Analytical approximation of the canard explosion in a van der Pol system with the nonlinear time transformation method

In the present work, we investigate the canard explosion in a van der Pol electronic oscillator, a fast transition from a small amplitude periodic orbit to a relaxation oscillation. To this aim we develop a new effective procedure, based on the nonlinear time transformation method, that uses elementary trigonometric functions. In fact, it is able to compute up to any desired order the approximation of the critical parameter value for which the transition occurs. Moreover, an approximation of the critical manifold in the phase space is also obtained simultaneously. On the other hand, we have previously proved the uniqueness of the perturbation solution. Our approach, that is an efficacious alternative to Melnikov method in the calculation of high-order coefficients, has one advantage with respect to the classical method, namely it approximates the critical manifold without discontinuities. Finally, our results strongly agree with those provided by numerical continuation methods.