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August  2021, 41(8): 3629-3650. doi: 10.3934/dcds.2021010

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

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

* Corresponding author: Sishu Shankar Muni

Received  May 2020 Revised  November 2020 Published  August 2021 Early access  January 2021

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.

Citation: Sishu Shankar Muni, Robert I. McLachlan, David J. W. Simpson. Homoclinic tangencies with infinitely many asymptotically stable single-round periodic solutions. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3629-3650. doi: 10.3934/dcds.2021010
##### References:
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Feudel, Complex dynamics in multistable systems, Int. J. Bifurcation Chaos, 18 (2008), 1607-1626.  doi: 10.1142/S0218127408021233.  Google Scholar [7] J. A. C. Gallas, Dissecting shrimps: Results for some one-dimensional physical systems, Physica A, 202 (1994), 196-223.   Google Scholar [8] J. M. Gambaudo and C. Tresser, Simple models for bifurcations creating horseshoes, J. Stat. Phys., 32 (1983), 455-476.  doi: 10.1007/BF01008950.  Google Scholar [9] N. K. Gavrilov and L. P. Silnikov, On three dimensional dynamical systems close to systems with structurally unstable homoclinic curve. I, Mat. Sb. (N.S.), 88 (1972), 475-492.   Google Scholar [10] N. K. Gavrilov and L. P. Silnikov, On three-dimensional dynamical systems close to systems with a structurally unstable homoclinic curve. II, Mat. Sb. (N.S.), 90 (1973), 139-156.   Google Scholar [11] 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.  Google Scholar [12] S. V. Gonchenko and L. P. Shil'nikov, Arithmetic properties of topological invariants of systems with nonstructurally-stable homoclinic trajectories, Ukr. Math. J., 39 (1987), 15-21.  doi: 10.1007/BF01056417.  Google Scholar [13] 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.  Google Scholar [14] S. V. Gonchenko, L. P. Shil'nikov and D. V. Turaev, Dynamical phenomena in systems with structurally unstable Poincaré homoclinic orbits, Chaos, 6 (1996), 15-31.  doi: 10.1063/1.166154.  Google Scholar [15] V. S. Gonchenko, Yu. 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.  Google Scholar [16] P. Hirschberg and C. R. Laing, Successive homoclinic tangencies to a limit cycle, Physica D, 89 (1995), 1-14.  doi: 10.1016/0167-2789(95)00211-1.  Google Scholar [17] V. Lakshmikantham and D. Trigiante, Theory of Difference Equations: Numerical Methods and Applications, Marcel Dekker, Inc., New York, 2002. doi: 10.1201/9780203910290.  Google Scholar [18] C. Mira, L. Gardini, A. Barugola and J. C. Cathala, Chaotic Dynamics in Two Dimensional Noninvertible Maps, World Scientific, 1996. doi: 10.1142/9789812798732.  Google Scholar [19] S. E. Newhouse, Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9-18.  doi: 10.1016/0040-9383(74)90034-2.  Google Scholar [20] C. N. Ngonghala, U. Feudel and K. Showalter, Extreme multistability in a chemical model system, Phys. Rev. E, 83 (2011), 056206. doi: 10.1103/PhysRevE.83.056206.  Google Scholar [21] J. Palis and F. Takens, Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations, Cambridge University Press, New York, 1993.   Google Scholar [22] S. Rahmstorf and J. Willebrand, The role of temperature feedback in stabilizing the thermohaline circulation, J. Phys. Oceanogr, 25 (1995), 787-805.  doi: 10.1175/1520-0485(1995)025<0787:TROTFI>2.0.CO;2.  Google Scholar [23] R. C. Robinson, An Introduction to Dynamical Systems. Continuous and Discrete, Prentice Hall, Upper Saddle River, NJ, 2004.  Google Scholar [24] L. P. Shil'nikov, A. L. Shil'nikov, D. V. Turaev and L. O. Chua, Methods of Qualitative Theory in Nonlinear Dynamics, Part I, volume 4, World Scientific Singapore, 1998. doi: 10.1142/9789812798596.  Google Scholar [25] 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, 28pp. doi: 10.1142/S0218127414501181.  Google Scholar [26] 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, 18pp. doi: 10.1142/S0218127414300183.  Google Scholar [27] D. J. W. Simpson, Unfolding codimension-two subsumed homoclinic connections in two-dimensional piecewise-linear maps, Int. J. Bifurcation Chaos, 30 (2020), 203006, 12pp. doi: 10.1142/S0218127420300062.  Google Scholar [28] D. J. W. Simpson and C. P. Tuffley, Subsumed homoclinic connections and infinitely many coexisting attractors in piecewise-linear maps, Int. J. Bifurcation Chaos, 27 (2017), 1730010, 20 pp. doi: 10.1142/S0218127417300105.  Google Scholar [29] S. Sternberg, On the structure of local homeomorphisms of Euclidean $n$-space, II, Amer. J. Math., 80 (1958), 623-631.  doi: 10.2307/2372774.  Google Scholar

