ODELF1

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Consider the system of equations {\begin{cases}x'=y-xf(x,y)\\y'=-x-yf(x,y)\end{cases}}\,

Find a Lyapunov function to determine the stability of the equilibrium solution (0,0)\,. Consider the three cases in your calculations: a) f(x,y)\, positive semidefinate, b) f(x,y)\, positive definite and c) f(x,y)\, negative definate.


{\begin{cases}x'=y-xf(x,y)\\y'=-x-yf(x,y)\end{cases}}\,

Let L=x^{2}+y^{2}\,.

{\frac  {\partial L}{\partial t}}=2x{\dot  {x}}+2y{\dot  {y}}\,

=2x\left[y-xf(x,y)\right]+2y\left[-x-yf(x,y)\right]\,

=-2x^{2}f(x,y)-2y^{2}f(x,y)\,

=-f(x,y)[2x^{2}+2y^{2}]\,

If f(x,y)\, is positive semidefinite, then {\frac  {\partial L}{\partial t}}\leq 0\, so the zero solution is uniformly stable.

If f(x,y)\, is positive definite, then {\frac  {\partial L}{\partial t}}<0\, so the zero solution is asymptotically stable.

If f(x,y)\, is negative definite, then {\frac  {\partial L}{\partial t}}>0\, so the zero solution is unstable.


Main Page : Ordinary Differential Equations : Lyapunov Functions