Jeremie Sagarra & Alexandre Perchat










In this workshop we will examine the topological structure of chaotic attractors in forced oscillators. The most important examples are the Van der Pol equation with nonlinear damping, and oscillators of the Duffing type having a nonlinear restoring force.


We consider various potential functions V(X) leading to

EQ 1: X'' + k X' + grad V = A sin wt

For the linear oscillator this potential is the harmonic potential V(X)= 0.5 *a* X2 proportional to the displacement squared. The forced oscillator with harmonic potential gives only periodic motion in the steady state, but various anharmonic potentials -corresponding to nonlinear restoring force- can lead to chaotic steady-state response. Here we consider qualitatively different shapes of anharmonic potential. we shall, for example, be concerned with whether the restoring force grad V increases more or less rapidly than a linear function of displacement (stiffening or softening spring, respectively), so that the potential rises more or less steeply than a quadratic. But we shall not be concerned with how much more or less rapidly; that is, we assume that the quantitative difference between cubic and quintic restoring force leads to mainly quantitative, not qualitative, differences in the observed dynamics and steady-state response.



The qualitative effect of a softening spring can be studied in the potential V(X)= 1/2 X2 - X4 , illustrated in the figure below:

FIG 1: An anharmonic potential with softening

Here the potential shape is like the harmonic potential for small displacement, but at larger displacement the restoring force is weakened. At the two smooth maxima of the potential, the restoring force is weakened. At the two smooth maximaof the potential, the restoring force becomes zero, and for larger displacements a trajectory can diverge to infinity, but we will only consider trajectories that remain within the potential well. In particular, oscillations that visit the portion of the well just inside the maxima experience the softening spring effect most strongly.

A chaotic steady-state response under these conditions was discovered by Huberman ad Crutchfield (1979). The figure below shows the exemple of a steady-state trajectory in the phase plane above, and below are eight Poincare sections of the chaotic attractor.

FIG 2: Forced oscillaions in an anharmonic potential with softening spring effect. Above, a single trajectory; below eight Poincare sections through one cycle.

The equation used to calculate the pictures is:

EQ 2: X'' + 0.4 X' + X - 4*X3= 0.1185 Sin 0.555t

and in th ephase plane, Y = X'. This chaotic response occurs in a rather small region of control space, perhaps because the softening spring effect in this potential is noticeable only in a narrow range of response amplitudes.

This chaotic response coexists with a smaller amplitude periodic limit cycle at the same control parameter values. AThe peridic motion is a chaotic one, in wich the peak-amplitude varies from one cycle to the next. The fluctuations in the peak amplitude are not large, but their size is unpredictable over long times.

For any trajectory follow the evolution from one starting point through precisely one circle of sinusoidal driving term wich is 0.1185 Sin 0.555t. The Poincare sections in the lower part of FIG 2 show that, in spite of this randomness, there is again a simple underlying structure. In fact, this is topologically the simplest chaotic attractor. The eight sections shown are taken at eight equally spaced angles through one complete cycle of the sinusoidal driving function, and sections progress clockwise in the plane. Most of the folding action occurs as the attractor passes the x axis, and is more easily seen in the three closely spaced Poincarre sections in FIG 4, viewed close up.

FIG 3: Close-up view of three successive Poincarre sections of the atractor of the forced oscillator equation 2, showing (top to bottom) the folding action.

The earliest section is on top; the middle section is the same as the one near the positive x axis in FIG 3; by the time of the bottom section, folding is well advanced. Imagining the three-dimensional bundle of steady-state trajectories in the (x, x',t) phase space, we see it forms an apparent ribbon or band which is smoothly and simply folded onto itself. This folded band is the simplest structure for a chaotic attractor; its experience was established by Rossler in simulation of a very different but equally simple system of ordinary differential equations.

The simple folding action repeated infinitely produces a fractal structure; but the folding brings layers together so quickly that the fractal structure is not visible without magnification. This rapid compression of layers allows us to more readily recognize the sim[;city of the underlying folding action.

As confirmation of the qualitative connection between a softening spring effect and the folded band chaotic attractor, we consider the potential V(X) = - 1/2 X2 +1/4 X4 , which models a vertical Euler support column loaded beyond its buckling point.

A small externalvibration converts these point attractors to limit cycle attractors in the forced system. For larger driving amplitude and cetain frequencies, such as in

EQ 3: X'' + 0.25 X' - X + X3 = 0.0191 Sin t

we find a small-amplitude limit cycle competing with a larger-amplitude chaotic motion, as illustrated in figure 4 in the left half of the phase plane.

FIG 4: Small-amplitude periodic attractor coexisting with a larger-amplitude chaotic steady state in a two-well potential oscillator, equation 3, representing a vibrating buckled beam.

The steady chaotic motion has an amplitude almost but not quite large enough to cross the potential barrier that seperates the two wells. By symmetry the same small limit cycle and large chaotic oscilation exist on the right side of the phase plane; here instead FIG 4 shows a sequence of four Poincarre sections of the chaotic attractor. These sections indicate that the chaotic atractor has the same simply folded band stucture as in equation 2. note however that the folding in figure 4 occurs only as the band approaches small displacement, which for equation 3 is where the softening spring effect occurs. Near maximum displacement, the buckled beam exhibits a hardening spring effect, and there is no folding as the attractor in FIG 4 passes maximum displacement. Thus we may associate the simple folding of FIG 2,3,4 with the softening spring effect, and expect that, at least for certain forcing amplitudes and frequencies, other softening spring oscillators may be found to have chaotic attractors. Indeed one has recently been identified by Virgin (1986) that leads to capsize of a vessel in regular.



