We report the discovery of a codimension-two phenomenon in the phase diagram of a second-order self-sustained nonlinear oscillator subject to a constant external periodic forcing, around which three regimes associated with the synchronization phenomenon exist, namely phase-locking, frequency-locking without phase-locking, and frequency-unlocking states. The triple point of synchronization arises when a saddle-node homoclinic cycle collides with the zero-amplitude state of the forced oscillator. A line on the phase diagram where limit-cycle solutions contain a phase singularity departs from the triple point, giving rise to a codimension-one transition between the regimes of frequency unlocking and frequency locking without phase locking. At the parameter values where the critical transition occurs, the forced oscillator exhibits a separatrix with a $\pi$ phase jump, i.e., a particular trajectory in phase space that separates two distinct behaviors of the phase dynamics. Close to the triple point, noise induces excitable pulses where the two variants of type-I excitability, i.e., pulses with and without $2\pi$ phase slips, appear stochastically. The impacts of weak noise and some other dynamical aspects associated with the transition induced by the singular phenomenon are also discussed.

中文翻译：

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强迫非线性振荡器中沿无界和有界相位振荡之间的跃迁的三重同步点，相位奇异性和激励性

我们报告了在受到外部恒定周期强迫作用的二阶自持非线性振荡器的相位图中，发现了一个共维二现象，该现象周围存在与同步现象相关的三种状态，即锁相，频率锁定而没有锁相和频率解锁状态。同步的三点出现在鞍节点同斜周期与强制振荡器的零振幅状态发生冲突时。相位图上的一条线（其中极限循环解包含一个相位奇点）偏离了三重点，从而在未锁定相位的情况下，在频率解锁和频率锁定的状态之间产生了余维一过渡。在发生临界转变的参数值处，强制振荡器表现出具有$\pi$相跳，即相空间中的一个特定轨迹，该轨迹将相动力学的两个不同行为分开。接近三点，噪声会诱发可激发的脉冲，其中有两种类型的I型可激发性，即带有和不带有脉冲的脉冲$\mathrm{2个}\pi$相位差，随机出现。还讨论了弱噪声的影响以及与奇异现象引起的过渡相关的其他一些动力学方面。