mathematics//dynamical systems//equilibrium and stability//repeller

A repeller is an equilibrium, or a set of states, that every nearby trajectory moves away from: a pencil balanced on its tip, a ball on top of a hill, a bicycle standing still. The system can rest there exactly, since the forces balance, but the smallest push grows, so it is never seen there for long unless something actively holds it. Holding a system at an unstable equilibrium is one of the main jobs of control: a self-balancing scooter, a rocket standing on its thrust, a fighter aircraft built unstable for agility.

A repeller is an equilibrium, or a set of states, that every nearby trajectory moves away from: a pencil balanced on its tip, a ball on top of a hill, a bicycle standing still. The system can rest there exactly, since the forces balance, but the smallest push grows, so it is never seen there for long unless something actively holds it. Holding a system at an unstable equilibrium is one of the main jobs of control: a self-balancing scooter, a rocket standing on its thrust, a fighter aircraft built unstable for agility.

For a linear system an equilibrium repels when every eigenvalue has a positive real part. It is then a source, called an unstable node when the eigenvalues are real and an unstable spiral when they are complex and the motion swings out in growing oscillations. When some eigenvalues are positive and others negative, the equilibrium is a saddle: trajectories come in along one direction and are thrown out along another. The inverted pendulum is a saddle, not a pure repeller.

Reversing time turns a repeller into an attractor: the trajectories are the same curves run the other way. Numerically this is how unstable equilibria and orbits are located, by integrating the equations backwards in time.

Feedback can move the eigenvalues of an unstable equilibrium into the left half of the complex plane and make it an attractor of the closed loop. The plant stays unstable; the loop does not, and it stays stable only for as long as the controller, the sensor and the actuator keep working (feedback control).

Saddles organise a phase portrait as much as attractors do. The curves that run into a saddle, its stable directions, are the borders between basins of attraction, the separatrices: two starting points on either side of one end on different attractors (attractor).