Nested mixed-mode oscillations, part II: Experimental and numerical study of a classical Bonhoeffer–van der Pol oscillator

Abstract

The dynamics of Bonhoeffer–van der Pol (BVP) oscillators are known to be equivalent to those of the FitzHugh–Nagumo model and have been extensively studied for many years. In a previous work (Inaba and Kousaka, 2020), we discovered nested mixed-mode oscillations (MMOs) generated by a piecewise-smooth driven Bonhoeffer–van der Pol oscillator. In this study, we focus on the MMOs that occur between the 1s- and 1s+1-generating regions for s=2 and 3 in a classical BVP oscillator where the nonlinear conductor is expressed as a third-order polynomial function, and we confirm the occurrence of nested mixed-mode oscillation-incrementing bifurcations (MMOIBs) that are at least doubly nested. Simple (un-nested), singly nested, and doubly nested MMOIBs generate [1s,1s+1×n]n+1, [A1,B1×n], and [A2,B2×n] MMO sequences, respectively, for successive n. A1=[1s,1s+1×m]m+1 and B1=[1s,1s+1×(m+1)]m+2 in the singly nested case for integers m and A2=[[1s,1s+1×l]l+1,[1s,1s+1×(l+1)]l+2×m](l+2)m+(l+1) and B2=[[1s,1s+1×l]l+1,[1s,1s+1×(l+1)]l+2×(m+1)](l+2)(m+1)+(l+1) in the doubly nested case for integers l and m. In particular, we numerically confirm that m=1 with s=2 and 3 cases for the singly nested MMOIBs and that l=1,m=1 with s=2 and 3 cases for the doubly nested MMOIBs. We also show that both the simple (un-nested) and nested MMOIB-generated MMOs have asymmetric Farey characteristics. In addition, we find that these numerical results are well-explained by effectively one-dimensional (1D) Poincaré return maps derived numerically from the dynamics of a constrained driven BVP oscillator that includes a diode with grazing–sliding characteristics. Finally, we verify these numerical results for the classical and constrained BVP oscillators in circuit experiments and derive the first return plots and 1D Poincaré return maps based on laboratory measurements.