Piecewise-smooth dynamical systems: Bouncing, slipping and switching: 1. Introduction Chris Budd

Most of the present theory of dynamical systems deals with smooth systems These systems are now fairly well understood Flows Maps

Can broadly explain the dynamics in terms of the omega-limit sets Fixed points Periodic orbits and tori Homoclinic orbits Chaotic strange attractors And the bifurcations from these Fold/saddle-node Period-doubling/flip Hopf

What is a piecewise-smooth system? Map Flow Hybrid Heartbeats or Poincare maps Rocking block, friction, Chua circuit Impact or control systems

PWS Flow PWS Sliding Flow Hybrid

Key idea … The functions or one of their nth derivatives, differ when Discontinuity set Interesting Discontinuity Induced Bifurcations occur when limit sets of the flow/map intersect the discontinuity set

Why are we interested in them? Lots of important physical systems are piecewise-smooth: bouncing balls, Newtons cradle, friction, rattle, switching, control systems, DC-DC converters, gear boxes … Newtons cradle

Beam impacting with a smooth rotating cam [di Bernardo et. al.]

Piecewise-smooth systems have behaviour which is quite different from smooth systems and is induced by the discontinuity Eg. period adding Much of this behaviour can be analysed, and new forms of Discontinuity Induced Bifurcations can be studied: border collisions, grazing bifurcations, corner collisions.

This course will illustrate the behaviour of piecewise smooth systems by looking at Some physical examples (Today) Piecewise-smooth Maps (Tomorrow) Hybrid impacting systems and piecewise-smooth flows (Sunday) M di Bernardo et. al. Bifurcations in Nonsmooth Dynamical Systems SIAM Rev iew, 50, (2008), M di Bernardo et. al. Piecewise-smooth Dynamical Systems: Theory and Applications Springer Mathematical Sciences 163. (2008)

Example I: The Impact Oscillator: a canonical piecewise-smooth hybrid system obstacle

Solution in free flight (undamped) xx

Periodic dynamics Chaotic dynamics Experimental Analytic

x dx/dt Chaotic strange attractor

Complex domains of attraction of the periodic orbits dx/dt x

Regular and discontinuity induced bifurcations as parameters vary Regular and discontinuity induced bifurcations as parameters vary. Period doubling Grazing

Grazing bifurcations occur when periodic orbits intersect the obstacle tanjentially: see Sunday for a full explanation

Grazing bifurcation x Partial period-adding Robust chaos

Chaotic motion x dx/dt t

Systems of impacting oscillators can have even more exotic behaviour which arises when there are multiple collisions. This can be described by looking at the behaviour of the discontinuous maps we study on Friday

Example II: The DC-DC Converter: a canonical piecewise-smooth flow

Sliding flow Sliding flow is also a characteristic of: III Friction Oscillators Coulomb friction

CONCLUSIONS Piecewise-smooth systems have interesting dynamics Some (but not all) of this dynamics can be understood and analysed Many applications and much still to be discovered Next two lectures will describe the analysis in more detail.