show all references

##### References:
 [1] C. C. Canavier, D. A. Baxter, J. W. Clark and J. H. Byrne, Nonlinear dynamics in a model neuron provide a novel mechanism for transient synaptic inputs to produce long-term alterations of postsynaptic activity, J. Neurophysiol, 69 (1993), 2252-2257.  doi: 10.1152/jn.1993.69.6.2252.  Google Scholar [2] N. G. de Brujin, Asymptotic Methods in Analysis, Dover, New York, 1981.  Google Scholar [3] 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.  Google Scholar [4] A. Delshams, M. Gonchenko and S. V. Gonchenko, On bifurcations of area-preserving and non-orientable maps with quadratic homoclinic tangencies, Regul. Chaotic Dyn., 19 (2014), 702-717.  doi: 10.1134/S1560354714060082.  Google Scholar [5] S. N. Elaydi, Discrete Chaos with Applications in Science and Engineering, Chapman and Hall., Boca Raton, FL, 2008.  Google Scholar [6] U. Feudel, Complex dynamics in multistable systems, Int. J. Bifurcation Chaos, 18 (2008), 1607-1626.  doi: 10.1142/S0218127408021233.  Google Scholar [7] J. A. C. Gallas, Dissecting shrimps: Results for some one-dimensional physical systems, Physica A, 202 (1994), 196-223.   Google Scholar [8] J. M. Gambaudo and C. Tresser, Simple models for bifurcations creating horseshoes, J. Stat. Phys., 32 (1983), 455-476.  doi: 10.1007/BF01008950.  Google Scholar [9] N. K. Gavrilov and L. P. Silnikov, On three dimensional dynamical systems close to systems with structurally unstable homoclinic curve. I, Mat. Sb. (N.S.), 88 (1972), 475-492.   Google Scholar [10] N. K. Gavrilov and L. P. Silnikov, On three-dimensional dynamical systems close to systems with a structurally unstable homoclinic curve. II, Mat. Sb. (N.S.), 90 (1973), 139-156.   Google Scholar [11] 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.  Google Scholar [12] S. V. Gonchenko and L. P. Shil'nikov, Arithmetic properties of topological invariants of systems with nonstructurally-stable homoclinic trajectories, Ukr. Math. J., 39 (1987), 15-21.  doi: 10.1007/BF01056417.  Google Scholar [13] 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.  Google Scholar [14] S. V. Gonchenko, L. P. Shil'nikov and D. V. Turaev, Dynamical phenomena in systems with structurally unstable Poincaré homoclinic orbits, Chaos, 6 (1996), 15-31.  doi: 10.1063/1.166154.  Google Scholar [15] V. S. Gonchenko, Yu. 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.  Google Scholar [16] P. Hirschberg and C. R. Laing, Successive homoclinic tangencies to a limit cycle, Physica D, 89 (1995), 1-14.  doi: 10.1016/0167-2789(95)00211-1.  Google Scholar [17] V. Lakshmikantham and D. Trigiante, Theory of Difference Equations: Numerical Methods and Applications, Marcel Dekker, Inc., New York, 2002. doi: 10.1201/9780203910290.  Google Scholar [18] C. Mira, L. Gardini, A. Barugola and J. C. Cathala, Chaotic Dynamics in Two Dimensional Noninvertible Maps, World Scientific, 1996. doi: 10.1142/9789812798732.  Google Scholar [19] S. E. Newhouse, Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9-18.  doi: 10.1016/0040-9383(74)90034-2.  Google Scholar [20] C. N. Ngonghala, U. Feudel and K. Showalter, Extreme multistability in a chemical model system, Phys. Rev. E, 83 (2011), 056206. doi: 10.1103/PhysRevE.83.056206.  Google Scholar [21] J. Palis and F. Takens, Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations, Cambridge University Press, New York, 1993.   Google Scholar [22] S. Rahmstorf and J. Willebrand, The role of temperature feedback in stabilizing the thermohaline circulation, J. Phys. Oceanogr, 25 (1995), 787-805.  doi: 10.1175/1520-0485(1995)025<0787:TROTFI>2.0.CO;2.  Google Scholar [23] R. C. Robinson, An Introduction to Dynamical Systems. Continuous and Discrete, Prentice Hall, Upper Saddle River, NJ, 2004.  Google Scholar [24] L. P. Shil'nikov, A. L. Shil'nikov, D. V. Turaev and L. O. Chua, Methods of Qualitative Theory in Nonlinear Dynamics, Part I, volume 4, World Scientific Singapore, 1998. doi: 10.1142/9789812798596.  Google Scholar [25] 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, 28pp. doi: 10.1142/S0218127414501181.  Google Scholar [26] 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, 18pp. doi: 10.1142/S0218127414300183.  Google Scholar [27] D. J. W. Simpson, Unfolding codimension-two subsumed homoclinic connections in two-dimensional piecewise-linear maps, Int. J. Bifurcation Chaos, 30 (2020), 203006, 12pp. doi: 10.1142/S0218127420300062.  Google Scholar [28] D. J. W. Simpson and C. P. Tuffley, Subsumed homoclinic connections and infinitely many coexisting attractors in piecewise-linear maps, Int. J. Bifurcation Chaos, 27 (2017), 1730010, 20 pp. doi: 10.1142/S0218127417300105.  Google Scholar [29] S. Sternberg, On the structure of local homeomorphisms of Euclidean $n$-space, II, Amer. J. Math., 80 (1958), 623-631.  doi: 10.2307/2372774.  Google Scholar
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
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 $\sigma > 1$
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
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
The basic structure of the phase space of the map (15)
The function (19) (with (20)) that we use as a convex combination parameter in (15)
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
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
. 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">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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