A constructive approach to robust chaos using invariant manifolds and expanding cones

Figure 1.  Initial portions of the stable and unstable manifolds of the fixed point $ Y $. Throughout this paper stable and unstable manifolds are coloured blue and red respectively

Figure 2.  The parameter region $ {\mathcal{R}} $: (3) and $ \phi > 0 $, where $ \phi $ is given by (10). The striped region indicates parameter values valid for Theorem 2.3. (This figure was created using $ \delta_L = 0.2 $ and $ \delta_R = 0.4 $.)

Figure 3.  Initial portions of the stable and unstable manifolds of the fixed point $ X $

Figure 4.  A phase portrait of (1) using the parameter values (15). This shows all periodic solutions (except $ Y $) up to period $ 20 $ (as black dots). These were computed via a brute-force search and the algorithm of [14] to generate all possible symbolic itineraries. The unstable manifold $ W^u(X) $ (coloured red but mostly obscured by the periodic solutions) was computed numerically by following it outwards from $ X $ until no further growth could be discerned

Figure 6.  The trapping region $ \Omega_{\rm trap} $

Figure 5.  The forward invariant region $ \Omega $ and its image $ f(\Omega) $

Figure 7.  The functions $ p $ (27), $ q $ (23), and $ r $ (24) for $ \tau > \delta + 1 $ and a fixed value of $ \delta \in (0, 1) $

Figure 8.  The slope maps (32). $ G_L(m) $ and $ G_R(m) $ are the slopes of $ A_L v $ and $ A_R v $, respectively, where $ v $ has slope $ m $