Hybrid dynamical systems

Physical systems can operate in a range of dynamic behaviour. The transitions between these behaviours can be modelled by an idealised instantaneous and discrete rule.

Consider the motion of swinging bell, where the clapper velocity changes instantaneously at impact. Under what conditions does an impact occur? How are the system’s dynamics, such as ringing patterns, affected by impacts? If parameters, such as material properties or excitation amplitude, are changed how are the dynamics affected? Can conditions for the system, which avoid or favour impacts, be derived?

Hybrid dynamical systems framework

In the hybrid system framework states (e.g. position and velocity) are piecewise smooth functions and the phase space is separated into admissible (inside the bell) and inadmissible (bell wall) regions, separated by a manifold (the wall) referred to as the discontinuity boundary. The motion in free flight can be modelled by a system of ODEs and interactions with the discontinuity boundary can be described by a discrete map. Interactions with the discontinuity boundary introduce nonlinearities and give rise to intriguing and novel

  • nonsmooth flows: periodically impacting orbits, chaotic impacting attractors, and chattering (orbits that interact with the discontinuity boundary infinitely often in a finite amount of time, while their amplitude converges to zero), and
  • discontinuity induced bifurcations (DIBs): invariant sets such as equilibria or smooth periodic orbits interacting (tangentially) with the discontinuity boundary, referred to as boundary equilibrium bifurcations and grazing bifurcations, respectively.

Unlike local bifurcations in smooth systems, which can be typically reduced to one or two dimensions (centre eigenspace dimension), DIBs are plagued by the curse of dimensionality [0]. That is, as the phase space dimension increases so does the number and nature of attractors arising from a DIB.
The conditions under which the variety of attractors are born and their nature are not known. Although no general theory exists, it is essential to derive these conditions for specific cases, typically low dimensional systems, and applications to contribute to a general theory and solutions to real world problems through accurate predictions and parameter analysis.

[0] Paul Glendinning and Mike R Jeffrey, Grazing-sliding bifurcations, border collision maps and the curse of dimensionality for piecewise smooth bifurcation theory. Nonlinearity, 2014, 28(1):263

For higher dimensional systems (>3) the mechanisms underlying DIBs are not well understood and they have been the focus of my research since my PhD studies. For this purpose I have focused on a system modelling rotating machines, where the motion of a driven disk is confined by a fixed circular boundary.
One of my contributions to this field is the discovery and analytical unfolding of a novel bifurcation termed non-smooth Hopf [1,2,3]. This result, supported by numerical computations, demonstrates that at a DIB multiple smooth and impacting orbits can coexist and coalesce and adds insight into how impacting orbits arise in the first place.
Numerical simulations have revealed complicated dynamics comprised of periodic, quasi-periodic and chaotic impacting orbits as well as parameter regions revealing various operation modes. From such an analysis consequence of the dynamics within the application context, such as machine wear, can be deduced [1].

[1] K. Mora, Non-smooth Dynamical Systems and Applications, PhD Thesis, University of Bath (UK), 2014
[2] K. Mora, C.J. Budd, Non-smooth Hopf-type and grazing bifurcations arising from impact/friction contact events,
In Extended Abstracts Spring 2016. Trends in Mathematics, vol 8., pp. 129-134, Birkhäuser, Cham, 2017 Nonsmooth Dynamics Conference, 1-5 Feb. 2016, Barcelona, Spain
(Proceedings, ResearchGate)
[3] K. Mora, C.J. Budd, P.S. Keogh, P. Glendinning, Non-smooth Hopf-type bifurcations in rotating machinery with impact and friction, Proc. of the Royal Society A, 2014, vol. 470, no. 2171, p. 20140490
(Journal, ResearchGate)

Resonance plays a critical role for the existence and stability of emerging impacting periodic orbits for a non-resonant system configuration [4]. This new mechanism is revealed by unfolding a two parameter grazing bifurcation and uncovers the transition from a two parameter-family of grazing periodic orbits to a smooth saddle node bifurcation of single impact periodic orbits.

[4] K. Mora, A. R. Champneys, A. D. Shaw, M. I. Friswell
Explanation of the onset of bouncing cycles in isotropic rotor dynamics; a grazing bifurcation analysis, Proc. of the Royal Society A, 2020, vol. 476, no. 2237, p. 20190549
(Journal, ResearchGate)

The dynamics and bifurcations of a physically more realistic impact and friction model based on Stronge’s energetic collision approach were analysed by my student Steffen Ridderbush in his Bachelor thesis [5].

[5] S. Ridderbusch, Dynamics in energy consistent
impact/friction models of rotating machines, Bachelor Thesis, Paderborn University (Germany), 2016

aperiodikHybrid dynamical systems