Mixed-mode oscillations from a constrained extended Bonhoeffer-van der Pol oscillator with a diode.

An extended Bonhoeffer-van der Pol (BVP) oscillator is a circuit that is naturally extended to a three-variable system from a two-variable BVP oscillator. A BVP oscillator is known to exhibit a canard explosion, and the extended BVP oscillator generates mixed-mode oscillations (MMOs). In this work, we considered a case study where the nonlinear conductor in the extended BVP oscillator includes an idealized diode. The idealized case corresponds to a degenerate case where one of the parameters tends to infinity, and circuit dynamics are represented using a constrained equation, and at the expense of the model's naturalness, i.e., in a case in which the solutions of the dynamics are defined only forward in time, the Poincaré return maps are constructed as one-dimensional (1D). Using these 1D return maps, we explain various phenomena, such as simple MMOs and MMO-incrementing bifurcations. In this oscillator, there exists a small amplitude oscillation, which emerges as a consequence of supercritical Hopf bifurcation, and there exists large relaxation oscillation which appears via canard explosion by changing the bifurcation parameter. Between these small and large amplitude oscillations, the MMO bifurcations exhibit asymmetric Farey trees. Furthermore, these theoretical results were verified using laboratory measurements and experiments.

Experimental and numerical study of nonsmooth maximum bounce height changes in a bouncing ball system.

The bouncing ball system is a simple mechanical collision system that has been extensively studied for several decades. In this study, we investigate the bouncing ball's dynamics both numerically and experimentally. We implement the system using a table tennis ball and paddle vibrated by a shaker. We focus on the relationship between the ball's maximum bounce height in the long time interval and the paddle's vibration frequency, observing several stepwise height changes for frequencies of 25-50 Hz, noting this significant characteristic in both our experiments and numerical simulations. We concentrate on this paddle frequency interval because the phenomenon is easy to handle in numerical simulations. Because the observed phenomenon has a simple order, it can be universal and appear in a large class of collision dynamics. Possibly, some researchers have investigated the bouncing ball system in which the nonsmooth maximum bounce height changes occur. However, they may have failed to notice the changes because the maximal height of the ball was not considered.

Experimental study of complex mixed-mode oscillations generated in a Bonhoeffer-van der Pol oscillator under weak periodic perturbation.

Bifurcations of complex mixed-mode oscillations denoted as mixed-mode oscillation-incrementing bifurcations (MMOIBs) have frequently been observed in chemical experiments. In a previous study [K. Shimizu et al., Physica D 241, 1518 (2012)], we discovered an extremely simple dynamical circuit that exhibits MMOIBs. Our model was represented by a slow/fast Bonhoeffer-van der Pol circuit under weak periodic perturbation near a subcritical Andronov-Hopf bifurcation point. In this study, we experimentally and numerically verify that our dynamical circuit captures the essence of the underlying mechanism causing MMOIBs, and we observe MMOIBs and chaos with distinctive waveforms in real circuit experiments.

Three-dimensional tori and Arnold tongues.

This study analyzes an Arnold resonance web, which includes complicated quasi-periodic bifurcations, by conducting a Lyapunov analysis for a coupled delayed logistic map. The map can exhibit a two-dimensional invariant torus (IT), which corresponds to a three-dimensional torus in vector fields. Numerous one-dimensional invariant closed curves (ICCs), which correspond to two-dimensional tori in vector fields, exist in a very complicated but reasonable manner inside an IT-generating region. Periodic solutions emerge at the intersections of two different thin ICC-generating regions, which we call ICC-Arnold tongues, because all three independent-frequency components of the IT become rational at the intersections. Additionally, we observe a significant bifurcation structure where conventional Arnold tongues transit to ICC-Arnold tongues through a Neimark-Sacker bifurcation in the neighborhood of a quasi-periodic Hopf bifurcation (or a quasi-periodic Neimark-Sacker bifurcation) boundary.

Sudden change from chaos to oscillation death in the Bonhoeffer-van der Pol oscillator under weak periodic perturbation.

In this paper, we analyze the sudden change from chaos to oscillation death generated by the Bonhoeffer-van der Pol (BVP) oscillator under weak periodic perturbation. The parameter values of the BVP oscillator are chosen such that a stable focus and a stable relaxation oscillation coexist if no perturbation is applied. In such a system, complicated bifurcation structure is expected to emerge when weak periodic perturbation is applied because the stable focus and the stable relaxation oscillation coexist in close proximity in the phase plane. We draw a bifurcation diagram of the fundamental harmonic entrainment. The bifurcation structure is complex because there coexist two bifurcation sets. One is the bifurcation set generated in the vicinity of the stable focus, and the other is that generated in the vicinity of the stable relaxation oscillation. By analyzing the bifurcation diagram in detail, we can explain the sudden change from chaos with complicated waveforms to oscillation death. We make it clear that this phenomenon is caused by a saddle-node bifurcation.