Zbadać stabilność w sensie Launowa
: 5 cze 2010, o 14:02
Zbadać stabilność w sensie Lapunova
\(\displaystyle{ \begin{cases} x'_{1}=x _{1}x _{2}- x_{1} ^{3} +x _{2} \\ x' _{2}=x _{1} ^{4}-x _{1}x _{2} ^{2}-x _{1} ^{3} \end{cases}}\)
\(\displaystyle{ V(x _{1},x _{2})=ax _{1} ^{4} +bx _{2} ^{2}}\)
Po rozwinięciu
\(\displaystyle{ V'=x _{1} ^{4}x _{2}(4a+2b)+x _{1} ^{3}x _{2}(4a-2b)-4ax _{1} ^{6} -2bx _{2} ^{3}x _{1}}\)
I nie wiem co dalej
\(\displaystyle{ \begin{cases} x'_{1}=x _{1}x _{2}- x_{1} ^{3} +x _{2} \\ x' _{2}=x _{1} ^{4}-x _{1}x _{2} ^{2}-x _{1} ^{3} \end{cases}}\)
\(\displaystyle{ V(x _{1},x _{2})=ax _{1} ^{4} +bx _{2} ^{2}}\)
Po rozwinięciu
\(\displaystyle{ V'=x _{1} ^{4}x _{2}(4a+2b)+x _{1} ^{3}x _{2}(4a-2b)-4ax _{1} ^{6} -2bx _{2} ^{3}x _{1}}\)
I nie wiem co dalej