A New Construction for Melnikov Chaos in Piecewise-Smooth Planar Systems

We consider a piecewise-smooth 2-dimensional system aligned x⃗=f⃗(x⃗)+ε g⃗(t,x⃗,ε ) aligned x → ˙ = f → ( x → ) + ε g → ( t , x → , ε ) where ε >0 ε > 0 is a small parameter and f⃗ f → is discontinuous along a curve Ω ⁰ Ω 0 . We assume that 0⃗ 0 → is a critical point for any ε ≥ 0 ε ≥ 0 , and that for ε =0 ε = 0 the system admits a trajectory γ ⃗(t) γ → ( t ) homoclinic to 0⃗ 0 → and crossing transversely Ω ⁰ Ω 0 in γ ⃗(0) γ → ( 0 ) . In a previous paper, we have shown that, also in an n -dimensional setting, the classical Melnikov condition is enough to guarantee the persistence of the homoclinic orbit to perturbations, but more recently we have found an open condition, a geometric obstruction which is not possible in the smooth case, which prevents chaos for 2-dimensional systems when