math.DS — Dynamical Systems
Existence of SRB measures for some Rössler systems
We prove that for the Rössler systemẋ = −(y + z),ẏ = x + ay,ż = b + z(x − c) with a = b = 0.2, the set of parameters c within 6 · 10 −5 of the classical value c = 5.7 for which the ow has a Hénon-like strange attractor supporting a SinaiRuelleBowen (SRB) measure has positive Lebesgue measure. The key step, proved with computer assistance, is that at an explicit parameter within 5.5 · 10 −5 of 5.7 the period-one saddle of the Poincaré map has a quadratic homoclinic tangency that unfolds generically; the conclusion then follows from the theorems of MoraViana, BenedicksYoung and BenedicksViana. The same method gives two further such tangencies near 5.7 and one at the band-merging parameter c ≈ 4.507. The proof reduces the generic-tangency conditions to nitely many inequalities, through explicit C 2 graph-transform bounds for the local invariant manifolds of a Poincaré map whose area contraction is of order 10 −11 10 −14 , and a scalar tangency function. The inequalities are checked with interval arithmetic in CAPD and rechecked with an independent implementation based on the Kv library; the analytic tangency criterion is formalised in Lean 4. Whether an SRB measure exists at c = 5.7 itself remains open.