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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 1tanhz+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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...Beyond the bristol book advances and perspectives in non smooth dynamics applications a b c d igor belykh rachel kuske maurizio porri david j w simpson department of mathematics statistics neuroscience institute georgia state university p o box atlanta usa school technology ga center for urban science progress mechanical aerospace engineering biomedical new york tandon brooklyn mathematical computational sciences massey palmerston north zealand dated december induced by switches impacts sliding other abrupt changes are pervasive physics biology yet systems with have historically received far less attention compared to their counterparts classic contains art review major results challenges study dynamical this paper we provide detailed made since cover hidden generalizations motion eects noise randomness multi scale approaches time dependent switching variety local global bifurcations also survey areas application including ecology climate which theory has been applied opens focus issue...

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