Why are limit cycles emphasized in qualitative analysis of differential equations?

[sharmeen]

i want to know that in qualitative analysis of differential equations why we give more importance to a limit cycle on any other trajectories to show the solution of a differential equation

 

[arildno]

Hmm..
i) Because it is "often" easier to find limit cycles than exact trajectories.

ii) In the case of mathematical modelling, because limit cycles are trajectories to which particular solutions approach, knowing the limit cycle enables us to predict how "stuff" actually "develop/move".

iii) From a "theoretical" point of view, classifying particular solutions in terms of which limit cycle they'll tend to, might be a handy device.