Periodic orbit vs. limit cycle
In a dynamical system, a solution can return to its starting state and go around the same loop again and again. A periodic orbit is such a loop. A limit cycle is a periodic orbit that is isolated, with no other periodic orbits nearby, so that nearby solutions spiral toward the limit cycle or away from it.
limit cycle ⊂ periodic orbit
Definitions
Level:
Missing definition: periodic_orbit
Missing definition: limit_cycle
Examples
| Example | periodic_orbit | limit_cycle | Why |
|---|---|---|---|
| The unit circle, for $\dot{x} = y$, $\dot{y} = -x$ (the harmonic oscillator) | ✓ | ✗ | The unit circle is a periodic orbit of the harmonic oscillator but not a limit cycle, because every circle around the origin is also a periodic orbit, so the unit circle is surrounded by other periodic orbits. |
Missing example: the_limit_cycle_of_the_van_der_pol_oscillator | |||
Missing example: the_circle_r_eq_1_for_rdot_eq_r_r_minus_1_dot | |||
Notes
- A limit cycle can be stable, unstable, or semistable (attracting from one side and repelling from the other).
- Linear systems have no limit cycles, because a multiple of a periodic solution of a linear system is another periodic solution, so periodic orbits of linear systems are never isolated.
- Conservative systems, such as the frictionless pendulum, have families of periodic orbits, and limit cycles arise in systems that gain and lose energy, such as the Van der Pol oscillator.