# American Institute of Mathematical Sciences

August  2022, 42(8): 4013-4030. doi: 10.3934/dcds.2022043

## Unfolding globally resonant homoclinic tangencies

 School of Fundamental Sciences, Massey University, Palmerston North, New Zealand

* Corresponding author: Sishu Shankar Muni

Received  September 2021 Revised  February 2022 Published  August 2022 Early access  April 2022

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.

Citation: Sishu Shankar Muni, Robert I. McLachlan, David J. W. Simpson. Unfolding globally resonant homoclinic tangencies. Discrete and Continuous Dynamical Systems, 2022, 42 (8) : 4013-4030. doi: 10.3934/dcds.2022043
##### References:
 [1] A. Delshams, M. Gonchenko and S. Gonchenko, On dynamics and bifurcations of area preserving maps with homoclinic tangencies, Nonlinearity, 28 (2015), 3027-3071.  doi: 10.1088/0951-7715/28/9/3027. [2] Y. Do and Y. C. Lai, Multistability and arithmetically period-adding bifurcations in piecewise smooth dynamical systems, Chaos, 18 (2008), 043107, 9 pp. doi: 10.1063/1.2985853. [3] N. K. Gavrilov and L. P. Silnikov, On three dimensional dynamical systems close to systems with structurally unstable homoclinic curve. I, Math. USSR Sbornik, 17 (1972), 467-485. [4] N. K. Gavrilov and L. P. Šilnikov, On three-dimensional dynamical systems close to systems with a structurally unstable homoclinic curve. II, Mat. Sb. (N.S.), 90, (1973), 167,139–156. [5] M. S. Gonchenko and S. V. Gonchenko, On cascades of elliptic periodic points in two-dimensional symplectic maps with homoclinic tangencies, Regul. Chaotic Dyns., 14 (2009), 116-136.  doi: 10.1134/S1560354709010080. [6] S. V. Gonchenko and L. P. Shilnikov, On two-dimensional area-preserving maps with homoclinic tangencies that have infinitely many generic elliptic periodic points, J. Math. Sci. (N. Y.), 128 (2005), 2767-2773.  doi: 10.1007/s10958-005-0228-6. [7] S. V. Gonchenko, L. P. Shil'nikov and D. V. Turaev, Quasiattractors and homoclinic tangencies, Comput. Math. Appl., 34 (1997), 195-227.  doi: 10.1016/S0898-1221(97)00124-7. [8] V. S. Gonchenko, Y. A. Kuznetsov and H. G. E. Meijer, Generalized Hénon map and bifurcations of homoclinic tangencies, SIAM J. Appl. Dyn. Syst., 4 (2005), 407-436.  doi: 10.1137/04060487X. [9] Y. A. Kuznetsov and H. G. E. Meijer, Numerical Bifurcation Analysis of Maps: From Theory to Software, Cambridge Monographs on Applied and Computational Mathematics, 34. Cambridge University Press, Cambridge, 2019.  doi: 10.1017/9781108585804. [10] S. S. Muni, R. I. McLachlan and D. J. W. Simpson, Homoclinic tangencies with infinitely many asymptotically stable single-round periodic solutions, Discrete Contin. Dyn. Syst., 41 (2021), 3629-3650.  doi: 10.3934/dcds.2021010. [11] S. E. Newhouse, Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9-18.  doi: 10.1016/0040-9383(74)90034-2. [12] J. Palis and F. Takens, Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations, Cambridge University Press, Cambridge, 1993. [13] L. P. Shilnikov, A. L. Shilnikov, D. V. Turaev and L. O. Chua, Methods of Qualitative Theory in Nonlinear Dynamics, World Scientific Publishing Co., Inc., River Edge, NJ, 1998. doi: 10.1142/9789812798596. [14] D. J. W. Simpson, Scaling laws for large numbers of coexisting attracting periodic solutions in the border-collision normal form, Int. J. Bifurcation Chaos, 24 (2014), 1450118, 28 pp. doi: 10.1142/S0218127414501181. [15] D. J. W. Simpson, Sequences of periodic solutions and infinitely many coexisting attractors in the border-collision normal form, Int. J. Bifurcation Chaos, 24 (2014), 1430018, 18 pp. doi: 10.1142/S0218127414300183. [16] D. J. W. Simpson and C. P. Tuffley, Subsumed homoclinic connections and infinitely many coexisting attractors in piecewise-linear continuous maps, Int. J. Bifurcation Chaos, 27 (2017), 1730010, 20 pp. doi: 10.1142/S0218127417300105. [17] S. Sternberg, On the structure of local homeomorphisms of euclidean $n$-space, II, Amer. J. Math., 80 (1958), 623-631.  doi: 10.2307/2372774.

