Van der Pol’s equation

Simple dynamical systems have proved valuable as models of certain classes of physical systems in many branches of science and engineering. In mechanics and electrical engineering Duffing s and van der Pol s equations have played important roles and in physical chemistry and chemical engineering much has been learned from the study of simple, even artificially simple, systems. In calling them simple we mean to imply that their formulation is as elementary as possible their behaviour may be far from simple. Models should have the two characteristics of feasibility and actuality. By the first we mean that a favourable case can be made for the proposed reaction, perhaps by some further elaboration of mechanism but within the framework of accepted kinetic principles. Thus irreversible reactions are acceptable provided that they can be obtained as the limit of a consistent reversible set. By actuality we mean that they are set in an actual context, as taking place in a stirred tank, on a catalytic surface or in a porous medium. It is not usually necessary to assume the reaction to take place in a closed system with certain components held constant presumably by being in excess. 

Example 13.2 Van der Pol s equations Van der Pol s equations provide a valuable framework for studying the important features of oscillatory systems. It describes self-sustaining oscillations in which eneigy is fed into small oscillations and removed from large oscillations. Consider the following system of ordinary differential equations called Van der Pol s equations... 

The initial conditions arejqfO) = 0.1 and> (0) =0.5, and a is constant. Figure 13.4 shows the solution for Van der Pol s equations (Eq. 13.11) with the time span 0 < / < 30 using aMATLAB code presented below. Figure 13.4a shows the periodic plots, while Figure 13.4b displays the limit cycle plots. 

Van der Pol s equations provide a valuable framework for studying the important features of oscillatory systems. 

One of the simplest nonlinear equation systems describing a circuit is Van der Pol s system (Hairer and Wanner, 2010) ... 

The stability diagram of the Mathieu equation in the plane of the parameters S and is called the Strutt-Ince diagram (van der Pol and Strutt 1928). This diagram can be transformed to the plane of the forcing amplitude r and the forcing frequency /= col2%. 

---