MHB Liapunov function (扁頭科學's question at Yahoo Answers)

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A suitable Lyapunov function of the form ax^2 + cy^2 can be constructed to analyze the stability of the critical point at the origin for the given differential equations. By evaluating the function with a = c = 1, the derivative L_v(f)(x,y) results in a negative expression, indicating that the function is a strict Lyapunov function. This confirms that the origin is asymptotically stable. The analysis shows that the chosen Lyapunov function effectively demonstrates the stability of the system. Therefore, the critical point at the origin is proven to be asymptotically stable.
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Here is the question:

Construct a suitable Liapunov function of the form ax^2+cy^2,
where a and c are to be determined. Then show that the critical
point at the origin is of the indicated type.
1. dx/dt = -x^3+xy^2, dy/dt = -2x^2y-y^3; asymptotically stable

Here is a link to the question:

Differential equation (Liapunov function)? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.
 
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Hello 扁頭科學,

If $f(x,y)=ax^2+cy^2$, then: $$L_v(f)(x,y)=\nabla f(x,y)\cdot (v_1,v_2)=(2ax,2cy)\cdot (-x^3+xy^2,-2x^2y-y^3)=\\-2ax^4+2ax^2y^2-4cx^2y^2-2cy^4$$ If $a=c=1$, we get $L_v(f)(x,y)=-2(x^4+x^2y^2+y^4)<0$ for all $(x,y)\ne (0,0)$. This means that $f$ is a Strict Lyapunov Function at $(0,0).$ As a consequence, $(0,0)$ is asymptotically stable.
 
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