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##### References:
 [1] A. Delshams, M. Gonchenko and S. Gonchenko, On dynamics and bifurcations of area preserving maps with homoclinic tangencies, Nonlinearity, 28 (2015), 3027-3071.  doi: 10.1088/0951-7715/28/9/3027. [2] Y. Do and Y. C. Lai, Multistability and arithmetically period-adding bifurcations in piecewise smooth dynamical systems, Chaos, 18 (2008), 043107, 9 pp. doi: 10.1063/1.2985853. [3] N. K. Gavrilov and L. P. Silnikov, On three dimensional dynamical systems close to systems with structurally unstable homoclinic curve. I, Math. USSR Sbornik, 17 (1972), 467-485. [4] N. K. Gavrilov and L. P. Šilnikov, On three-dimensional dynamical systems close to systems with a structurally unstable homoclinic curve. II, Mat. Sb. (N.S.), 90, (1973), 167,139–156. [5] M. S. Gonchenko and S. V. Gonchenko, On cascades of elliptic periodic points in two-dimensional symplectic maps with homoclinic tangencies, Regul. Chaotic Dyns., 14 (2009), 116-136.  doi: 10.1134/S1560354709010080. [6] S. V. Gonchenko and L. P. Shilnikov, On two-dimensional area-preserving maps with homoclinic tangencies that have infinitely many generic elliptic periodic points, J. Math. Sci. (N. Y.), 128 (2005), 2767-2773.  doi: 10.1007/s10958-005-0228-6. [7] S. V. Gonchenko, L. P. Shil'nikov and D. V. Turaev, Quasiattractors and homoclinic tangencies, Comput. Math. Appl., 34 (1997), 195-227.  doi: 10.1016/S0898-1221(97)00124-7. [8] V. S. Gonchenko, Y. A. Kuznetsov and H. G. E. Meijer, Generalized Hénon map and bifurcations of homoclinic tangencies, SIAM J. Appl. Dyn. Syst., 4 (2005), 407-436.  doi: 10.1137/04060487X. [9] Y. A. Kuznetsov and H. G. E. Meijer, Numerical Bifurcation Analysis of Maps: From Theory to Software, Cambridge Monographs on Applied and Computational Mathematics, 34. Cambridge University Press, Cambridge, 2019.  doi: 10.1017/9781108585804. [10] S. S. Muni, R. I. McLachlan and D. J. W. Simpson, Homoclinic tangencies with infinitely many asymptotically stable single-round periodic solutions, Discrete Contin. Dyn. Syst., 41 (2021), 3629-3650.  doi: 10.3934/dcds.2021010. [11] S. E. Newhouse, Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9-18.  doi: 10.1016/0040-9383(74)90034-2. [12] J. Palis and F. Takens, Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations, Cambridge University Press, Cambridge, 1993. [13] L. P. Shilnikov, A. L. Shilnikov, D. V. Turaev and L. O. Chua, Methods of Qualitative Theory in Nonlinear Dynamics, World Scientific Publishing Co., Inc., River Edge, NJ, 1998. doi: 10.1142/9789812798596. [14] D. J. W. Simpson, Scaling laws for large numbers of coexisting attracting periodic solutions in the border-collision normal form, Int. J. Bifurcation Chaos, 24 (2014), 1450118, 28 pp. doi: 10.1142/S0218127414501181. [15] D. J. W. Simpson, Sequences of periodic solutions and infinitely many coexisting attractors in the border-collision normal form, Int. J. Bifurcation Chaos, 24 (2014), 1430018, 18 pp. doi: 10.1142/S0218127414300183. [16] D. J. W. Simpson and C. P. Tuffley, Subsumed homoclinic connections and infinitely many coexisting attractors in piecewise-linear continuous maps, Int. J. Bifurcation Chaos, 27 (2017), 1730010, 20 pp. doi: 10.1142/S0218127417300105. [17] S. Sternberg, On the structure of local homeomorphisms of euclidean $n$-space, II, Amer. J. Math., 80 (1958), 623-631.  doi: 10.2307/2372774.
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
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 $\mu = {\bf 0}$
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 $(x,y) = (1,1)$
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
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)
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)
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)
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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