^{1,a)}and Lai-Sang Young

^{1,b)}

### Abstract

This paper contains a numerical study of the periodically forced van der Pol system. Our aim is to determine the extent to which chaotic behavior occurs in this system as well as the nature of the chaos. Unlike previous studies, which used continuous forcing, we work with instantaneous kicks, for which the geometry is simpler. Our study covers a range of parameters describing nonlinearity, kick sizes, and kick periods. We show that horseshoes are abundant whenever the limit cycle is kicked to a specific region of the phase space and offer a geometric explanation for the stretch-and-fold behavior which ensues.

We revisit a much studied situation, namely the periodically forced van der Pol system, and offer new insight into how chaotic behavior is created. We find that when the sinusoidal forcing in the original problem is replaced by periodic kicks, the geometry becomes more transparent. Results of a numerical study of such systems are reported in this article. Our aim is not to conduct a detailed study of a specific regime, but to investigate the extent to which chaotic behavior occurs, the conditions under which it occurs, and the dynamical mechanisms that lead to horseshoes and strange attractors—as parameters describing nonlinearity, kick sizes, and kick periods are varied. Following a strategy proposed in earlier work, we use the geometric relation between the kicked cycle and the strong stable foliation of the unforced system as a tool for detecting chaos and are able to identify the source of much of the potentially chaotic behavior. We find that when the limit cycle is kicked to a neighborhood of the origin, horseshoes occur for substantial ranges of parameters. The invariant Cantor sets are, however, highly localized, as they are defined by extremely strong expansion on very small regions of the phase space, with most other orbits of the periodically kicked vdP system tending to sinks. Thus, geometric complexity is clearly present, though it affects only a rather small set of initial conditions.

L.S.Y. was supported by NSF Grant DMS-1101594.

I. INTRODUCTION

II. STRATEGY FOR DETECTING CHAOTIC BEHAVIOR IN KICKED OSCILLATORS

III. DETECTING CHAOS IN THE vdP SYSTEM

A. Reduced 1D maps for

B. Candidate parameters for horseshoes and attractors

IV. DETAILS FOR A CASE STUDY

A. Numerical confirmation of horseshoes

B. Partial confirmation for attractors

V. INCREASING NONLINEARITY

A. Geometry of strong stable foliations

B. A conjecture for all

VI. SUMMARY AND DISCUSSION

## Figures

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.

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.

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).

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 .

(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 .

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 .

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).

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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