The dynamics of synchronization of two-stage van der Pol generator

Results of numerical simulation of self-oscillations synchronization process in two-cascade ring generator van der Pol by harmonic signal are presented. Studies were carried out within the framework of the DT- model of the dynamic system. The model was developed on the basis of the principle of compliance within the framework of the method of slowly changing amplitudes of characteristics of a discrete system with characteristics of an analog prototype. Shortened equations for complex oscillation amplitudes in generator stages are obtained. It was found that in an autonomous system there is an effect of bistability of amplitudes. In the synchronization mode with an external harmonic signal, solutions of shortened equations made it possible to calculate amplitude-frequency and phase-frequency characteristics of synchronous oscillations. It is shown that transitions between bistable states are observed in the synchronous oscillation holding band. Differences of frequency characteristics of synchronization of classical and two-stage oscillators van der Pol were analyzed.

Natural Objects

Tudor claimed that in his seminal work Untitled, electronic components were discovered as “natural objects” as they were patched in a feedback loop creating a giant oscillator which generated sounds without exterior input. But his description of the piece as “part of a never-ending series of discovered works” calls into question its very status as a standalone “work.” Turning to its performance history appears to only complicate the puzzle. Despite his aim to perform everything live, the proliferation of modular instruments forced Tudor to record the output of his setup in advance and use this recording as input source in subsequent performances. He would later create Toneburst, which realized the no-input principle without the aid of pre-recorded sources. Shortly before his death, Tudor revived Toneburst for other musicians of the Merce Cunningham Dance Company to perform. A close examination of recordings reveals that the same three source tapes were used not only in all performances of Untitled, but also in all performances of Toneburst after its revival. This surprising discovery, along with Tudor’s use of the linguistic indeterminacy inherent in the title of Untitled to solve a conundrum he faced in the revival, is used to depict the complex oscillation between work and performance in Tudor’s live-electronic music.

On the Complex Oscillation of Meromorphic Solutions of Nonhomogeneous Linear Differential Equations with Meromorphic Coefficients

In this paper, we study the higher order differential equation f k + B f = H , where B is a rational function, having a pole at ∞ of order n > 0 , and H ≡ 0 is a meromorphic function with finite order, and obtain some properties related to the order and zeros of its meromorphic solutions.