The opposite of a softening spring is a stiffening spring, in which the restoring force rises more rapidly than a linear function of the displacement, and the potential rises more rapily than the harmonic potential V(X)=1/2aX2.

The engineer used to dealing with practical problems might be concerned by the case of zero linear stiffness, corresponding to loading a column precisely to the buckling point, is somehow atypical of stiffening springs generally. Some reassurance is offered in figure 5, which shows a poincare section of the chaotic attractor computed from :

EQ 4: X'' + 0.05 X' + aX + X3 = 7.5 Sin t

where a = -0.2 on the left, a = 0 in the middle and a = 0.2 on the right. The effect on the chaotic attractor of introduction a small linear stiffness is difficult to detect. In fact, even for larger amounts of linear stiffness the general appearance of figure 3 would change mainly by distotions and displacements rather than by disappearance of the various response regions. For this region we associate the chaotic responses with the stiffening spring effect.

FIG 5: Poincarre sections of the forced Duffing-type oscillator equation 4 for different values of the linear stiffness. Left, a=-0.2; middle, a=0, right, a=+0.2

FIG 6: Poincarre sections of the forced Duffing oscillator. With increasing damping from left to right and top to bottom.

Within each of the control-space regions of chaotic response in the Figure there may be considerable variation in the structure of the attracting set. This is illustrated in the figure, which shows Poincare sections of chaotic attractors. The form with the simplest appearance occurs with the largest value of damping. The overall folding action in three dimensions is similar in all three cases, but with higher damping the layers formed by folding are more rapidly compresses together. Thus the succession of layers remains more visible with less damping. This illustrates a more general principle that the folding and mixing action of a chaotic oscillator can be seen in simplest form with maximum dissipation.

The folding action in region with large damping turns out to be similar to the folded band that we associated with the softening spring. We note that in the flattened S-shape appearance of the highlyd amped case, two successive folds are visible. The most recent fold produced the more open bend of the S, while the fold during the previous forcing cycle produced the bend that appears more tightly compressed.



Another important and qualitative different anharmonic oscillation occurs when the potential has more than one well.One example is the Euler column loaded past the buckling point, with V(x) = - 1/2 ax2+ bx4 and two potential wells. In addition to the chaotic response associated earlier with the oscillator jumps back and forth between between the two wells in an erratic manner. An important example of multiple potential wells is the periodic potential V(x) = -cos x, an infinite lattice of wells having minima at x = 2 * pi *n, n = ..., -2 ,-1 ,0 ,1 , 2 ...This potential models the nonlinear oscillations of a pendulum and is of interest in the study of Josephson junctions and of charge-density waves in plasma.

FIG 7: A steady-state chaotic trajectory in the phase plane of the vibrating buckled beam equation 5 below with negative linear stiffness and moderately large forcing amplitude.

The figure 7 shows a chaotic trajectory of the two-well potential oscillator :

EQ 5: X'' + 0.4 X'-X + X3 = 0.4sint in the (X, X' ) phase plane.

Successive Poincare sections of this attractor, and of a chaotic attractor of the equation 6: X'' + 0.5 X' + sin X = 1.1sin 0.5t are shown progressing top to bottom in the figure 8; in each case the sections are for equally spaced angles through one half of a forcing cycle.

Beause of a symmetry in the 2 last equations under the transformation :

X ---> - X X' ---> - X' t ----> t + pi / omega

the section at the bottom of each column that is exactly one half-cycle advanced from top section is also identical to the top section rotated 180 degrees about the origin in yhe ( X, X' = Y ) phase plane . The second half of the forcing cycle is therefore described by rotating each picture in a column by 180 degrees. Note that with the periodic potential of the last equation the phase space is periodic in x as well as in t. In the left column of the figure 8 the infinitely many identical intervals 2*pi*n < x < 2*pi*(n+1) are all plotted as one interval 0 < x < 2*pi; trajectories above the x axis may move off the picture to the right and re-enter at the left edge and conversly.

FIG 8: Poincarre sections of two forced oscillators, progressing top to bottom with time through one half-cycle. Left column, the preiodic potential, eq. 6, right column, the two-well potential (buckeled beam), eq. 5

Chaotic attractors of thes types were first reported by Huberman (1980) for the infinite well , and by Moon and Holmes (1979) in the two well case. We have chosen parameters for figure 8 involving larger dissipation in both cases, thus gaining more compression of the fractal layers and a simpler picture. This helps reveal the similarity of the two well and infinite well attractors.

The folding action of the two-well attractor is carefully analysed in Holmes and Whitley (1983). Continuous animation of the infinite well attractor can be seen in the movie of Crutchfield (1984). A detailed control-space diagram of the infinite-well pendulum oscillator was obtained by MacDonald and Plischke in the amplitude-frequency control plane, with a fixed value of damping somewhat lower than we have chosen. Their control-space diagram is highly complex, perhaps due in part to the smaller dissipation chosen by them.

As holmes has observed, the attractor of the infinite-well potential shown in the left column of figure 8 also has some claim to being called a Birkhoff attractor.