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Homoclinic tangencies with infinitely many asymptotically stable single-round periodic solutions

  • * Corresponding author: Sishu Shankar Muni

    * Corresponding author: Sishu Shankar Muni 
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  • We consider a homoclinic orbit to a saddle fixed point of an arbitrary $ C^\infty $ map $ f $ on $ \mathbb{R}^2 $ and study the phenomenon that $ f $ has an infinite family of asymptotically stable, single-round periodic solutions. From classical theory this requires $ f $ to have a homoclinic tangency. We show it is also necessary for $ f $ to satisfy a 'global resonance' condition and for the eigenvalues associated with the fixed point, $ \lambda $ and $ \sigma $, to satisfy $ |\lambda \sigma| = 1 $. The phenomenon is codimension-three in the case $ \lambda \sigma = -1 $, but codimension-four in the case $ \lambda \sigma = 1 $ because here the coefficients of the leading-order resonance terms associated with $ f $ at the fixed point must add to zero. We also identify conditions sufficient for the phenomenon to occur, illustrate the results for an abstract family of maps, and show numerically computed basins of attraction.

    Mathematics Subject Classification: Primary: 37G25; Secondary: 37G15, 39A23.

    Citation:

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  • Figure 1.  A sketch of tangentially intersecting stable [blue] and unstable [red] manifolds of a saddle fixed point of a two-dimensional map. Note that the tangential intersections form a homoclinic orbit

    Figure 2.  A sketch of the stable [blue] and unstable [red] manifolds of the origin for a $ C^\infty $ map $ f $ satisfying (3). A homoclinic orbit is indicated with black dots in the case $ 0<\lambda<1 $ and $ \sigma > 1 $

    Figure 3.  Selected points of an $ {\rm SR}_k $-solution (single-round periodic solution satisfying Definition 2.1) in the case $ \lambda > 0 $ and $ \sigma > 0 $. The region $ {\mathcal{N}}_\eta $ is shaded

    Figure 4.  The stability of a period-$ n $ solution of $ f $ in terms of the trace $ \tau $ and determinant $ \delta $ of the Jacobian matrix $ D f^n $ evaluated at one point of the solution

    Figure 5.  The basic structure of the phase space of the map (15)

    Figure 6.  The function (19) (with (20)) that we use as a convex combination parameter in (15)

    Figure 7.  Parts of the stable [blue] and unstable [red] manifolds of the origin for the map (15) with (16)–(21). Panels (a)–(d) correspond to (22)–(25) respectively. In each panel the region $ h_0 < y < h_1 $ is shaded

    Figure 8.  Asymptotically stable $ {\rm SR}_k $-solutions of (15) with (16)–(21). Panels (a)–(d) correspond to (22)–(25) respectively. Points of the stable $ {\rm SR}_k $-solutions are indicated by triangles and coloured by the value of $ k $ (as indicated in the key). In panels (a) and (b) the solutions are shown for $ k = 0 $ (a fixed point in $ y > h_1 $) up to $ k = 15 $. In panel (c) the solutions are shown for $ k = 0,2,4,\ldots,14 $ and in panel (d) the solutions are shown for $ k = 1,3,5,\ldots,15 $. In each panel one saddle $ {\rm SR}_k $-solution is shown with circles (with $ k = 14 $ in panel (c) and $ k = 15 $ in the other panels). In panels (b) and (c) asymptotically stable double-round periodic solutions are shown with diamonds

    Figure 9.  Basins of attraction for the asymptotically stable $ {\rm SR}_k $-solutions shown in Fig. 8. Specifically each point in a $ 1000 \times 1000 $ grid is coloured by that of the $ {\rm SR}_k $-solution to which its forward orbit under $ f $ converges to

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