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Three Frogs oscilator

By Snow_Trash on Feb 5, 2020
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8.6.9 (Japanese tree frogs) Many thanks to Bard Ermentrout for suggesting the following exercise. An isolated male Japanese tree frog will call nearly periodically. When two frogs are placed close together (say, 50 cm apart), they can hear each other calling and tend to adjust their croak rhythms so that they call in alternation, half a cycle apart—a form of phase-locking known as antiphase synchronization. So what happens when three frogs interact? This situation frustrates them; there’s no way all three can get half a cycle away from everyone else. Aihara et al. (2011) found experimentally that in this case, the three frogs settle into one of two distinctive patterns (and they occasionally seem to switch between them, probably due to noise in the environment). One stable pattern involves a pair of frogs calling in unison, with the third frog calling approximately half a cycle out of phase from both of them. The other stable pattern has the three frogs maximally out of sync, with each calling one-third of a cycle apart from the other two. Aihara et al. (2011) explored a coupled oscillator model of these phenomena, the essence of which is contained in the following systems for two frogs, θ1 = ω + H (θ2 − θ1 ) θ = ω + H (θ − θ ), 2 1 2 and three frogs, θ1 = ω + H (θ2 − θ1 ) + H (θ3 − θ1 ) θ = ω + H (θ − θ ) + H (θ − θ ) 2 1 2 3 2 θ3 = ω + H (θ1 − θ3 ) + H (θ2 − θ3 ). Here Ri denotes the phase of the calling rhythm of frog i, and the function H quantifies the interaction between any two of them. For simplicity we’ll assume all the frogs are identically coupled (same H for all of them) and have identical natural frequencies X . Furthermore, assume that H is odd, smooth, and 2Q-periodic. a) Rewrite the systems for both two and three frogs in terms of the phase differ- ences φ = θ1 − θ2 and ψ = θ2 − θ3 . b) Show that the experimental results for two frogs are consistent with the simplest possible interaction function, H ( x )  a sin x , if the sign of a is chosen appro- priately. But then show that this simple H cannot account for the three-frog results. c) Next, consider more complicated interaction functions of the form H ( x ) = a sin x + b sin 2 x . For the three-frog model, use a computer to plot the phase portraits in the (φ, ψ ) plane for various values of a and b. Show that for suitable choices of a and b, you can explain all the experimental results for two and three frogs. That is, you can find a domain in the ( a, b ) parameter space for which the system has:

i) a stable antiphase solution for the two-frog model; ii) a stable phase-locked solution for the three-frog model, in which frogs 1 and 2 are in sync and approximately Q out of phase from frog 3; iii) a co-existing stable phase-locked solution with the three frogs one- third of a cycle apart. d) Show numerically that adding a small even periodic component to H does not alter these results qualitatively. Caveat: The three-frog model studied here is more symmetrical than that con- sidered by Aihara et al. (2011). They assumed unequal coupling strengths because in their experiments one frog was positioned midway between the other two. The frogs at either end therefore interacted less strongly with each other than with the frog in the middle.

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