Bursting oscillations are ubiquitous in multi-time scale systems and have attracted widespread attention in recent years. However, research on experimental demonstration of the bursting oscillations induced by delayed bifurcation is very rarely reported. In this paper, a parametrically driven Rucklidge system is introduced and a distinct delayed behavior is observed when the time-varying parameter passes through the pitchfork bifurcation point. Different bursting patterns induced by such a delayed behavior are numerically investigated under different excitation amplitudes based on the fast–slow analysis method. Furthermore, in order to reproduce the bursting electronic signals and explore the underlying formation mechanisms experimentally, a real physical circuit of the parametrically driven Rucklidge system is developed by using off-the-shelf electronic devices. The real-time measurement results such as time series, phase portraits and transformed phase portraits are in good qualitative agreement with those obtained from the numerical computations. The experimental evidence to verify bursting oscillations induced by delayed pitchfork bifurcation is thus provided in this study.

中文翻译：

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Rucklidge 系统的爆破振荡和实验验证

爆发振荡在多时间尺度系统中无处不在，近年来引起了广泛关注。然而，关于延迟分叉引起的爆发振荡的实验证明研究很少报道。在本文中，引入了参数驱动的 Rucklidge 系统，并在时变参数通过干草叉分叉点时观察到明显的延迟行为。基于快慢分析方法，在不同的激发幅度下对这种延迟行为引起的不同爆发模式进行了数值研究。此外，为了重现爆发的电子信号并通过实验探索潜在的形成机制，使用现成的电子设备开发了参数驱动 Rucklidge 系统的真实物理电路。时间序列、相图和变换相图等实时测量结果与数值计算得到的结果具有很好的定性一致性。因此，本研究提供了验证延迟干草叉分叉引起的突发振荡的实验证据。