I have recently been studying continuous dynamical systems whose phase space can be divided into a number of regions. Inside each of these the flow is smooth, but there is a discrete jump in the flow just at the boundaries. In the mathematical description, the right hand side of the differential equation is different for different regions of the phase space of the dynamical variables.

Note: I don't mean something trivial like systems which exhibit smoothness in different regions of physical space separated by boundaries, like differently heated gases in partitions, or water in contact with vapour etc. The different regions I mention are regions in the phase space of the dynamical systems. So imagine a set of continuous-time differential equations defining a flow which is segregated in its phase space into regions in which the evolution of the equations is piecewise smooth.

I also don't mean phase transition. There is no variation of order parameter or bifurcations here. The piecewise smoothness exists in the dynamical phase space for a fixed value of the system parameters.

I have been studying them in an engineering context of a mechanical device in which there is a sudden change in the velocity of a moving part when it hits something. But it struck me that such piecewise smooth systems should be found in many scenarios, from other areas of physics, maybe certain quantum phenomena, to biological systems that can be studied with the theory of dynamical systems.

Some examples of the kind of systems I am looking for are:

  • Quantum mechanics: the Muffin-Tin potential is a quantum model where the potential (the right side of the differential equation) is approximated to be piecewise defined.
  • Classical mechanics: the hard impacting oscillator (oscillator with a rigid wall at an end restricting the amplitude, like the devices I was studying).
  • Theoretical computer science: Hybrid automata and reachability problems which are further piecewise linear.

I am curious to apply my understanding of the mechanical system to such systems.

So, what are other dynamical systems in nature which exhibit piecewise smooth behaviour?

  • 2
    $\begingroup$ Overly broad and clearly big list. Vote to close. $\endgroup$
    – genneth
    May 25, 2012 at 12:53
  • $\begingroup$ You basically gave the definition of a shock wave : "Shock waves are characterized by an abrupt, nearly discontinuous change in the characteristics of the medium." (from wikipedia). When studying e.g. the air flow through a supersonic boom, you have smooth solution inside and outside of the cone, and you have to reconnect the two solution at the wave front, where some quantity are conserved and some others aren't. $\endgroup$ May 25, 2012 at 12:58
  • $\begingroup$ It also corresponds to phase transition in thermodynamics. This is a huge domain so I tend to agree with @genneth. $\endgroup$ May 25, 2012 at 13:02
  • $\begingroup$ @FrédéricGrosshans, no it doesn't correspond to phase transitions. The jump occurs when you change the order parameter of the system in that case. Here there is a jump between two regions of the phase space for the same fixed system parameter values. $\endgroup$ May 25, 2012 at 13:23
  • 1
    $\begingroup$ @FrédéricGrosshans yes, but neither of shock waves or water in contact with vapour is a dynamical system in which there are changing macroscopic variables. I'll edit my question to elaborate. $\endgroup$ May 25, 2012 at 14:35

1 Answer 1


You might be interested in in Pulse-coupled oscillators. See for example Mirrollo&Strogatz,1990:

They investigate a set of identical oscillators each described by a single phase variable $\phi_i \in [0,1]$ with $\dot{\phi_i}=1$. When $\phi_i = 1$ the oscillator resets to zero and sends out a spike which causes an instantaneous phase jump in all other oscillators which is given by a transfer function $h(\phi)$. In that paper they show that a special class of transfer functions will cause the oscillators to synchronize for all initial conditions.

This model with finite delays between sending of spikes and reception is used to model neural networks see for example Jahnke et al, 2008 and the references therein.


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