Kyle C. A. Wedgwood,

The top panel shows a projection of the moving orthonormal system from the full (v,u,a,b) space onto (v,u,b). Around the point Γ(θ), where θ is the phase, we establish an orthonormal basis in a subspace of (v,u,b). As θ evolves, so does this coordinate system, as shown by the moving black lines, which represent the moving orthonormal basis. In this movie, we choose some initial conditions off-cycle, shown by the blue orbit. The ρ coordinates along the moving coordinate system are shown in the bottom panel.

In this movie, we show the case in which both the shear force and the rate of attraction back to cycle are linear. The limit-cycle is first unravelled so that it may be represented by a straight line. We choose P(θ)=sin(θ) as our forcing function and apply it instantaneously at t=0. We then allow the resulting image of the kicked orbit to evolve under the flow between kicks until t=1 (in arbitrary units). As the curve relaxes back to the cycle, we see that the shear forcing causes a fold in the curve to develop. The accumulation of such folds over successive forcing periods can ultimately give rise to chaotic dynamics, which would not be observed in the corresponding phase-only model. The thinner black lines represent the isochrons of the system which, in this simple example, are straight lines with slope -λ/σ.

The red curve represents the underlying periodic orbit of the system ẋ=f(x), with f taken for the ML model. Every T units of time, we apply a kick taking v → v-A, where A=2.0, whilst leaving w unchanged, to all phase-points, giving the blue curve. The movie then shows the evolution of all of these phase-points. Please note that this movie does not show trajectories of the system, but the image of points starting on the limit cycle, under the action of the kick composed with the flow generated by ẋ=f(x). This movie show the action of 4 such kicks. We observe that trajectories spend a long time near the saddle node to the bottom left of the figure, so that these trajectories travel slower than those close to the limit cycle. As we apply more kicks, we see the folds developing and accumulating.