated system the 620 nonlinear phenomena of hidden dynamics may be washed out, while they may flourish 621 in a better behaved or better modelled (i.e. small κ) system.

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That discontinuities yield strange dynamics is unsurprising, and the idea of ' (and hence to many works deriving from it) apply solely to the linear (or convex) 630 combination found by assuming g ≡ 0. The nonlinear approach with g = 0 permits 631 us to explore the different dynamics possible at the discontinuity, and thus to explore 632 the many other systems that make up Filippov's full theory of differential inclusions.

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When nonsmooth systems do surprising things, we usually find we can make sense of 634 them by extending our intuition for smooth systems to the switching layer, where, as 635 in smooth systems, nonlinearity cannot be ignored.

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Finally, there are currently no standard numerical simulation codes that can han-637 dle discontinuous systems with complete reliability, event detection being insufficient 638 to take full account of all their singularities and issues of non-uniqueness (see e.g.

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[19, 18]). It is hoped that by capturing the ghosts of switching -in the form of 640 nonlinear discontinuity -such codes may soon be developed.