©2019 by Mark Levi. Proudly created with


Here is a sample of my papers dealing with different problems from the same general area.

My main result  of recent few years on this topic is based on the observation that  a class of  van der Pol--type relaxation oscillators resembling an original forced van der Pol equation exhibit chaotic behavior with zero probability: Lebesque almost all initial conditions give riseto simple periodic behavior, while only exceptional (physically unobservable) solutions are ``chaotic".

The note [39] describes a previously unnoticed  basic stretching mechanism in the phase space of forced relaxation oscillators. This stretching takes place for most initial conditions (in the sense of Lebesgue measure), and it is somewhat surprising that this simple basic mechanismhas been overlooked in the half century  since the original work by  Cartwright and Littlewood. 

The rigorous result of [] disproved an experiment reported in a physics journal. More generally, I showed that the Josephson equation of the form $\beta\ddot\phi+\dot\phi +\sin\phi = p(t)$ (here $p(t)$ is  any periodic function) exhibits only periodic or quasiperiodic behavior, but no chaos, as long as        $\beta <{1\over 4}$. The interesting point about this result is that it specifies the numerical range of the parameter, rather than asking for $ \beta $ to be ``sufficiently small".  This result has answered some specific questions that   some experimentalists in Josephson thermometry were interested in. 

The paper [22] explains geometrically an interesting period--adding phenomenon in a nonlinear circuit which was observed experimentally in 1927 by  van der Pol. The paper explained also some  experimental findings whose mechanism was not known: for instance, why the frequency lock--in plateaus overlap in one parameter range and why they are separated by ``chaotic'' gaps in another.  

In [59]  it is shown that for a certain class of relaxation oscillations there is a hidden linearity when the relaxation parameter becomes small.  Specifically, it is well known that the Poincar\'e map of a periodically forced van der Pol type oscillator is well approximated by a circle map. All the work on the subject starting with that of Cartwright, Littlewood and Levinson   amounts to understanding this map.

Now the gist of [59] is the  proof of the fact that  this circle map is in fact close to piecewise linear (when the relaxation parameter is small)  for a certain class of oscillators.  This shows that the system is as simple as it can possibly be, contrary to what may have been expected on the basis of the work of the past 50 years. 

My thesis [6] is a study of the van der Pol-type relaxation oscillators with periodic forcing. The main result of this paper is a  geometric description of the  Poincar\'e map (this geometry was not known from the earlier papers) and the qualitative analysis using symbolic dynamics and other methods.