A Homoclinic Orbit in a Planar Singular ODE - A Computer Assisted Proof

We consider a family of 2-dimensional ODEs of the form Δξ(z)z ′ = fξ(z) depending on a real parameter ξ which was investigated by Vladimirov [Rep. Math. Phys., 61 (2008), pp. 381–400]. In this system, there exist stationary points pξ which belong to the set of zeros of Δξ. We prove, using rigorous numerics, the existence of a homoclinic orbit to pξ for some parameter value ξ = ξh. Due to the singularity of the system it takes a finite time to travel along this orbit, and this property gives rise to a compacton-like traveling wave in some hydrodynamic system describing relaxing media. Our approach could be used to prove similar results in other singular systems as well.