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This letter investigates the period-doubling cascades of canards, generated in the extended Bonhoeffer-van der Pol oscillator. Canards appear by Andronov Hopf bifurcations (AHBs) and it is confirmed that these AHBs are always supercritical in our system. The cascades of period-doubling bifurcation are followed by mixed-mode oscillations. The detailed two-parameter bifurcation diagrams are derived, and it is clarified that the period-doubling bifurcations arise from a narrow parameter value range at which the original canard in the non-extended equation is observed.

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In this paper, we explain how to compute bifurcation parameter values of periodic solutions for non-autonomous nonlinear differential equations. Although various approaches and tools are available for solving this problem nowadays, we have devised a very simple method composed only of basic computational algorithms appearing in textbooks for beginner's, i.e., Newton's method and the Runge-Kutta method. We formulate the bifurcation problem as a boundary value problem and use Newton's method as a solver consistently. All derivatives required in each iteration are obtained by solving variational equations about the state and the parameter. Thanks to the quadratic convergence ability of Newton's method, accurate results can be quickly and effectively obtained without using any sophisticated mathematical library or software. If a discontinuous periodic force is applied to the system, we can use the same strategy to solve the bifurcation problem. The key point of this method is deriving a differentiable composite map from the various information about the problem such as the location of sections, the periodicity, the Poincaré mapping, etc.