Wrinkled tori and bursts due to resonant temporal forcing (Q5942846)

In this mainly numerical article the authors study the effect of resonant temporal forcing on a system undergoing a Hopf bifurcation with \(D_4\) (square) symmetry. The forcing has a large impact on the behaviour of the system if its frequency is near a strong resonance with the Hopf frequency. Indeed, bursts with large dynamical range occur. This occurrence is related to the presence of heteroclinic connections to infinity.NEWLINENEWLINENEWLINEThe focus in the article is on a particular (but representative) example in which an attracting quasiperiodic solution that exists in absence of forcing `wrinkles' into chaos as the amplitude of the forcing increases. The overall behaviour is governed by the approach of the resulting attractor to solutions at infinity. Windows of stable periodic solutions are also found and their corresponding attractor is destroyed still at finite amplitude by a boundary crisis. If the strength of the forcing is increased, the solution looses stability in a torus bifurcation, producing a quasiperiodic solution which apparently comes close to infinity, forming a sequence of bursts of infinite dynamic range.NEWLINENEWLINENEWLINEOther aspects of the complicated observed dynamics are related to the presence of a new type of gluing bifurcation, whose existence is not only observe numerically but also proved analytically in the Appendix.