13 Bifurcations and Chaos

Models contain parameters: a harvest rate, a damping coefficient, a reaction rate, a forcing amplitude. As a parameter varies, the phase portrait usually changes only by small deformations, but at special values it changes qualitatively: equilibria appear or disappear, exchange stability, or give birth to periodic orbits. These changes are bifurcations, and they are where interesting behaviour begins: the collapse of the fishery in Example 1.6.2, the onset of oscillation in a chemical reaction, the buckling of a beam. This section classifies the basic local bifurcations, proves the condition for their occurrence, and ends with the most dramatic qualitative behaviour a smooth system can have, deterministic chaos, which is possible only in three or more dimensions.

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