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People Bouncing on Trampolines: Dramatic Energy Transfer, a Table-Top Demonstration, Complex Dynamics and a Zero Sum Game

Figure 7

Stability of symmetric passive bouncing.

The four eigenvalues of the Jacobian (Floquet multipliers) corresponding to the mapping of the state over one period of the periodic motion: symmetric bouncing for symmetric masses. The product of the eigenvalues was equal to 1 (with an error of about ). Two eigenvalues are equal to . In the intermediate regime shown, all four eigenvalues, two of which are complex conjugates and reciprocals of each other, have unit absolute values. At other regimes, one eigenvalue has magnitude greater than one, implying linear instability.

Figure 7