1. The problem statement, all variables and given/known data Example: x'=y-x^3 y'=-x-y^3 2. Relevant equations 3. The attempt at a solution Linear system x'=y y'=-x Is stable because Det(P-[itex]\lambda[/itex]E)=[itex]\lambda[/itex]2+1 [itex]\lambda[/itex]1,2=+-i So if Im not mistaken,than Ishould use Lyapunov stability,because the linear system is stable and I cant say anything about original system. ( I dont know why i cant tell anything about the original system, I just now it like "algorithm") So The Lyapunov function in general looks like V=ax^2+by^2 So V'=2axx'+2byy' I substitute x' and y' from original system: V'=2axy-2ax^4-2byx-2by^4 So my book says that xy is not relevent and in order to get rid of them 2a-2b=0 -> a=1 and b=1 So V=x^2+y^2 Now I have the function which will allow me to determine stabilty. V'=2xx'+2yy' Again I do the same - take x' and y' from original system V'=2xy-2x^4-2yx-2y^4=-2(x^4-y^4) Can i say the function is asympt.stable because V'=-V ? In which cases i have to use Lyapunov stability,linearization is not enough ( or its not so easy to determine) ? And do I have to determine a and b constants allways or I can just use V=x^2+y^2?