In this paper, the mechanism of system solutions approaching infinity is explored based on a modified Rayleigh–Duffing oscillator with two slow-varying periodic excitations. System solutions approaching infinity is a new novel route to bursting oscillation, and are not reported yet. The system can be separated into a fast subsystem and a slow subsystem according to the slow–fast analysis method. We find that there is a critical value for the fast subsystem, which limits the original region of the stable equilibrium point and the stable limit cycle, the right of which is the divergent region. When the control parameter slowly varies closely to the critical value $$\delta _{\mathrm{CR}} $$ δ CR , both the stable equilibrium point and the stable limit cycle quickly leave the original region and approach positive infinity. The mechanism of two different bursting forms called bursting oscillation of point/point and bursting oscillation of cycle/cycle induced by system solutions approaching infinity are explored. This paper provides a new possible route to bursting oscillation unrelated to bifurcations and deepens the comprehension of bursting dynamics behaviours. Lastly, the accuracy of our study is verified by overlapping the transformed phase portraits onto the bifurcation diagrams.

中文翻译：

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在改进的瑞利-达芬振荡器中由接近无穷大的系统解引起的两种爆发模式

在本文中，基于具有两个慢变周期激励的修正瑞利-达芬振荡器，探索了接近无穷大的系统解的机制。接近无穷大的系统解决方案是一种新的突发振荡途径，目前尚未有报道。根据慢-快分析方法，系统可以分为快速子系统和慢速子系统。我们发现快速子系统有一个临界值，它限制了稳定平衡点和稳定极限环的原始区域，右边是发散区域。当控制参数缓慢变化接近临界值$$\delta _{\mathrm{CR}} $$ δ CR 时，稳定平衡点和稳定极限环都迅速离开原始区域并接近正无穷大。探讨了点/点的突发振荡和循环/周期的突发振荡由接近无穷大的系统解引起的两种不同的突发形式的机制。本文为与分岔无关的爆裂振荡提供了一种新的可能途径，并加深了对爆裂动力学行为的理解。最后，通过将变换后的相图重叠到分叉图上来验证我们研究的准确性。