Experimental verification of periodic pulling in a nonlinear electronic oscillator

Abstract

An experimental investigation of entrainment and the phenomenon known as periodic pulling is described. Periodic pulling refers to the incomplete entrainment of an oscillatory nonlinear system by a periodically varying driving force. The process whereby the system s oscillation frequency is pulled toward the driving frequency stops short of complete synchronization and is interrupted at regular intervals. In this way, periodic pulling produces pulselike amplitude and frequency modulation in the system s oscillatory response. Associated with these combined, nonsinusoidal modulations is an asymmetric, or single-sided, spectral peak made up of spectral components at frequencies incommensurate with the undriven {spontaneous) and driving frequencies. The system being investigated uses a unijunction transistor as the nonlinear element in an electrical circuit. The behavior of the system is shown to resemble that predicted for a forced van der Pol oscillator with an adjustable nonlinear restoring force.For certain ranges of driving frequency and amplitude, when the system is not entrained to the driving force, periodic pulling is shown to alter the system's oscillatory response significantly from that expected for conventional amplitude modulation involving two sinusoids. The essential features of the periodic pulling phenomenon are observed and compared, with good agreement, to a model.

Evaluating the Spectrum of Unlocked Injection Frequency Dividers in Pulling Mode

“…This result [9] has been reported in literature to explain experimental observations of pulling in microwave solid-state oscillators [10], in a unijunction transistor based oscillator [11], and in many papers dealing with the study of plasma instabilities and with periodically driven oscillating plasma systems (see [11,12], and references therein). These systems are well modeled by the van der Pol equation and exhibit a variety of dynamical phenomena observed in forced oscillators of van der Pol type [13][14][15].…”

“…The occurrence of the periodic pulling is easily recognized by the characteristic aspect of the pulled oscillations, usually OPEN ACCESS called beats, which exhibit a simultaneous modulation of amplitude and frequency with a pulse-like envelope of the amplitude. A theoretical investigation of the oscillatory response in the pulling mode of driven oscillators is given in a number of papers (see [1][2][3][4][5][6][7][8][9][10][11][12] and references therein) starting from the pioneering investigation of Rjasin [6], who first performed a harmonic analysis of beats to establish the spectral composition. Later on, an approximate, but physically insightful, treatment of the pulling was given in a celebrated paper of Adler [7], who obtained an analytical expression for the phase difference between the forcing and the system response neglecting the amplitude modulation.…”

Lock-in and quasiperiodicity in a forced hydrodynamically self-excited jet

“…When forced, however, the global mode has a natural frequency, f * n , whose value can shift from its unforced value (f n , without an asterisk), as we will see in § 3.2. According to Hilborn (2000), this shifting, or pulling, as it is often called (Koepke & Hartley 1991), of the natural frequency is a characteristic feature of forced nonlinear oscillators. In the literature, it is quantified using both the bare winding number, f n /f f , and the dressed winding number, f * n /f f .…”

Nonlinear interaction of drift waves with driven plasma currents

“…For larger frequency mismatches phase slippage occurs more frequently. The phase slippage and the broadening of the peak are evidence for incomplete synchronization and periodic pulling [10,14,[21][22][23].…”