Applying Bendixson-Dulac Theorem in circle with radius 3Show that the system dxdt=y+15x5−x3,;;;dydt=−x+13x2y3has a centre at the origin, but that there are no closed orbits lying inside the circle whose equation is x2+y2=3.I have shown there is a centre at the origin, but am struggling to find an appropriate function to use in the Bendixson-Dulac theorem. Any help would be much appreciated, thank you.

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Kayla Fitzgerald · Accepted Answer

Step 1If you apply the theorem with a general function φ(x,y), you would find that:∂(φf)∂x+∂(φg)∂y=x2(x2+y2−3)φ+fφx′+gφy′.Step 2The first term is almost everywhere positive in the circle (given ψ is also almost everywhere positive). If only there was such a function φ(x,y)&amp;gt;0 which has φx′=φy′=0.

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