A dynamic system that leaves itself long enough for itself often comes into a state of equilibrium (for example, a rolling ball comes to a halt). It can also come in a periodic "state" (for example, the clockwise of a clock). The path that the system then describes in the state space is called a stable limit cycle. The third possible state is chaos (for example, the atmosphere). Characteristics
A limit cycle therefore has two essential properties. First, it is a periodic state of the system. In addition, there is a stable limit cycle that converges the system to the bicycle for all starting conditions near the bicycle. When the system removes from the bike, one speaks of an unstable limit cycle. The stable limit cycle thus has a convergence area. That consists of all states from which the system converges to the bike.
The convergence area never covers the entire state space. Within each bicycle there is always at least one equilibrium. This condition is usually unstable and gives the system a special feature. If the system is in this equilibrium, it will remain of course. But, the smallest disturbance is enough to make it converge to the stable limit cycle. You can imagine this as the pendulum of the clock that stems exactly in the middle. A small tick is enough to make the clock run.
Limit cycles arise from equilibrium states by so-called Hopf bifurcations. In such bifurcation, the equilibrium state becomes unstable because you change a parameter of the system. At that moment the limit cycle is created. You can easily imagine this when spinning the loop (parameter change). If the clock is sufficiently excited, the clock can tap. The disappearance of limit cycles occurs in the same Hopf bifurcation, but in reverse order.
The limit cycle is a central concept of bifurcation theory (dynamic system theory, chaos theory) and plays an essential role in the emergence of chaos. The complexity of limit cycles increases (or decreases) by period doubling bifurcations. When these period doubles accumulate, chaos eventually results.
A limit cycle can not cut itself (in which direction should the system continue?). Obviously, a system with limit cycles has at least two variables, otherwise images in the flat plane do not work. It is less obvious that generally, a limit cycle is displayed with two well-chosen system variables. Examples Stable limit cycle of a sling bell. The pendulum always gets a push on the way. Signed are: the limit cycle (yellow) and two paths that converge to the limit cycle (blue and rose).
There are many elegant dynamic systems with limit cycles. Less elegant, but easy to understand is the loopwheel. The pendulum of such a clock makes a periodic movement and continues to follow as long as there is energy.
The state of this system is described by two variables that change in time: the deviation (x) and the velocity (v) of the pendulum. Simply simulate this system with a muted harmonic oscillator that gets a push in a fixed place in its job (on the way). The result is shown in the figure. The two variables (x and v) go through a distorted circle in the state space (yellow line). This is the system's cycle cycle. If you follow the bike then x will increase and decrease periodically. This describes the movement of the pendulum.
The bike is stable. From a state within the cycle, the system converges to the cycle (dark blue line). This is the case when the pendulum first became (almost) quiet. If the amplitude of the pendulum motion increases for any reason (e.g., someone bumps counterclockwise), the system converges back to the bicycle (rose line).
Other examples that work in the same way are: striking your heart, moving a tree in the wind, or the emergence of a vortex.
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