An exact homoclinic orbit and its connection with the Rössler system

•A three parameter unfolding of a triple-zero bifurcation is considered.•A blow-up leads to a generalized Michelson system.•In this system an exact homoclinic orbit is determined.•Several codimension-two homoclinic bifurcation are numerically detected.•The Rössler system exhibits these degeneracies that imply chaotic dynamics.

In this Letter we consider a three parameter unfolding of a linear degeneracy corresponding to a triple-zero eigenvalue of an equilibrium point. Using blow-up techniques we obtain a system where an exact homoclinic connection is determined. The numerical continuation of this global connection shows that it exhibits three different kinds of codimension-two degeneracies. Finally, these same codimension-two homoclinic bifurcations are detected in the Rössler system, ensuring in this way the existence of chaotic dynamics.