Difference between limit cycle and periodic orbit
Answers
Answered by
1
Definition
Periodic Orbit for a Vector Field
Consider a system of ordinary differential equations dxdt=f(x),x∈Rn(n≥2) or dxdt=f(x,t),x∈Rn(n≥1), corresponding to an autonomous or non-autonomous vector field, respectively. A non-constant solution to such a system, x(t) , is periodic if there exists a constant T>0 such that
x(t)=x(t+T)
for all t . The period of this solution is defined to be the minimum such T . The image of the periodicity interval [0,T] under x in the state space Rn is called the periodic orbit or cycle.
Limit Cycle
A periodic orbit Γ on a plane (or on a two-dimensional manifold) is called a limit cycle if it is the α-limit set or ω-limit set of some point z not on the periodic orbit, that is, the set of accumulation points of either the forward or backward trajectory through z , respectively, is exactly Γ . Asymptotically stable and unstable periodic orbits are examples of limit cycles.
Example (Guckenheimer and Holmes, 1983; Strogatz 1994)
The figure shows the periodic orbit which exists for the vector field dxdt=αx−y−αx(x2+y2)dydt=x+αy−αy(x2+y2), where α>0 is a parameter. Transforming to radial coordinates, we see that the periodic orbit lies on a circle with unit radius for any α>0 : drdt=αr(1−r2),dθdt=1. This periodic orbit is a stable limit cycle for α>0 and unstable limit cycle for α<0 . When α=0 , the system above has infinite number of periodic orbits and no limit cycles.
Periodic Orbit for a Vector Field
Consider a system of ordinary differential equations dxdt=f(x),x∈Rn(n≥2) or dxdt=f(x,t),x∈Rn(n≥1), corresponding to an autonomous or non-autonomous vector field, respectively. A non-constant solution to such a system, x(t) , is periodic if there exists a constant T>0 such that
x(t)=x(t+T)
for all t . The period of this solution is defined to be the minimum such T . The image of the periodicity interval [0,T] under x in the state space Rn is called the periodic orbit or cycle.
Limit Cycle
A periodic orbit Γ on a plane (or on a two-dimensional manifold) is called a limit cycle if it is the α-limit set or ω-limit set of some point z not on the periodic orbit, that is, the set of accumulation points of either the forward or backward trajectory through z , respectively, is exactly Γ . Asymptotically stable and unstable periodic orbits are examples of limit cycles.
Example (Guckenheimer and Holmes, 1983; Strogatz 1994)
The figure shows the periodic orbit which exists for the vector field dxdt=αx−y−αx(x2+y2)dydt=x+αy−αy(x2+y2), where α>0 is a parameter. Transforming to radial coordinates, we see that the periodic orbit lies on a circle with unit radius for any α>0 : drdt=αr(1−r2),dθdt=1. This periodic orbit is a stable limit cycle for α>0 and unstable limit cycle for α<0 . When α=0 , the system above has infinite number of periodic orbits and no limit cycles.
Similar questions