The discrete circle map is the archetypical example of a driven periodic system, showing a complex resonance structure under a change of the forcing frequency known as the devil's staircase. Adler's equation can be seen as the direct continuous equivalent of the circle map, describing locking effects in periodic systems with continuous forcing. This type of locking produces a single fundamental resonance tongue without higher-order resonances, and a devil's staircase is not observed. We show that, with harmonically modulated forcing, nonlinear oscillations close to a Hopf bifurcation generically reproduce the devil's staircase even in the continuous case. Experimental results on a semiconductor laser driven by a modulated optical signal show excellent agreement with our theoretical predictions. The locking appears as a modulation of the oscillation amplitude as well as the angular oscillation frequency. Our results show that by proper implementation of an external drive, additional regions of stable frequency locking can be introduced in systems which originally show only a single Adler-type resonance tongue. The induced resonances can be precisely controlled via the modulation parameters.

中文翻译：

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通过非线性系统的调制强迫在霍夫夫分叉附近普遍产生魔鬼楼梯

离散圆图是驱动周期系统的典型示例，它显示了在被称为魔鬼阶梯的强迫频率变化下的复杂共振结构。阿德勒方程可以看作是圆图的直接连续等价物，描述了具有连续强迫的周期系统中的锁定效应。这种类型的锁定产生单个基音共鸣舌，而没有高阶共鸣，并且没有观察到魔鬼的阶梯。我们显示出，通过谐波调制的强迫，即使在连续情况下，接近霍普夫分叉的非线性振荡也可以重现魔鬼的阶梯。由调制光信号驱动的半导体激光器的实验结果与我们的理论预测非常吻合。锁定表现为振荡幅度以及角振荡频率的调制。我们的结果表明，通过适当地使用外部驱动器，可以在最初仅显示单个Adler型共振舌的系统中引入稳定的频率锁定的其他区域。可以通过调制参数精确控制感应的共振。