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.

Figure 1

Circuit diagram of the Bonhoeffer–van der Pol oscillator.

Figure 2

One-parameter bifurcation diagram near a supercritical AHB point ($k_1=0.2$).

Figure 3

One-parameter bifurcation diagram near a subcritical AHB point ($k_1=0.9$).

Figure 4

Coexisting stable focus (filled circle), unstable limit cycle (dashed line), and stable relaxation oscillation (solid line) ($k_1=0.9,B_0=0.21$).

Figure 5

The BVP oscillator under weak periodic perturbation.

Figure 6

One-parameter bifurcation diagram of $T_{B_1}$ and the corresponding largest Lyapunov exponent.

Figure 7

Quasiperiodic attractor ($B_1=0.0008$). (a) The attractor in the phase plane. (b) Time series.

Figure 8

Chaotic attractor for small $B_1$ ($B_1=0.002$). (a) The attractor in the phase plane. (b) Time series.

Figure 9

Oscillation death ($B_1=0.01$). (a) The attractor in the phase plane. (b) Time series.

Figure 10

Stable relaxation oscillation ($B_1=0.07$). (a) The attractor in the phase plane. (b) Time series.

Figure 11

Chaotic oscillation ($B_1=0.0415$). (a) The attractor in the phase plane. (b) Time series.

Figure 12

Two-parameter bifurcation diagrams. (a) Bifurcation sets related to the stable relaxation oscillation. (b) Bifurcation sets related to the stable focus.

Figure 13

The superimposed bifurcation diagram.

Figure 14

Chaotic oscillation ($\omega =1.424,B_1=0.018$). (a) The attractor in the phase plane. (b) Time series.

Figure 15

Structure of the fixed-point manifold of Fig. 12b.