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

Unfolding globally resonant homoclinic tangencies

Sishu Shankar Muni , , Robert I. McLachlan and  David J. W. Simpson

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.

Keywords: Multistability, homoclinic tangency, smooth maps, asymptotically stable, periodic solutions.
Mathematics Subject Classification: Primary: 37G25; Secondary: 37G15, 39A23.
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.

show all references

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.

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
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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} $
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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) $
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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
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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)
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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)
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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)
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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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