"Alternating positive and negative feedback produces a special form of stability represented by endless oscillation between two polar states or conditions." (John Gall, "Systemantics: The Systems Bible", 2002)
"This synergistic character of nonlinear systems is precisely what makes them so difficult to analyze. They can't be taken apart. The whole system has to be examined all at once, as a coherent entity. As we've seen earlier, this necessity for global thinking is the greatest challenge in understanding how large systems of oscillators can spontaneously synchronize themselves. More generally, all problems about self-organization are fundamentally nonlinear. So the study of sync has always been entwined with the study of nonlinearity." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)
"A moderate amount of noise leads to enhanced order in excitable systems, manifesting itself in a nearly periodic spiking of single excitable systems, enhancement of synchronized oscillations in coupled systems, and noise-induced stability of spatial pattens in reaction-diffusion systems." (Benjamin Lindner et al, "Effects of Noise in Excitable Systems", Physical Reports. vol. 392, 2004)
"In negative feedback regulation the organism has set points to which different parameters (temperature, volume, pressure, etc.) have to be adapted to maintain the normal state and stability of the body. The momentary value refers to the values at the time the parameters have been measured. When a parameter changes it has to be turned back to its set point. Oscillations are characteristic to negative feedback regulation […]" (Gaspar Banfalvi, "Homeostasis - Tumor – Metastasis", 2014)
"A limit cycle is an isolated closed trajectory. Isolated means that neighboring trajectories are not closed; they spiral either toward or away from the limit cycle. If all neighboring trajectories approach the limit cycle, we say the limit cycle is stable or attracting. Otherwise the limit cycle is unstable, or in exceptional cases, half-stable. Stable limit cycles are very important scientifically - they model systems that exhibit self-sustained oscillations. In other words, these systems oscillate even in the absence of external periodic forcing." (Steven H Strogatz, "Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering", 2015)
"Among the differences that will always be with you are the small overshoots and oscillations just before and after the vertical jumps in the square waves. This is called 'Gibbs ripple' and it will cause an overshoot of about 9% at the discontinuities of the square wave no matter how many terms of the series you add. But [...] adding more terms increases the frequency of the Gibbs ripple and reduces its horizontal extent in the vicinity of the jumps." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)
"Before considering the wave equation for mechanical waves, you should understand the difference between the motion of individual particles and the motion of the wave itself. Although the medium is disturbed as a wave goes by, which means that the particles of the medium are displaced from their equilibrium positions, those particles don’t travel very far from their undisturbed positions. The particles oscillate about their equilibrium positions, but the wave does not carry the particles along – a wave is not like a steady breeze or an ocean current which transports material in bulk from one location to another. For mechanical waves, the net displacement of material produced by the wave over one cycle, or over one million cycles, is zero. So, if the particles aren’t being carried along with the wave, what actually moves at the speed of the wave? […] the answer is energy." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)
"But the presence of √−1 (the rotation operator between the two perpendicular numbe rlines in the complex plane) in the exponent causes the expression e^ix to move from the real to the imaginary number line. As it does so, its real and imaginary parts oscillate in a sinusoidal fashion […] So the real and imaginary parts of the expression e^ix oscillate in exactly the same way as the real and imaginary components of the rotating phasor […]" (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)
"Wessel and his fellow explorers had discovered the natural habitat of Leibniz’s ghostly amphibians: the complex plane. Once the imaginaries were pictured there, it became clear that their meaning could be anchored to a familiar thing - sideways or rotary motion - giving them an ontological heft they’d never had before. Their association with rotation also meant that they could be conceptually tied to another familiar idea: oscillation." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)
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