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Beyond the Bristol Book: Advances and Perspectives in Non-Smooth Dynamics and Applications
Beyond the Bristol Book: Advances and Perspectives in Non-Smooth
Dynamics and Applications
1, a) 2, b) 3, c) 4, d)
Igor Belykh, Rachel Kuske, Maurizio Porfiri, and David J.W. Simpson
1)Department of Mathematics and Statistics & Neuroscience Institute, Georgia State University, P.O. Box 4110,
Atlanta, Georgia, 30302-410, USA
2)School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30313,
USA
3)Center for Urban Science and Progress, Department of Mechanical and Aerospace Engineering,
and Department of Biomedical Engineering New York University, Tandon School of Engineering, Brooklyn,
New York 11201, USA
4)School of Mathematical and Computational Sciences, Massey University, Palmerston North, 4410,
New Zealand
(Dated: 8 December 2022)
Non-smooth dynamics induced by switches, impacts, sliding, and other abrupt changes are pervasive in
physics, biology, and engineering. Yet, systems with non-smooth dynamics have historically received far less
attention compared to their smooth counterparts. The classic ‘Bristol book’ [M. di Bernardo, C.J. Budd,
A.R. Champneys, P. Kowalczyk. Piecewise-smooth Dynamical Systems. Theory and Applications. Springer-
Verlag, 2008] contains a 2008 state-of-art review of major results and challenges in the study of non-smooth
dynamical systems. In this paper, we provide a detailed review of progress made since 2008. We cover hidden
dynamics, generalizations of sliding motion, the effects of noise and randomness, multi-scale approaches,
systems with time-dependent switching, and a variety of local and global bifurcations. Also, we survey new
areas of application, including neuroscience, biology, ecology, climate sciences, and engineering to which the
theory has been applied.
This paper opens the 2022 Focus Issue on Non- generate nearly any type of behavior, via a huge range
Smooth Dynamics. We review advances in the of discontinuity-induced bifurcations, some with smooth
theory of piecewise-smooth and non-smooth dy- counterparts, like fold-type or Hopf-type bifurcations,
namical systems beyond the extensive coverage of but others specific to non-smooth phenomena, e.g., graz-
the high-impact ‘Bristol book’ that was published ing and sliding. While the theory of smooth dynami-
in 2008. We also highlight the contributions to cal systems dates back to Poincar´e’s time, systematic
this Focus Issue that articulate the role of non- efforts to understand non-smooth dynamics and bifur-
smoothdynamicsandcoverawiderangeoftopics cations have only been performed more recently.
including Filippov systems, discontinuity-induced Notwithstanding valuable early contributions by An-
bifurcations, vibro-impact systems, pulse-coupled 1 2 3,4 5,6
systems, switching networks, and applications in dronov et al. , Neimark , Filippov , Feigin , and oth-
ers (see §1.7 of Jeffrey7), progress on non-smooth dy-
mechanics and biomechanics. namicsunderwentrapidaccelerationinthe1990s, ignited
by the fundamental work of Nordmark and collaborators
8–10
on impact oscillators and discontinuity maps . Re-
I. INTRODUCTION searchers at Bristol, UK, and nearby Bath were central
to many of these developments, and took the extra step
Non-smooth dynamics, appearing as switches, im- of collating the state-of-the-art theory at the time into a
11
pacts, sticking, sliding, and chatter require careful for- graduate-level textbook published in 2008. The book
mulation and treatment due to the essential piecewise was completely novel, it standardized terminology, and it
or discontinuous features. Piecewise-smooth and non- madenon-smoothdynamicsmainstreambyshowinghow
smooth dynamical systems represent a vast research area standard dynamical systems ideas, when appropriately
in nonlinear science, related to systematic mathematical generalized, provide the key to understanding physical
analysis and modeling of non-smooth dynamics and bi- problems in diverse disciplines. As of this writing, the
furcations, possibly in the presence of uncertainty and book has over 2,000 citations in Google Scholar. In view
stochasticity. The introduction of non-smoothness can of its lofty place in non-smooth dynamics literature, we,
and many others, refer to it simply as the ‘Bristol book’.
