How to prove that limit cycle can exist in 2D curl field?

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Wernbergbo9d · Accepted Answer

Step 1A limit cycle cannot exist in a 2D curl vector field. A 2D curl vector field is a hamiltonain vector field and as such it has a first integral, so if you have at least one closed curve trajectory, then there will be a whole smooth foliation of closed curve trajectories in an open neighbourhood of the given closed curve trajectory.Indeed, let the vector field in question isX=P(x,y)∂∂x+Q(x,y)∂∂yassume the γ0 is a closed curve trajectory tangent to the vector field, i.e. γ0 is the trajectory of a periodic solution of the system of ODEsdxdt=P(x,y)dydt=Q(x,y)We assume that the vector field X is smooth and well defined in a large enough open domain in the plane that contains γ0. Now, γ0 is a smooth embedded submanifold of the plane, diffeomorphic to the unit circle, and it bounds a simply connected closed compact domain K(γ0) which is diffeomorphic to the closed unit disc. Let us take a small enough neighbourhood D(γ0) of K(γ0), diffeomorphic to an open disc of the plane. Then the vector field X is smooth and well defined on the this simply connected open domain D(γ0) and γ0⊂D(γ0)So from now on all arguments will take place in D(γ0)Define the smooth differential one formω=−Q(x,y)dx+P(x,y)dyand observe thatω(X)≡0Now,dω=dBig(−Q(x,y)dx+P(x,y)dyBig)=−dQ(x,y)∧dx+dP(x,y)∧dy=∂Q∂y(x,y)dy∧dx+∂P∂x(x,y)dx∧dy=∂Q∂y(x,y)dx∧dy+∂P∂x(x,y)dx∧dy=(∂Q∂y(x,y)dx+∂P∂x(x,y))dx∧dy≡0Since D(γ0) is simply connected, more precisely it deformation retracts onto a point, there exists a smooth function f(x, y) defined at least on D(γ0) such thatdf=ωHence,P(x,y)=∂f∂y(x,y)Q(x,y)=−∂f∂x(x,y)which means that the original vector field can be written asX=∂f∂y(x,y)∂∂x−∂f∂x(x,y)∂∂yand is therefore hamiltonian with first-integral (hamiltonian) f(x,y) so every trajectory in the domain D(γ0) follows a level curve of the function f, i.e.f(x,y)=cfor some constant c. Since γ0⊂f(x,y)=c0,then the trajectories of the vector field X in a small enough tubular neighbourhood of γ0 are also closed curve trajectories.

How to prove that limit cycle can exist

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