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Periodic, nonperiodic, and chaotic solutions for a class of difference equations with negative feedback

Kennedy Benjamin B.

Opuscula Mathematica

2023

Vol. 43, no. 4

507--546

We study the scalar difference equation [formula], where f is nonincreasing with negative feedback. This equation is a discretization of the well-studied differential delay equation x′(t) = f(x(t − 1)). We examine explicit families of such equations for which we can find, for infinitely many values of N and appropriate parameter values, various dynamical behaviors including periodic solutions with large numbers of sign changes per minimal period, solutions that do not converge to periodic solutions, and chaos. We contrast these behaviors with the dynamics of the limiting differential equation. Our primary tool is the analysis of return maps for the difference equations that are conjugate to continuous self-maps of the circle.

Stabilizator prądu termoemisji elektronowej dla wysokich energii elektronów

Sikora J., Szczepaniak L.

Przegląd Elektrotechniczny

2008

R. 84, nr 3

273-274

W pracy jest przedstawiony układ stabilizacji prądu termoemisji elektronowej z detekcją prądu termoemisji w obwodzie zasilania katody, przeznaczony zwłaszcza dla wysokich energii elektronów.

Electron emission stabilizers are widely used in many vacuum devices with a hot filament, such as ionization gauges, mass spectrometers, electron microscopes. The reported electron emission stabilizers cannot be easily applied in co-operation with high accelerating voltage sources because of the use of convenient negative feedback loop (NFL) connecting a filament supply circuit with an anode supply circuit. This work describes an emission current stabilizer with an emission current detecting system operated in a filament supply circuit and generated a signal of negative feedback. Consequently, the anode is not connected to NFL but only to the accelerating voltage source, which makes the use of high accelerating voltage simple and convenient.