Geometry of folding in relation to the -foliation. Images of and are shown, the kick being indicated by black arrows, as are and for , both multiples of the period of the limit cycle .Since , the action of where is a multiple of the period of amounts to sliding down a -leaf. Thus for the kick shown, the folding of the limit cycle under is very much in evidence.
Strong stable foliation for (green), the limit cycle (blue), and kicked images of the cycle of size (respectively, dashed red and black). Our time parametrization of the limit cycle is also shown. It is evident that smaller kicks will not lead to interesting behavior, as the kick map followed by projection along stable leaves sends the limit cycle bijectively onto itself. Kicks that are too large (not shown) are also not likely to produce chaotic behavior. Kicks in between, especially those that send the limit cycle near the origin, are more likely to produce horseshoes.
Here we show the lift of for , 2.2, 2.5, 2.6, 2.65, 2.8, 3.2, and 6 as a mapping from to . For , the map is monotonic (plot1). As increases, a local minimum develops, the graph acquiring a sharp “cusp-like” turn. This “cusp” becomes more and more pronounced as increases (plots 2-4). Shortly after the kick-image of the limit cycle clears the origin, changes from a degree 1 map to a degree 0 map (this happens between plots 4 and 5). As continues to grow, the cusp slowly smooths itself out (plots 6-8).
(a) The region R (cyan) is mapped by into a horseshoe. Two“horizontal” strips, and (blue), which together comprise , are shown, as are the two “vertical” strips and (magenta), the union of which is . Under is mapped onto , while is mapped onto . (b) Stable (red) and unstable (blue) cones for and .
Graphs of (blue) and (cyan). Also shown are the boxes (larger) and (smaller). Notice that is strictly larger than , with a small piece sticking out at the bottom, while is contained strictly inside .
Foliation and limit cycle for (top left), (top right), (bottom left). The bottom right plot shows the limit cycle together with the slow manifold for . This plot is shown in coordinates, with the slow manifold being given by the curve .
Evidence for horseshoe for , 3, and 4. (a) The left panel shows the limit cycle and (red) and (cyan) for . The right panel zooms in to near 0, with the x–coordinate magnified by 100 while maintaining the direction of weaker expansion parallel to that shown (in pink) in the upper right corner. Any horizontal line segment between y = 0 and y = 0.09 will meet at least twice and at least once, with the meeting point with the latter sandwiched between the first two. We argue that this will lead to a horseshoe (see text). (b) Corresponding plots for , except that now the horizontal magnification is 2000. The range of y for a horizontal line segment to have the same intersections as above is now twice as large, ranging from 0 to 0.18. The two plots on the right show and for , suggesting a similar “handshake” between the two curves as in (a) and (b).
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