London Douglas

Answered question

2022-04-01

Prove instability using Lyapunov function
${x}^{\prime }={x}^{3}+xy$
${y}^{\prime }=-y+{y}^{2}+xy-{x}^{3}$

Answer & Explanation

German Ferguson

Beginner2022-04-02Added 18 answers

The dominant term is ${x}^{6}+{x}^{3}y+{y}^{2}$. This term is positive (complete the square). All other terms are small when x,y, are small, so they can be controlled. Here are the details.
Write
${V}^{\prime }\left(x,y\right)={x}^{6}+{x}^{4}y+{y}^{2}-{y}^{3}-x{y}^{2}+{x}^{3}y$
$={x}^{6}+\left(1+x\right){x}^{3}y+\left(1-x-y\right){y}^{2}$
$\ge {x}^{6}+\left(1+x\right){x}^{3}y+\frac{1}{2}{y}^{2}\phantom{\rule{1em}{0ex}}\text{if}|x|+|y|\le \frac{1}{4}$
$\ge {x}^{6}-\frac{5}{4}{|x|}^{3}|y|+\frac{1}{2}{y}^{2}\phantom{\rule{1em}{0ex}}\text{since}|x|\le \frac{1}{4}$
$={\left({|x|}^{3}-\frac{5}{8}|y|\right)}^{2}+\frac{7}{64}{y}^{2}$
$\ge 0$
with equality iff $x=y=0$

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