Sudden Jumps in the Logistic Map with Periodic Modulation: Theory and Experiment
When the logistic map is augmented with a periodic modulation, bistability as well as sudden jumps occur in response to changes of the control parameters λ and μ. Using the inverse-curve method for graphical iterations of one-dimensional maps, we are able to specify the conditions for bistability and identify three basic mechanisms for the bifurcation discontinuities. These mechanisms are (1) basin crossing, (2) curve separation, and (3) transfer crisis. In order to verify experimentally the findings from computer iterations, we use an anharmonic electrical oscillator circuit with a junction diode as the nonlinear element, driven with a superposition of sinusoidal and square-wave signals of amplitudes λ and μ, respectively. The experiments confirm, on a qualitative level, our theoretical predictions.
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