Trapping Region for ODE System.x′=2x+y−2x3−3xy2,y′=−2x+4y−4y3−2x2y,

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Mey9ci0 · Accepted Answer

Step 1For V(x,y)=2x2+y2 one gets 16x2y2 instead of 20x2y2, soV˙(x,y)=8(x2+y2)−8(x4+2x2y2+y4),=8(x2+y2)−8(x2+y2)2=8(x2+y2)(1−x2−y2)Now you want to find α and β (with 0&amp;lt;α&amp;lt;β) such that for the set A={x,y∣V(x,y)=α} it holds that V˙(x,y)&amp;gt;0 and for the set B={x,y∣V(x,y)=β} it holds that V˙(x,y)&amp;lt;0 Namely, this would mean that all initial conditions on the level set A of the Lyapunov function would flow towards higher level sets and that all initial conditions on the level set B of the Lyapunov function would flow towards lower level sets. Thus all initial conditions starting on either A or B should end up in some region in-between A and B.It is worth noting that the level sets of V(x,y) do not overlap with levels sets of V˙(x,y). So for x2+y2=1, where V˙(x,y)=0, does not correspond with a level set of V(x,y) and thus x2+y2=1 would not be a limit cycle.

Trapping Region for ODE System. \(\displaystyle{x}'={2}{x}+{y}-{2}{x}^{{3}}-{3}{x}{y}^{{2}},\) \(\displaystyle{y}'=-{2}{x}+{4}{y}-{4}{y}^{{3}}-{2}{x}^{{2}}{y},\)

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