Limit cycle
A limit cycle or limit cycle is in mathematics and the theory of dynamical systems is an isolated periodic solution of an autonomous differential equation system.
Looking at the solutions of the differential equation system as curves in the phase space, the limit cycle is a closed curve ( cycle) on the neighboring trajectories in the limit of infinite time to run either or removed. If neighboring solutions are also periodic solutions themselves, so it is no limit cycle, since it is not an isolated periodic solution.
Run adjacent trajectories in the limit of infinite time to the limit cycle, the limit cycle is a one-dimensional attractor and is called stable. The other hand, removing neighboring trajectories in the limit of infinite time (or run in the limit infinite negative time to the limit cycle ), then the limit cycle is called unstable.
Introduction by two examples
Dynamic systems are systems of autonomous differential equations of the form:
The solution of these differential equations are called trajectories and describe the behavior / development of the system in time t. In the theory of dynamical systems, the asymptotic stability of such solutions, ie their behavior in the temporal limit of interest. If, now in the limit oscillatory solutions of the system, as they are reflected in the cyclical, closed curve in the phase space resist. It is called the limit cycle. If there are other trajectories (with different initial conditions ) of this closed curve for large times t ever closer ( " into spirals " ), so it is an attractor ( " attractor ") or stable limit cycle. A classic example is the Van der Pol oscillator, the phase space portrait is shown in the figure above. To remove all trajectories for the limit cycle, it is called unstable or repeller.
A simple mathematical pendulum, however, does have periodic trajectories, but no limit cycle. This can be seen in the adjacent figure out that there are no trajectories approach the cycles, or remove them, ie the cycles do not occur in isolation. Not isolated here means that in each environment the cycle again there are other cycles.
Stability
The stability of a limit cycle of period T is determined by its Floquet multipliers. The limit cycle corresponds to a fixed point in the Poincaré mapping. The Poincaré mapping is obtained by a cut ( Poincaré section) in phase space, so that the limit cycle, the sectional plane perpendicular pierces with its period (see blue section plane in the illustration ).
The stability of the limit cycle now corresponds to the stability of the fixed point of his Poincaré mapping P.
Let x * be the fixed point of P so true. For a point x, which is close to the fixed point, that is the picture now applies
Assuming that is small, P may be considered as linear in the vicinity of x * (DP (x *) is the Jacobian matrix of x * P ) and it follows that
The eigenvalues of DP ( x *) then determine the stability of x * and called Floquet multipliers limit cycle. If all the multipliers are less than 1 in magnitude, then the limit cycle is asymptotically stable if the following holds for all i, then the limit cycle is unstable. A Floquet multiplier is always 1 and corresponds to the direction of movement on the limit cycle. This multiplier is called Goldstone mode.
In addition to stable and unstable limit cycles there are also semi-stable limit cycles, ie external trajectories spiral towards the limit cycle and trajectories within the limit cycle spirals from the limit cycle away (or vice versa).
Hopf bifurcation
Limit cycles arise generically from Hopf bifurcations. Consider a solution of a system of differential equations with a free parameter, varying these constantly, so bifurcation can occur, ie the solution under consideration is changing qualitatively. From a fixed point may arise and the other way around a limit cycle. A simple example is the turning on of the laser. For this purpose, the pump current of 0 mA is highly regulated to a few mA. As long as the pump current is below the threshold current is he not light, ie the radiated electric field and a stable fixed point. Once the threshold is exceeded, the laser begins to luminaires and it is. Here there is a stable limit cycle of period and the bifurcation occurring at a supercritical Hopf bifurcation.