Van der Pol oscillator

In dynamics, the Van der Pol oscillator is a non-conservative oscillator with non-linear damping. It evolves in time according to the second-order differential equation: In dynamics, the Van der Pol oscillator is a non-conservative oscillator with non-linear damping. It evolves in time according to the second-order differential equation: where x is the position coordinate—which is a function of the time t, and μ is a scalar parameter indicating the nonlinearity and the strength of the damping. The Van der Pol oscillator was originally proposed by the Dutch electrical engineer and physicist Balthasar van der Pol while he was working at Philips. Van der Pol found stable oscillations, which he subsequently called relaxation-oscillations and are now known as a type of limit cycle in electrical circuits employing vacuum tubes. When these circuits were driven near the limit cycle, they become entrained, i.e. the driving signal pulls the current along with it. Van der Pol and his colleague, van der Mark, reported in the September 1927 issue of Nature that at certain drive frequencies an irregular noise was heard, which was later found to be the result of deterministic chaos. The Van der Pol equation has a long history of being used in both the physical and biological sciences. For instance, in biology, Fitzhugh and Nagumo extended the equation in a planar field as a model for action potentials of neurons. The equation has also been utilised in seismology to model the two plates in a geological fault, and in studies of phonation to model the right and left vocal fold oscillators. Liénard's theorem can be used to prove that the system has a limit cycle. Applying the Liénard transformation y = x − x 3 / 3 − x ˙ / μ {displaystyle y=x-x^{3}/3-{dot {x}}/mu } , where the dot indicates the time derivative, the Van der Pol oscillator can be written in its two-dimensional form: Another commonly used form based on the transformation y = x ˙ {displaystyle y={dot {x}}} leads to: Two interesting regimes for the characteristics of the unforced oscillator are: One can also write a time-independent Hamiltonian formalism for the Van der Pol oscillator by augmenting it to a four-dimensional autonomous dynamical system using an auxiliary second-order nonlinear differential equation as follows: Note that the dynamics of the original Van der Pol oscillator is not affected due to the one-way coupling between the time-evolutions of x and y variables. A Hamiltonian H for this system of equations can be shown to be