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

  • *Corresponding author: D. J. W. Simpson

    *Corresponding author: D. J. W. Simpson
The authors were supported by Marsden Fund contract MAU1809, managed by Royal Society Te Apārangi
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  • Chaotic attractors in the two-dimensional border-collision normal form (a piecewise-linear map) can persist throughout open regions of parameter space. Such robust chaos has been established rigorously in some parameter regimes. Here we provide formal results for robust chaos in the original parameter regime of [S. Banerjee, J.A. Yorke, C. Grebogi, Robust Chaos, Phys. Rev. Lett. 80(14):3049-3052, 1998]. We first construct a trapping region in phase space to prove the existence of a topological attractor. We then construct an invariant expanding cone in tangent space to prove that tangent vectors expand and so no invariant set can have only negative Lyapunov exponents. Under additional assumptions we characterise an attractor as the closure of the unstable manifold of a fixed point and prove that it satisfies Devaney's definition of chaos.

    Mathematics Subject Classification: Primary: 37G35; Secondary: 39A28.

    Citation:

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  • 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 $

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