New global bifurcation diagrams for piecewise smooth systems: transversality of homoclinic points does not imply chaos
It is well-known that chaos arises if a smooth dynamical system which admits a transversal homoclinic point is perturbed. This paper presents a new result that this phenomenon may not exist for some piecewise smooth two-dimensional systems subjected to a periodic or almost periodic perturbation. The authors assume that the fixed point lies on the discontinuity surface and investigate two scenarios where the unperturbed homoclinic orbit crosses transversally the discontinuity surface. For the piecewise smooth systems where there is sliding close to the fixed point, they prove that chaos does not exist even though there is a transversal homoclinic point and the perturbation is periodic. On the other hand, for piecewise smooth systems where there is no sliding close to the fixed point, they conjecture that chaos is possible if there is a transversal homoclinic point and the perturbation is periodic.
scientific article; zbMATH DE number 6987666