equilibria: stable & unstable

dynamical systems begins with the notion of equilibria — states that remain constant over time. In the simplest cases, one distinguishes between stable and unstable equilibria, though to specify what exactly these mean requires care (and math).

one simple example of stable and unstable equilibria is found in a rigid-rod pendulum, rotating about a pivot point (with or, as here, without, friction). letting the system evolve from a variety of initial conditions reveals a stable equilibrium (bottom) and an unstable equilibrium (top).

stable & unstable equilibria in a rigid-rod pendulum.

there’s a lot to discuss about this system. skipping ahead (for the sake of seeing something wonderful), let’s consider what happens when you give this system a shake, vibrating the pivot point up & down. the simulation indicates that past a criticla threshold of amplitude/frequency, shaking the pendulum converts the unstable equilibrium to a stable equilibrium.

shaken, not stirred.

that’s swell. but does the mathematics prove that this is the expected behavior? well, that’s one of the reasons why you take a dynamical systems course.

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