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A shrinking target theorem for ergodic transformations of the unit interval
Unfolding globally resonant homoclinic tangencies
School of Fundamental Sciences, Massey University, Palmerston North, New Zealand |
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.
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. |








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