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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 and Continuous Dynamical Systems, 2021, 41 (8) : 3629-3650. doi: 10.3934/dcds.2021010
References:
[1]

C. C. CanavierD. A. BaxterJ. 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.

[2]

N. G. de Brujin, Asymptotic Methods in Analysis, Dover, New York, 1981.

[3]

A. DelshamsM. 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.

[4]

A. DelshamsM. 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.

[5]

S. N. Elaydi, Discrete Chaos with Applications in Science and Engineering, Chapman and Hall., Boca Raton, FL, 2008.

[6]

U. Feudel, Complex dynamics in multistable systems, Int. J. Bifurcation Chaos, 18 (2008), 1607-1626.  doi: 10.1142/S0218127408021233.

[7]

J. A. C. Gallas, Dissecting shrimps: Results for some one-dimensional physical systems, Physica A, 202 (1994), 196-223. 

[8]

J. M. Gambaudo and C. Tresser, Simple models for bifurcations creating horseshoes, J. Stat. Phys., 32 (1983), 455-476.  doi: 10.1007/BF01008950.

[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. 

[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. 

[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.

[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.

[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.

[14]

S. V. GonchenkoL. 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.

[15]

V. S. GonchenkoYu. 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.

[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.

[17]

V. Lakshmikantham and D. Trigiante, Theory of Difference Equations: Numerical Methods and Applications, Marcel Dekker, Inc., New York, 2002. doi: 10.1201/9780203910290.

[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.

[19]

S. E. Newhouse, Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9-18.  doi: 10.1016/0040-9383(74)90034-2.

[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.

[21] J. Palis and F. Takens, Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations, Cambridge University Press, New York, 1993. 
[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.

[23]

R. C. Robinson, An Introduction to Dynamical Systems. Continuous and Discrete, Prentice Hall, Upper Saddle River, NJ, 2004.

[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.

[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.

[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.

[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.

[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.

[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.

show all references

References:
[1]

C. C. CanavierD. A. BaxterJ. 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.

[2]

N. G. de Brujin, Asymptotic Methods in Analysis, Dover, New York, 1981.

[3]

A. DelshamsM. 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.

[4]

A. DelshamsM. 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.

[5]

S. N. Elaydi, Discrete Chaos with Applications in Science and Engineering, Chapman and Hall., Boca Raton, FL, 2008.

[6]

U. Feudel, Complex dynamics in multistable systems, Int. J. Bifurcation Chaos, 18 (2008), 1607-1626.  doi: 10.1142/S0218127408021233.

[7]

J. A. C. Gallas, Dissecting shrimps: Results for some one-dimensional physical systems, Physica A, 202 (1994), 196-223. 

[8]

J. M. Gambaudo and C. Tresser, Simple models for bifurcations creating horseshoes, J. Stat. Phys., 32 (1983), 455-476.  doi: 10.1007/BF01008950.

[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. 

[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. 

[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.

[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.

[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.

[14]

S. V. GonchenkoL. 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.

[15]

V. S. GonchenkoYu. 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.

[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.

[17]

V. Lakshmikantham and D. Trigiante, Theory of Difference Equations: Numerical Methods and Applications, Marcel Dekker, Inc., New York, 2002. doi: 10.1201/9780203910290.

[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.

[19]

S. E. Newhouse, Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9-18.  doi: 10.1016/0040-9383(74)90034-2.

[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.

[21] J. Palis and F. Takens, Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations, Cambridge University Press, New York, 1993. 
[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.

[23]

R. C. Robinson, An Introduction to Dynamical Systems. Continuous and Discrete, Prentice Hall, Upper Saddle River, NJ, 2004.

[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.

[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.

[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.

[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.

[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.

[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.

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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