In the book of Goldbeter which I have mentioned in several recent posts a concept which occurs repeatedly is that of entrainment. While looking for some more information about this topic I found a paper of Russo, di Bernardo and Sontag (PloS Computational Biology 4, e1000739) which gives an insightful treatment of the subject. The basic idea is to consider two systems which are coupled in some way and to consider the influence of oscillations in one system on the behaviour of the other. It is easy to see how this might be translated into a problem expressed in terms of dynamical systems. A classical example related to this is contained in a story about Christiaan Huygens who was, among other things, the inventor of the pendulum clock in the mid 17th century. Apparently he did not construct clocks himself but had them made by others according to his plans. The well-known story is that he noticed that when two pendulum clocks were placed next to each other the phase of their oscillations became synchronized with, say, one always at the leftmost point of its swing when the other was at the rightmost. Another example is that of the circadian rhythm. There is a 24 hour rhythm in our body and it is interesting to know whether it comes from an intrinsic oscillator or not. Experiments with subjects isolated from the usual rhythm of day and night show that there is an intrinsic oscillator but that its period is closer to 25 hours. Under normal circumstances its period is brought to 24 hours due to the cycle of day and night by entrainment.

The particular mathematical set-up considered in the paper of Russo et. al. is the following. Consider an autonomous dynamical system containing some parameters. Now replace one or more of those parameters by functions of time with period . If solutions of the original system have a suitable tendency to converge to a stationary solution for a given choice of the parameters then solutions of the resulting non-autonomous system converge to periodic solutions of period . In the papers there are nice plots of numerical simulations which give a striking picture of this behaviour. The central result of the paper is a theorem which guarantees this type of behaviour under certain hypotheses. As pointed out in the paper verifying these hypotheses has some similarity to finding a Lyapunov function for an autonomous system. The positive side is that if it can be done it is possible to get strong conclusions. The negative side is that verifying the hypotheses is generally a matter of trial and error. There is no algorithm available for doing that.

The criterion is dependent on the choice of a matrix norm. This is used to define a quantity called the matrix measure of a matrix . The criterion is that the Jacobian of the function defining the dynamical system should have a matrix measure which is bounded above by a negative constant. In that case the system is said to be infinitesimally contracting. The matrix measure is defined by a limiting procedure, , but for particular choices of the matrix norm it is possible to calculate in a purely algebraic way. I have no intuitive feeling for what this definition means.

February 25, 2012 at 8:20 pm |

Hi!

I did not look up in the paper you quote, but with the few details you wrote here, I have the following intuition…

This matrix measure is the derivative of the function which maps a matrix on its matrix norm, taken at the identity in direction of A. The condition says the slope in this direction is negative, meaning going this way, you’ll find contractions. Furthermore, the flow near an equilibrium for small t has leading order id+tA, i.e. it runs in the direction mentioned above.

My intuition wisely ignored the non-autonomous thing, and why you require the uniform negativity …

Another thing is: if you think of A diagonal, ยต(A) negative just means that it has only stable eigenvalues. (at least with the usual matrix norm)

And last but not least, I had to think about the experience Bernold showed in class. The one with the metronomes sitting on a board, the board on two rolling cans, and (anti)synchronization taking place or not, depending on the original tuning of the metronomes. I love it!

February 26, 2012 at 9:54 am |

Hi Juliette,

Thanks for your comment. The class demonstration you mention sounds very interesting. I should talk to Bernold about this subject when I see him again.