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. 2012 Jun 7;7(6):e38402. doi: 10.1371/journal.pone.0038402

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Wu et al.

This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are properly credited.

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(A) Establishment of a coordinate system on the half line with the origin . Here, is the equilibrium point and is transversal to the vector field in the neighborhood of . Note that both and depend continuously on ; (B) Curves of the Poincaré map . Each intersection between the curves and the black line corresponds to a fixed point of as well as to a limit cycle of the system (8). For Hz, the curve has no intersection with the black line, so that there is no limit cycle. At higher values of , the curve moves upward; it first intersects with the black line at , where a single semi-stable limit cycle emerges. As increases to Hz, two bifurcated limit cycles appears. Here, one cycle is stable characterized by the quantity at one fixed point, and the other cycle is unstable with the quantity at the other fixed point.

Inline graphic — (A) Establishment of a coordinate system on the half line with the origin . Here, is the equilibrium point and is transversal to the vector field in the neighborhood of . Note that both and depend continuously on ; (B) Curves of the Poincaré map . Each intersection between the curves and the black line corresponds to a fixed point of as well as to a limit cycle of the system (8). For Hz, the curve has no intersection with the black line, so that there is no limit cycle. At higher values of , the curve moves upward; it first intersects with the black line at , where a single semi-stable limit cycle emerges. As increases to Hz, two bifurcated limit cycles appears. Here, one cycle is stable characterized by the quantity at one fixed point, and the other cycle is unstable with the quantity at the other fixed point.