Solved: "Prove instability using Lyapunov functionx′=x3+xyy′=−y+y2+xy−x3"

London Douglas

Answered question

2022-04-01

Prove instability using Lyapunov function

x^{'} = x^{3} + x y

y^{'} = - y + y^{2} + x y - 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^{'} (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}

\geq x^{6} + (1 + x) x^{3} y + \frac{1}{2} y^{2} if | x | + | y | \leq \frac{1}{4}

\geq x^{6} - \frac{5}{4} {| x |}^{3} | y | + \frac{1}{2} y^{2} since | x | \leq \frac{1}{4}

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

\geq 0

with equality iff

x = y = 0

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