The van der Pol oscillator is a model for an electrical circuit which is a simple ODE with a stable periodic solution. The equation is where is a parameter. Only non-negative values of will be considered here. The sign of can be changed by replacing by . For the system reduces to the harmonic oscillator. It is common to reduce the equation to a first order system by defining . The resulting system is , . It is an example of the Liénard system which in general reads . We just need to choose and . In the general case defining a new variable where gives the first order system , . There is a good general theory of the existence of unique stable periodic solutions for Liénard systems. This shows that for any positive value of the van der Pol oscillator has a unique stable limit cycle. The origin is an unstable stationary point. It is of interest to consider the limits and . In the first limit the periodic solution converges to the circle of radius two. In the other limit it converges to what is called a relaxation oscillation. This is a closed curve which consists of two pieces of the nullcline of the original equation and two straight lines of constant joining them. To get an intuitive explanation of this take the quotient of the two equations to get . The only way to make this consistent with tending to infinity is to either let the expression in the brackets tend to zero or to let tend to zero. For large the two straight line pieces are traversed much faster than the other two. Thus the solution spends most of its time on the curved pieces and the period is determined in leading order by the time spent on these pieces. On the curved pieces it is possible (at least formally) to derive a simplified equation by substituting the relation into the evolution equation for . The result is . This equation, determines the limiting behaviour of the period. The limiting value can be determined explicitly. Detailed information can also be obtained about the asymptotic behaviour in the limit . This can be done by the method of averaging which actually originated in the work of van der Pol which was developed further by Krylov and Bogoliubov.
How is the existence and uniqueness of the limit cycle for the van der Pol system proved? It is relatively elementary to prove that solutions repeatedly cross one of the axes at arbitrarily late times. Consider the mapping from one crossing point to the next. A periodic solution corresponds to a fixed point of this mapping. It is shown that the difference in the value is positive for certain initial data and negative for others. It follows by continuity that it is zero for some choice of data and this gives the existence of a periodic solution. Its uniqueness is shown by using a discrete symmetry of the equation.
The van der Pol equation appears to occupy a central position in the field of nonlinear oscillations. For many years now the book ‘Mathematical Biology’ of Murray has been on my bookshelf. I have only read small parts of it in any detail and now I realized that a variety of ideas related to those which are important for the van der Pol equation are explained in a very helpful way there. The van der Pol equation was a basis for the FitzHugh-Nagumo equation, a system of two ODE which is a kind of caricature of the Hodgkin-Huxley model of nerve conduction. Another system which shows relaxation oscillations is the Field-Noyes model. It is a three-dimensional system and so analytically harder to handle than FitzHugh-Nagumo. It is a model for the Belousov-Zhabotinski model of oscillatory chemical reactions. I intend to return to the FitzHugh-Nagumo model in a future post.