I have a two dimensional piecewise smooth ODE and I was wondering if there are some known results for disproving the existence of homoclinic and etheroclinic loops (chain of etheroclinic orbits). I am trying to use an energy function that is nonincreasing along the flow.
My scenario is the following: I have a state space $\Omega$ and a continuous, lipschitz function $E(\cdot)$ that is nonincreasing along the flow of my ODE. Let's assume I proved the following facts:
- $E$ is constant on the sets $A_1, ..., A_n$. These sets also contain the equilibria.
- There is no trajectory that connects any $A_i$ to $A_j$ that is entirely connected in the union $\bigcup A_i$
- Everywhere else the energy $E$ strictly decreases along the solutions.
- On each $A_i$ the dynamics is simple, i.e. I can exclude existence of homoclinic and etheroclinic orbits in $A_i$
Is this enough to show that is not possible to have homoclinic, etheroclinic loops?
For example, for the homoclinic case, I could argue (by contradiction) that I can always take 2 points $x$ (on the 'unstable' direction of the saddle $P$) and $y$ (on the 'stable' direction of the saddle $P$) arbitrarily close to $P$ such that $E(x)=E(y)$. Furthermore as the solution passing through $x$ and $y$ 'gets' out of $\bigcup A_i$, it will hold that $E(y)<E(x)$ therefore proving the argument. For the etheroclinic loops, I would argue in a similar way!