TY - JOUR

T1 - Mixed-mode oscillation-incrementing bifurcations and a devil's staircase from a nonautonomous, constrained Bonhoeffer-van der Pol oscillator

AU - Takahashi, Hiroaki

AU - Kousaka, Takuji

AU - Asahara, Hiroyuki

AU - Stankevich, Nataliya

AU - Inaba, Naohiko

N1 - Publisher Copyright: © The Author(s) 2018. Published by Oxford University Press on behalf of the Physical Society of Japan.

PY - 2018/10/1

Y1 - 2018/10/1

N2 - In this study, we analyze mixed-mode oscillation-incrementing bifurcations (MMOIBs) generated in the nonautonomous, constrained Bonhoeffer-van der Pol oscillator proposed by Kousaka et al. [Physica D 353-354, 48 (2017)]. Specifically, we investigate MMOIBs occurring in the 14-15 and 11-12 regions. These two kinds of MMOIBs exhibit qualitatively different MMO-bifurcation structures. The former MMOIBs successively occur many times, while the latter exhibit finite MMOIBs. In the latter case, standard MMOIBs occur only five times, and are then followed by another type of MMOIB. However, the following MMOIBs are also only generated seven times and the solution finally settles down into a 20 attractor. We clarify the exact reason for these phenomena by analyzing 1D Poincaré return maps derived from the constrained dynamics. By focusing on the initial successive MMOIBs, we create asymmetric Farey trees that occur between 14 and 15 by analyzing the 1D Poincaré return map. We find that there exist two sets of successive MMOIBs between 14 and 15. In particular, we rigorously define the MMO increment-terminating tangent bifurcations, toward which the MMOIBs accumulate and terminate. Furthermore, we uncover a nested bifurcation structure caused by MMOIBs. This occurs inside a short interval in the 14-15 region and accumulates toward another MMO increment-terminating tangent bifurcation point. These three types of successively generated MMOIBs accumulate in different ways toward the MMO increment-terminating tangent bifurcation points. We also analyze the behavior of the "firing number," which varies with the MMOIBs. In particular, we theoretically explain why a firing number that exhibits a devil's staircase has higher values in chaos-generating regions than in MMO-generating regions.

AB - In this study, we analyze mixed-mode oscillation-incrementing bifurcations (MMOIBs) generated in the nonautonomous, constrained Bonhoeffer-van der Pol oscillator proposed by Kousaka et al. [Physica D 353-354, 48 (2017)]. Specifically, we investigate MMOIBs occurring in the 14-15 and 11-12 regions. These two kinds of MMOIBs exhibit qualitatively different MMO-bifurcation structures. The former MMOIBs successively occur many times, while the latter exhibit finite MMOIBs. In the latter case, standard MMOIBs occur only five times, and are then followed by another type of MMOIB. However, the following MMOIBs are also only generated seven times and the solution finally settles down into a 20 attractor. We clarify the exact reason for these phenomena by analyzing 1D Poincaré return maps derived from the constrained dynamics. By focusing on the initial successive MMOIBs, we create asymmetric Farey trees that occur between 14 and 15 by analyzing the 1D Poincaré return map. We find that there exist two sets of successive MMOIBs between 14 and 15. In particular, we rigorously define the MMO increment-terminating tangent bifurcations, toward which the MMOIBs accumulate and terminate. Furthermore, we uncover a nested bifurcation structure caused by MMOIBs. This occurs inside a short interval in the 14-15 region and accumulates toward another MMO increment-terminating tangent bifurcation point. These three types of successively generated MMOIBs accumulate in different ways toward the MMO increment-terminating tangent bifurcation points. We also analyze the behavior of the "firing number," which varies with the MMOIBs. In particular, we theoretically explain why a firing number that exhibits a devil's staircase has higher values in chaos-generating regions than in MMO-generating regions.

UR - http://www.scopus.com/inward/record.url?scp=85057192333&partnerID=8YFLogxK

U2 - 10.1093/ptep/pty099

DO - 10.1093/ptep/pty099

M3 - Article

AN - SCOPUS:85057192333

VL - 2018

JO - Progress of Theoretical and Experimental Physics

JF - Progress of Theoretical and Experimental Physics

SN - 2050-3911

IS - 10

M1 - 103A02

ER -