Including homoclinic connections and T-point heteroclinic cycles in the same global problem for a reversible family of piecewise linear systems

• Global behavior is analyzed in a family of reversible piecewise linear systems.
• The aim is to prove the existence of certain homoclinic orbits and T-point cycles.
• A problem including these two different objects as particular cases is constructed.
• This problem leads to a common theorem of existence and local uniqueness.
• The analytical proof of this result is given.

Apart from being a complex task even in piecewise linear systems, the proof of the existence of homoclinic connections and T–point heteroclinic cycles must be usually carried out in separate ways because they are obviously different dynamical objects. Despite this, some features of the system may narrow the disparities between such global bifurcations and help us to look for alternative methods to analyze them.In this work, taking advantage of the reversibility and some geometrical features of a piecewise linear version of the Michelson system, we construct, by adding a suitable parameter, a global problem that includes homoclinic connections and T-point heteroclinic cycles as particular cases. Moreover, this problem leads to a common result for the existence and local uniqueness of these global bifurcations, whose proof is also given.