be an autonomous ordinary differential equation defined by the vector field then a solution of the system is a cycle(or periodic solution) if it is a closed solution which is not an equilibrium point. The period of a cycle is the smallest positive such that .
Let be the flow defined by the above ODE and be the metric of then:
A cycle, , is a stable cycle if for all there exists a neighborhood of such that for all , .
A cycle, , is unstable cycle if it is not a stable cycle.
A cycle, , is asymptotically stable cycle if for all where is a neighborhood of , .[PL]
then the above autonomous ordinary differential equations with initial value condition has a solution which is a stable cycle. Namely the solution defined by
which has a period of .
|Date of creation||2013-03-22 15:00:51|
|Last modified on||2013-03-22 15:00:51|
|Last modified by||Daume (40)|
|Synonym||stable periodic solution|
|Synonym||unstable periodic solution|
|Synonym||asymptotically stable periodic solution|
|Defines||asymptotically stable cycle|