Period-doubling cascades of canards from the extended Bonhoeffer–van der Pol oscillator

Abstract

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.

Stern–Brocot trees in cascades of mixed-mode oscillations and canards in the extended Bonhoeffer–van der Pol and the FitzHugh–Nagumo models of excitable systems

“…An exception is an interesting work published recently in this journal by Sekikawa et al [23]. Among other things, these authors reported two-parameter bifurcation diagrams of the period-doubling bifurcations associated with canards [24] in an extended Bonhoeffervan der Pol (BVP) oscillator shown schematically in Fig.…”

Globally exponential synchronization criterion of chaot-ic oscillators using active control

“…The multi-scroll chaotic attractors exhibited by electric circuits have received considerable attention of the researchers theoretically as well as experimentally [23]. The three-dimensional BVP [23] is one of the elec-1 C tric circuits, which realizes chaos and shows very rich, complex dynamic behavior and yet has potential applications. An extended BVP oscillator can show a stable oscillator as well as a stable threshold phenomenon [23].…”

“…The three-dimensional BVP [23] is one of the elec-1 C tric circuits, which realizes chaos and shows very rich, complex dynamic behavior and yet has potential applications. An extended BVP oscillator can show a stable oscillator as well as a stable threshold phenomenon [23]. The extended BVP oscillator has been observed numerically and experimentally in certain systems and has attributes of great utility to medicine, chemical kinetics, neuroscience, electrical circuits and secure communications [15], [23][24][25].…”

“…An extended BVP oscillator can show a stable oscillator as well as a stable threshold phenomenon [23]. The extended BVP oscillator has been observed numerically and experimentally in certain systems and has attributes of great utility to medicine, chemical kinetics, neuroscience, electrical circuits and secure communications [15], [23][24][25]. The extended BVP model consists of an inductor, two capacitors and a linear resistor.…”

Importance of sufficient precision in stable dynamics for the numerical computation of canards in singularly perturbed systems

“…Since the dynamics near such a quasi-threshold are unstable and expanding, canards produce a chaotic behavior if they are forced even by very small signals [8]. The so-called mixed mode oscillations (MMOs) [9][10][11] are intensively related to canards.…”