Periodic orbits in the Rössler system

From the Sharkovskii theorem to periodic orbits for the Rössler system

“…However, in our recent paper [5], where we consider two sets of parameters in the Rössler system with attracting periodic orbits: 3-periodic for the first set and 5-periodic for the other, we proved the existence of the periods implied by the Sharkovskii theorem quite easily. Just as in [12], the proof relies on the construction of a set of covering relations, but due to the fact that orbit we started with was attracting the construction was much simpler in this case.…”

“…For this end we recall the notion of covering for h-sets in R N from [12,11,5]. It is, in fact, a particular case of the similar notion from [13], but with exactly one exit (or 'unstable') direction.…”

“…In [5] we proved the existence of m-periodic orbits of all m succeeding n in Sharkovskii order (1) in the Rössler system [8] with an attracting n-periodic orbit (16)…”

“…In Section 5 we prove the main theorem. Finally in Section 6 we apply our result to the Rössler system from [5].…”

“…As an application of Theorem 2 we study the Rössler system [5] for different values of parameters with attracting periodic orbits of 3, 5 and 6. We verify the assumptions of Theorem 2, hence we obtain an infinite number of periodic orbits.…”