"Large" strange attractors in the unfolding of a heteroclinic attractor

Figure 1.  Example of a Misiurewicz-type map $ h: {\mathbb{S}}^1 \rightarrow {\mathbb{R}} $. For $ \delta>0 $, the set $ C_{\delta} $ is a neighbourhood of the set of critical points $ C $. These maps are among the simplest examples with non-uniform expansion

Figure 2.  Illustration of the transition map $ \Psi_{2 \rightarrow 1} $ from $ {\text{Out}}(O_2) $ to $ {\text{In}} (O_1) $. The graph of $ \Phi_2(x, 0) $ may be seen as the first hit of $ W^u(O_2) $ to $ {\text{In}} (O_1) $

Figure 3.  Graph of $ k(x) = -K_\omega \ln (x) $ and illustration of the sequences $ (\lambda_n)_n $ and $ (\lambda_{(n, a)})_n $ for a fixed $ a \in [0, 2\pi[ $

Figure 4.  Graph of the map $ h_a $ for $ K_\omega \gg K_\omega^0 $ with $ q = 2 $ (number of critical points). Indicated is a superstable periodic orbit of period 2 as it contains a critical point

Figure 5.  Adapted bifurcation diagram of [25] for the family (4): below the graph of $ t_2 $, the dynamics is governed by an attracting invariant 2-dimensional torus. Above the graph of $ t_1 $, one observes rotational horseshoes. In between, rotational horseshoes exist but they may be not observable in numerics; the dynamics is probably dominated either by a sink or bistability between a sink and an attracting torus

Figure 6.  Illustration (based on plots obtained by using Maple) of $ h_a(x) \pmod{2\pi} $ for $ a = 0.1 $, $ \xi = 0 $ and $ \Phi_2(x, 0) = \sin x $ ($ \Rightarrow q = 2 $). (a) $ K_\omega \approx 0 $ – attracting torus. (b) $ K_\omega = 0.5 $ – attracting torus inside a resonance wedge with a saddle and a sink. (c): $ K_\omega = 2 $ – torus breakdown. (d): $ K_\omega = 5 $ – mixing properties ("big lobe'' of [30])