A movie showing the accumulation of folds in the kicked ML model. The thin black line represents the underlying periodic orbit of the system x˙=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. 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 x˙=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 (MOV 2118 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>