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LC oscillators. The 9th publication. nanyoly mendez. EES Section 1. 3rd part

COUPLED OSCILLATORS

For mathematicians, the simplest description is when each oscillator affects everyone else in the system and force coupling increases with the phase difference between oscillators. In this case, the interaction between two oscillators moving in sync is minimal.

The timing is the most familiar in the organization of coupled oscillators, one example is the case referred to the male fireflies, which are struggling to attract females crossing above. Males gather at dusk, and between the flashes at the beginning are not coordinated, closed at night begin to form and grow core of sync. Finally, whole trees come to click on a concert, silent and hypnotic "for hours.

But this large-scale coupled oscillation has long resisted the attempts of mathematical analysis. Fireflies are an oscillating system paradigm "coupled impulses" (the interaction takes place at the sudden flash of another firefly which modify each its own pace). This coupling pulse is very common in biology, such is the synchrony the chirping of crickets or the connection through the neuronal action potentials (voltage spikes). Steven Strogatz created an idealized mathematical model for the fireflies and other pulse-coupled systems. That showed that in certain circumstances, oscillators implemented in different times just always synchronized. The show is based on the notion of "absorption", shorthand for the idea that if an oscillator drives another above the threshold, they remain synchronized forever, that is, a sequence of absorption processes and to eventually concatenate together in solidarity to all oscillators.

But the timing is not inevitable, as it often does not reach coupled oscillators synchronize. And the explanation lies in the symmetry breaking, which means that a symmetric state individually as synchrony is replaced by several states less symmetrical, but taken together they refunded the original symmetry. In reality, coupled oscillators are a great source of symmetry breaking.

The timing is the most obvious case of the concatenation of phases, in which many oscillators are molded to the same pattern, but not necessarily in one step. By coupling two identical oscillators it is possible synchrony (Phase difference equal to zero), and antisincronía (phase difference of half a cycle). A biological example is the kangaroo hop along through the plains of Australia: their hind legs oscillate periodically and hit the ground at the same time. But if a man runs after the kangaroo, his feet touch the ground alternately. Clearly, the range of possibilities expands dramatically when the network consists of many oscillators.

Ian Stewart in 1985 developed a mathematical classification of patterns of networks of coupled oscillators. The classification is to combine the theory of groups (symmetries of a collection of objects) with the call Hopf bifurcation (an overview of how the oscillators are "set up").

The equations that describe certain systems behave in a peculiar way when the system is removed from its point of rest. Instead of a slow return to equilibrium or quickly away from him toward instability, range. The point where this transition occurs is called bifurcation point because the behavior of the system is divided into two branches: an unstable resting state coexists with a stable oscillation. Hopf showed that systems whose linearized form is experiencing such kind of forks are limit cycle oscillators (having an amplitude and waveform feature).

Nanyoly Mendez

Solid-State Electronics Section 1