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Unfolding globally resonant homoclinic tangencies

  • * Corresponding author: Sishu Shankar Muni

    * Corresponding author: Sishu Shankar Muni 
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  • Global resonance is a mechanism by which a homoclinic tangency of a smooth map can have infinitely many asymptotically stable, single-round periodic solutions. To understand the bifurcation structure one would expect to see near such a tangency, in this paper we study one-parameter perturbations of typical globally resonant homoclinic tangencies. We assume the tangencies are formed by the stable and unstable manifolds of saddle fixed points of two-dimensional maps. We show the perturbations display two infinite sequences of bifurcations, one saddle-node the other period-doubling, between which single-round periodic solutions are asymptotically stable. The distance of the bifurcation values from global resonance generically scales like $ |\lambda|^{2 k} $, as $ k \to \infty $, where $ -1 < \lambda < 1 $ is the stable eigenvalue associated with the fixed point. If the perturbation is taken tangent to the surface of codimension-one homoclinic tangencies, the scaling is instead like $ \frac{|\lambda|^k}{k} $. We also show slower scaling laws are possible if the perturbation admits further degeneracies.

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

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  • Figure 1.  A homoclinic tangency for a saddle fixed point of a two-dimensional map. In this illustration the eigenvalues associated with the fixed points are positive, i.e. $ 0 < \lambda < 1 $ and $ \sigma > 1 $. A coordinate change has been applied so that in the region $ {\mathcal{N}} $ (shaded) the coordinate axes coincide with the stable and unstable manifolds. The homoclinic orbit $ \Gamma_{\rm HC} $ is shown with black dots. A typical single-round periodic solution is shown with blue triangles

    Figure 2.  A sketch of codimension-one surfaces of homoclinic tangencies (green) and where $ \lambda(\mu) \sigma(\mu) = 1 $ (purple). The vectors $ {\bf n}_{\rm tang} $ and $ {\bf n}_{\rm eig} $, respectively, are normal to these surfaces at the origin $ \mu = {\bf 0} $

    Figure 3.  A phase portrait of (49) with (54) and $ \mu = {\bf 0} $. The shaded horizontal strip is where the middle component of (49) applies. We show parts of the stable and unstable manifolds of $ (x,y) = (0,0) $. Note the unstable manifold has very high curvature at $ (x,y) \approx (0,1.1) $ because (49) is highly nonlinear in the horizontal strip. For the given parameter values (49) has an asymptotically stable, single-round periodic solutions of period $ k+1 $ for all $ k \ge 1 $. These are shown for $ k = 1,2,\ldots,15 $; different colours correspond to different values of $ k $. The map also has an asymptotically stable fixed point at $ (x,y) = (1,1) $

    Figure 4.  Panel (a) is a numerically computed bifurcation diagram of (49) with (54) and $ \mu_2 = \mu_3 = \mu_4 = 0 $. The triangles [circles] are saddle-node [period-doubling] bifurcations of single-round periodic solutions of period $ k+1 $. Panel (b) shows the same points but with the horizontal axis scaled in such a way that the asymptotic approximations to these bifurcations, given by the leading-order terms in (45) and (46), appear as vertical lines

    Figure 5.  Panel (a) is a numerically computed bifurcation diagram of (49) with (54) and $ \mu_1 = \mu_3 = \mu_4 = 0 $. Panel (b) shows convergence to the leading-order terms of (47) and (48)

    Figure 6.  Panel (a) is a numerically computed bifurcation diagram of (49) with (54) and $ \mu_1 = \mu_2 = \mu_4 = 0 $. Panel (b) shows convergence to the leading-order terms of (55) and (56)

    Figure 7.  Panel (a) is a numerically computed bifurcation diagram of (49) with (54) and $ \mu_1 = \mu_2 = \mu_3 = 0 $. Panel (b) shows convergence to the leading-order terms of (57) and (58)

    Figure 8.  A two-dimensional slice of the four-dimensional parameter space of (49) defined by fixing $ \mu_3 = \mu_4 = 0 $. The remaining parameter values are given by (54) except we have set $ a_{1,0} = 0 $ to simplify the numerical computations. For each $ 15 \le k \le 20 $ we show the region bounded by curves of saddle-node and period-doubling bifurcations where (49) has an asymptotically stable period-$ (k+1) $ solution. Intersections of these regions are indicated by successively darker shades of grey

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