London Douglas

2022-04-01

Prove instability using Lyapunov function

${x}^{\prime}={x}^{3}+xy$

$y}^{\prime}=-y+{y}^{2}+xy-{x}^{3$

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

${V}^{\prime}(x,y)={x}^{6}+{x}^{4}y+{y}^{2}-{y}^{3}-x{y}^{2}+{x}^{3}y$

$={x}^{6}+(1+x){x}^{3}y+(1-x-y){y}^{2}$

$\ge {x}^{6}+(1+x){x}^{3}y+\frac{1}{2}{y}^{2}{\textstyle \phantom{\rule{1em}{0ex}}}\text{if}\left|x\right|+\left|y\right|\le \frac{1}{4}$

$\ge {x}^{6}-\frac{5}{4}{\left|x\right|}^{3}\left|y\right|+\frac{1}{2}{y}^{2}{\textstyle \phantom{\rule{1em}{0ex}}}\text{since}\left|x\right|\le \frac{1}{4}$

$={({\left|x\right|}^{3}-\frac{5}{8}\left|y\right|)}^{2}+\frac{7}{64}{y}^{2}$

$\ge 0$

with equality iff$x=y=0$

Write

with equality iff

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