sync

Author:Steven H Strogatz
Language: eng
Format: epub
Published: 2014-12-17T16:00:00+00:00

The conclusion, then, is that the state space for this simplest of Josephson arrays is equivalent to the surface of a torus. Every point on the torus corresponds to an electrical state of the array, and vice versa. As time passes and the array changes its state from moment to moment, the corresponding point on the torus glides smoothly from place to place, like a speck of dust carried on the surface of a gentle stream. The flow pattern for this imaginary stream—its whorls and eddies, backwaters and torrents—are all inherent in the circuit equations for the array. Given the present values of the phases, the equations dictate how they will change in the next instant.

The equations are nonlinear, so we couldn't hope to solve them explicitly, but we thought it might be possible to infer the overall qualitative features of the flow pattern. For example, stagnation points (places on the torus where the speck gets stuck) would correspond to states of electrical equilibrium for the array, with all currents and voltages constant in time. The stability of such states could be assessed by imagining the speck to be nudged away from them; if it always returns, as if it were being sucked down a drain, the equilibrium state is stable. Or suppose the flow pattern contains a closed loop, an eddy around which a speck can circulate endlessly, always revisiting its starting position after a certain amount of time. Such a loop would signify a periodic, repetitive form of behavior—an electrical oscillation in the array. Kurt and I knew that such loops were bound to occur, but we didn't know anything about their stability, whether they'd funnel nearby states into themselves or not.

The simplest loop is the synchronous oscillation, where the phases of both junctions are equal at all times. The corresponding trajectory flows along the main diagonal of the square. It starts in the lower left corner, then travels northeast until it exits at the top right corner, where it instantly returns to the lower left (since 360 degrees and 0 degrees correspond to the same phase). When viewed on the square, the trajectory appears to jump discontinuously from one corner to the other, but on the torus—the true state space for the system—there is no jump. The transition is seamless.

When we analyzed the overall flow pattern, we were shocked to find that every other trajectory repeats itself in a similar way. Every solution is periodic. On the face of it, that might not sound so surprising. A pendulum swinging to and fro would always repeat its behavior, at least in the simple textbook case where there's no friction in its bearings and no air resistance. In that case, it doesn't matter whether you start the pendulum swinging in a large arc or a small one—either way, it always repeats. The same is true for other kinds of "conservative" mechanical systems, hypothetical idealizations where all forms of friction and dissipation are imagined to vanish, and mechanical energy is perfectly conserved, with none lost to heat.

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