By John Edward Campbell

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**Extra info for A Course of Differential Geometry**

**Example text**

One way to verify and exhibit this effect is to make a plot similar to the Poincar´e section in Fig. 7 but with v plotted versus the logarithm of time rather than versus x at those times when z (mod 2π) has a particular value such as π/2. This amounts to projecting the Poincar´e section onto the v-axis so that a set of points projects onto another set of points. Any section with a dimension greater than or equal to one will project onto a line. Thus, if the attractor is a limit cycle with a dimension of one, its Poincar´e section has a dimension of zero, whereas a chaotic system will have a dimension greater than two with a Poincar´e section that projects onto a line.

1994), but for the present purpose they are deemed less elegant than systems that involve functions with only two linear regions and can thus be represented by a single absolute value nonlinearity. 7 Signum Oscillators Whereas the absolute value is a continuous piecewise-linear function, discontinuous functions sometimes arise in models of real applications. An example is a harmonic oscillator with Coulomb damping, as would result from a friction force that is independent of velocity and position, like sliding friction and unlike air resistance, which increases with velocity (Den Hartog, 1930; Squire, 1985; Peters and Pritchett, 1997).

The minor axis of this ellipse expands (or more likely contracts) at a rate given by the third Lyapunov exponent λ3 . From the definition, it is clear that λ1 ≥ λ2 ≥ λ3 , and this convention is universal. As an example, Fig. 18 shows a succession of Poincar´e sections √ for the conservative forced pendulum in Eq. 02 containing 106 initial conditions distributed throughout its volume. Even in a single transit (N = 1, corresponding to a time lapse of ∆t = 2π), the ball has distorted into a highly elongated but very thin ellipsoid.

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