However, the theory of non-smooth dynamics has de-
veloped further since 2008. The main purpose of this pa-
a)Electronic mail: ibelykh@gsu.edu peristoreviewtheseadvances. Theadvancesarediverse,
b)Electronic mail: rachel@math.gatech.edu some theoretical, others breaking into new areas of ap-
c)Electronic mail: mporfiri@nyu.edu plication. Some reviews and additional books have been
d)Electronic mail: d.j.w.simpson@massey.ac.nz compiled. Of particular note is the work of Jeffrey7 —
Beyond the Bristol Book: Advances and Perspectives in Non-Smooth Dynamics and Applications 2
another Bristol book that extends Filippov’s framework
to systems with multiple switches and explains the oc-
currence of novel dynamics in physically-motivated reg-
ularizations of nonsmooth models.
Wealsobrieflysurveyarticlesinthepresentfocusissue
that brings together applied mathematicians, physicists,
and engineers to display recent advances in the theory
andapplications of non-smooth dynamical systems. Top-
ics covered range from the dynamics and bifurcations of
piecewise-smooth and impacting systems, including non-
classical sliding homoclinic and grazing bifurcations, to FIG. 1. A phase portrait of a two-dimensional non-smooth
the constructive role of non-smoothness in the stability system of the form (1). Evolution on the switching manifold
and control of switched networks with an eye towards h(x) = 0 is termed sliding motion. Sliding motion usually
applications in biology and engineering. ends when the system state reaches a point of tangency (vis-
ible fold).
The idea of organizing this focus issue was inspired
by a non-smooth dynamics minisymposium held at the
virtual 2021 SIAM Conference on Applications of Dy- 7
Jeffrey . Here, things are clearer with (1) rewritten as
namical Systems. This focus contains a collection of re-
search papers from a broad spectrum of topics related to x˙ = [1 − H(h(x))]fL(x)+H(h(x))fR(x), (2)
modeling, analysis, and control of non-smooth dynami-
cal networks. We hope that this collection will generate where H is the Heaviside function. Hidden dynamics can
significant interest among the mathematics, physics, and appear when H is replaced with a smooth approximation
12–14
engineering audiences of the journal. Junior researchers that is non-monotone . This occurs, for example,
might also find this collection useful as an inspiration to in friction models to capture the shape of the Stribeck
start graduate research in this exciting field of research. curve15,16 that accounts for the extra break-away force
that in-contact objects require to begin slipping.
The resulting non-monotone model typically involves
dynamics that are qualitatively different to those of (2).
II. HIDDEN DYNAMICS The lack of monotonicity can cause a shift in bifurcation
17,18
values, or introduce new bifurcations . It can intro-
Muchof the Bristol book is dedicated to the dynamics duce oscillations in the boundary layer in scenarios where
and bifurcations of ordinary differential equation (ODE) (2) has roughly uni-directional sliding motion. Taken to
systems of the form anextreme, non-monotonesmoothingcanconvertsliding
19
( motion into chaos .
fL(x), h(x) < 0, When multiple switching conditions are involved,
x˙ = f (x), h(x) > 0. (1) monotone smoothing is sufficient to generate new
R 19,20 21
dynamics . As shown by Harris and Ermentrout
Here, the system state x(t) ∈ Rn evolves according to this occurs for the Wilson-Cowan neuron model with dis-
one of two vector fields, f , and f , as governed by the continuous firing rate functions. The model is
L R
sign of a smooth function h : Rn → R. This represents u˙ = −u+H(u−av−b),
thesimplestformulationofstate-dependentswitchingbe- τv˙ = −v +H(u−cv−d), (3)
tweentwomodesofevolution. Solutionscanbecomecon-
strained to the switching manifold h(x) = 0, Fig. 1. This where u(t) represents the average activity of a large neu-
is sliding motion, usually formulated as the solution to ral network, v(t) is a recovery variable, and H is again the
a convex combination of f and f in accordance with
L R Heaviside function. The intersection of the two switching
3,4
Filippov . manifolds, u = av + b and u = cv + d, is a steady-state
A more realistic model might incorporate hysteresis solution that loses stability and emits a stable limit cycle
or time-delay in the function h, or smooth the vector as the parameter τ is increased, Fig. 2-a. By replac-
field over a narrow region (boundary layer) containing ing each H(z) with 1 tanh z+1 (here z is a dummy
the switching manifold. If the addition of such complexi- 2 ε
variable), the system is now smooth but the analogous
ties has little or no bearing on the qualitative features of transition occurs much earlier in a (classical) Hopf bifur-
the dynamics, it is probably better to work with the sim- cation, Fig. 2-b. By taking ε → 0 we recover (3), yet the
pler model (1). This is often indeed the case and serves Hopf bifurcation value converges to τ ≈ 0.1373, which is
to illustrate the importance of understanding the dynam- substantially earlier than the bifurcation value for (3) of
ics and bifurcations of such systems. However, in many τ ≈ 0.5240. We conclude that, for intermediate values
situations, new dynamics arise. of τ, arbitrarily steep monotone smoothing causes the
Understanding the causes and consequences of such steady-state solution to lose stability and stable, small-
hidden dynamics has recently been championed by amplitude, oscillations to be created.
