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Canards are interesting nonlinear phenomena that have generated intense research interest since their discovery in the late 20th century. We are interested here in how canard-generating dynamics are influenced by extremely weak periodic perturbations that cause the formation of saddle-node bifurcations in the fundamental harmonic entrainment region. In a previous study, we discovered that another entrainment region exists within the fundamental harmonic entrainment region surrounded by the second saddle-node bifurcation curves. We found that two pairs of stable and saddle canards coexist in this second entrainment region under such weak periodic perturbation. Moreover, the stable and saddle canards are matched pairwise; i.e., each stable canard quite closely resembles a corresponding saddle canard. Calculation of the correlation coefficients of the four canards revealed two similar solutions on the order of 0.9999⋯ between the two pairs of similar canards. In contrast, the correlation coefficients of the dissimilar canards differ from unity in proportion to the difference between the given bifurcation parameter value and the parameter values at the saddle-node bifurcation points. Approximately, they take values from 0.998 to 0.975. These contrasts are noteworthy. Similar bifurcation phenomena were observed in the 1/2-subharmonic entrainment region. We hypothesize that the two pairs of stable and saddle canards are invariant with respect to a slight shift of time at the saddle-node bifurcation points, and we numerically prove that such a property approximately holds at the bifurcation points.

Bifurcation transitions between a 1D invariant closed curve (ICC), corresponding to a 2D torus in vector fields, and a 2D invariant torus (IT), corresponding to a 3D torus in vector fields, have been the subjects of intensive research in recent years. An existing hypothesis involves the bifurcation boundary between a region generating an ICC and a region generating an IT. It asserts that an IT would be generated from a stable fixed point as a consequence of two Hopf (or two Neimark–Sacker) bifurcations. We assume that this hypothesis may puzzle many researchers because it is difficult to assess its validity, although it seems to be a reasonable bifurcation scenario at first glance. To verify this hypothesis, we conduct a detailed Lyapunov analysis for a coupled delayed logistic map that can generate an IT, and indicate that this hypothesis does not hold according to numerical results. Furthermore, we show that a saddle-node bifurcation of unstable periodic points does not coincide with the bifurcation boundary between an ICC and an IT. In addition, the bifurcation boundaries of torus doubling do not coincide with a period-doubling bifurcation of unstable periodic points. To conclude, torus bifurcations have no relation with the bifurcations of unstable periodic points. Additionally, we exactly derive a quasi-periodic Hopf bifurcation boundary introducing a double Poincaré map.

Bifurcation transitions between a 1D invariant closed curve (ICC), corresponding to a 2D torus in vector fields, and a 2D invariant torus (IT), corresponding to a 3D torus in vector fields, have been the subjects of intensive research in recent years. An existing hypothesis involves the bifurcation boundary between a region generating an ICC and a region generating an IT. It asserts that an IT would be generated from a stable fixed point as a consequence of two Hopf (or two Neimark–Sacker) bifurcations. We assume that this hypothesis may puzzle many researchers because it is difficult to assess its validity, although it seems to be a reasonable bifurcation scenario at first glance. To verify this hypothesis, we conduct a detailed Lyapunov analysis for a coupled delayed logistic map that can generate an IT, and indicate that this hypothesis does not hold according to numerical results. Furthermore, we show that a saddle-node bifurcation of unstable periodic points does not coincide with the bifurcation boundary between an ICC and an IT. In addition, the bifurcation boundaries of torus doubling do not coincide with a period-doubling bifurcation of unstable periodic points. To conclude, torus bifurcations have no relation with the bifurcations of unstable periodic points. Additionally, we exactly derive a quasi-periodic Hopf bifurcation boundary introducing a double Poincaré map.

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.