A movie showing the stretch-and-fold action brought about by shear forces. In this movie, we show both the shear forces 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 generated by system (15) 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. Since the isochrons of the system are unchanged between kicks, we observe that phase points simply traverse the isochron they are kicked to as they relax back to cycle (MOV 769 kB)
Wedgwood K, Lin K, Thul R, Coombes S
CC BY 2.0
<cite style="font-style:normal">Wedgwood K, Lin K, Thul R, Coombes S (2013). "<a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3582465">Phase-Amplitude Descriptions of Neural Oscillator Models</a>". <i>Journal of Mathematical Neuroscience</i>. <a href="https://en.wikipedia.org/wiki/Digital_object_identifier" class="extiw" title="w:Digital object identifier">DOI</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1186/2190-8567-3-2">10.1186/2190-8567-3-2</a>. <a href="https://en.wikipedia.org/wiki/PMID" class="extiw" title="w:PMID">PMID</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pubmed/23347723">23347723</a>. <a href="https://en.wikipedia.org/wiki/PubMed_Central" class="extiw" title="w:PubMed Central">PMC</a>: <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3582465">3582465</a>.</cite>