Beyond the Bristol Book: Advances and Perspectives in Non-Smooth Dynamics and Applications 3
FIG. 2. The upper plots show a bifurcation diagram and representative phase portraits of the non-smooth system (3) with
a = 2, b = 0.05, c = 0.25, and d = 0.3, as given by Harris and Ermentrout21. The lower plots are for the smooth system
obtained by replacing the Heaviside functions with hyperbolic tangent functions as explained in the text (using ε = 0.005).
Stable solutions are colored blue; unstable solutions are colored red. The green curves are switching manifolds in the upper
plots and nullclines in the lower plots. The bifurcation diagrams show the v-value of the steady-state solution and minimum
and maximum v-values of the limit cycle.
36
Adeeper understanding of the dynamics and bifurca- cently been proposed. Jeffrey identifies a ‘canopy’ con-
tions of (1) can be gained by smoothing with monotone vex combination that is, in a sense, the simplest. Dieci
22,23 37
functions . The smoothed model is inherently slow- andDifonzo instead take the barycentric mean. Kakla-
fast and, in this way, slow-fast systems and piecewise- manos and Kristiansen38 smooth the system, then define
smooth systems are closely related (see also Section IX). sliding motion by taking the nonsmooth limit. Jeffrey
24 39
For example, folded nodes of slow-fast systems can, et al. applyperturbations(hysteresis, time-delay, noise,
when the limit to the Heaviside function is taken, be- andnumericaldiscretization) and take the zero perturba-
25
cometwo-folds of piecewise-smooth systems (a two-fold tion limit. Such a procedure gives different results for the
of (1) is a point on h(x) = 0 at which both fL(x) and different types of perturbations renewing the remarks of
40
fR(x) have a tangency to h(x) = 0 and certain generic- Utkin , in the context of relay control, that the most ap-
ity conditions are satisfied7). Two-folds were considered propriate definition for sliding motion depends critically
4 26
by Filippov and Teixeira , but only recently analysed on the physical properties of the system under consider-
in more detail27–31. Single folds have been investigated ation.
32 33
by smoothing , as have planar two-folds , including Related to this problem, friction models with suffi-
the non-uniqueness of trajectories that enter two-folds34. ciently many degrees of freedom (DoF) naturally involve
A contraction analysis based on regularization was also switching manifolds that are codimension-two (instead of
used to study the stability of different classes of switched codimension-one). Some theory for the dynamics and bi-
Filippov systems35. furcations of such systems has recently been developed
by Antali and St´ep´an41,42.
III. GENERALISATIONS AND EXTENSIONS OF
SLIDING MOTION IV. LOCAL BIFURCATIONS OF NON-SMOOTH ODES
The discontinuous neuron model (3) is one of many As parameters are varied, interactions between invari-
non-smooth models that involve multiple switching man- ant sets and switching manifolds produce a wide variety
ifolds. To specify sliding motion along the intersection of novel bifurcations collectively known as discontinuity-
of two switching manifolds, Filippov’s approach to con- induced bifurcations. The simplest type of discontinuity-
structing a tangent convex combination can fail to define induced bifurcation is arguably a boundary equilibrium
a unique solution. Several ways to remedy this have re- bifurcation that occurs when an equilibrium of a smooth
Beyond the Bristol Book: Advances and Perspectives in Non-Smooth Dynamics and Applications 4
FIG.3. Twoboundary-equilibriumbifurcations not described FIG. 4. Phase portraits and a bifurcation diagram of (4).
in the Bristol book. In (a), an unstable node transitions to a Thisisaminimalmodelofanautomaticpilotwherethevessel
stable pseudo-equilibrium. In (b), a stable node and a saddle heading φ(t) is controlled through a parameter β that governs
pseudo-equilibrium collide and annihilate. how the rudder switches between two allowed positions. The
switching manifold is colored green for crossing regions, blue
for the attracting sliding region, and red for the repelling
component of the system collides with a switching man- sliding region. These regions are bounded by folds shown
ifold. Discontinuity-induced bifurcations are described as black triangles.
in Chapter 5 of the Bristol book, which, following
43
Kuznetsov et al. , chronicles ten topologically distinct,
54
generic, boundary equilibrium bifurcations in the two- cycle as a function of parameters . The amplitude grows
dimensional setting. Unfortunately, two cases were over- asymptotically linearly when the dynamics is piecewise-
looked, shown in Fig. 3. These cases were only described linear to leading order, while if two folds are involved
later44,45, and they serve to illustrate the difficulty in at- the amplitude is usually asymptotically proportional to
tempting a comprehensive classification of bifurcations of the square-root of the parameter change, as in Fig. 4. An
non-smooth systems46. interesting exception is a two-fold perturbed by hysteresis
55
Indeed, for systems with more than two dimensions, which gives a cube-root scaling law . For non-smooth
boundary equilibrium bifurcations can create chaotic systems that are C1 but not C2, a modification to the
attractors47,48, and even multiple attractors49. This sug- standard Hopf bifurcation non-degeneracy coefficient is
56,57
gests that future developments in the bifurcation the- required .
ory of high-dimensional non-smooth systems may bene- Also in recent years, there have been many studies
fit from focusing on weaker results that apply generally that aim to count or bound that number of limit cycles
rather than a large number of strong results for particular possible in various classes of non-smooth systems; see
50 58
situations . Llibre and Zhang and references within. The unfold-
Boundary equilibrium bifurcations can mimic Hopf bi- ings of several codimension-two bifurcations have been
59–61
furcations by converting a stable equilibrium into a sta- derived , as has the three-dimensional unfolding of
ble limit cycle43,51, but there are many other mecha- the simultaneous occurrence of Hopf, saddle-node, and
62
nisms, unique to non-smooth systems, that can achieve boundary equilibrium bifurcations .
52
this transition . Two folds, each shifting along a switch-
ing manifold as parameters are varied, can collide, inter-
change positions, and generate a limit cycle. As shown V. GLOBAL BIFURCATIONS
in Fig. 4, such a phenomenon occurs for the automatic
pilot model Global bifurcation theory for systems with disconti-
¨ ˙ ˙ nuities remains quite undeveloped. Di Bernardo and
φ+φ=−H(φ+βφ), (4) 63
Hogan provided an extensive review in 2010. Per-
1
given in the classic book of Andronov et al. . The desired haps, the first major focus of existing studies is on de-
heading of φ = 0 for the ship or vessel is achieved when riving conditions under which global bifurcations in non-
the control parameter β is positive. If the value of β smooth ODEs are qualitatively similar to their classi-
64 48
is decreased through zero, two folds collide and a stable cal (smooth) counterparts . Novaes and Teixeira de-
limit cycle is created. Only recently has this type of rived a version of Shilnikov saddle-focus theorem whereby
53
bifurcation been analyzed in a general setting . a sliding saddle-focus homoclinic loop yields a count-
Every type of Hopf-like bifurcation involves a scaling able infinity of sliding saddle periodic orbits. Belykh
65
law for the amplitude and period of the bifurcating limit et al. constructed an analytically tractable non-smooth
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