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December  2020, 40(12): 6815-6836. doi: 10.3934/dcds.2020295

How to identify a hyperbolic set as a blender

1. 

Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1142, New Zealand

2. 

Graduate School of Business Administration, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo 186-8601, Japan

* Corresponding author: h.m.osinga@auckland.ac.nz

Received  August 2019 Revised  June 2020 Published  December 2020 Early access  August 2020

A blender is a hyperbolic set with a stable or unstable invariant manifold that behaves as a geometric object of a dimension larger than that of the respective manifold itself. Blenders have been constructed in diffeomorphisms with a phase space of dimension at least three. We consider here the question of how one can identify, characterize and also visualize the underlying hyperbolic set of a given diffeomorphism to verify whether it actually is a blender or not. More specifically, we employ advanced numerical techniques for the computation of global manifolds to identify the hyperbolic set and its stable and unstable manifolds in an explicit Hénon-like family of three-dimensional diffeomorphisms. This allows to determine and illustrate whether the hyperbolic set is a blender; in particular, we consider as a distinguishing feature the self-similar structure of the intersection set of the respective global invariant manifold with a plane. By checking and illustrating a denseness property, we are able to identify a parameter range over which the hyperbolic set is a blender, and we discuss and illustrate how the blender disappears.

Citation: Stefanie Hittmeyer, Bernd Krauskopf, Hinke M. Osinga, Katsutoshi Shinohara. How to identify a hyperbolic set as a blender. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6815-6836. doi: 10.3934/dcds.2020295
References:
[1]

A. BackJ. GuckenheimerM. MyersF. Wicklin and P. Worfolk, DsTool: Computer assisted exploration of dynamical systems, Notices Amer. Math. Soc., 39 (1992), 303-309. 

[2]

P. G. BarrientosA. Raibekas and A. A. P. Rodrigues, Chaos near a reversible homoclinic bifocus, Dynamical Systems, 34 (2019), 504-516.  doi: 10.1080/14689367.2019.1569592.

[3]

C. BonattiS. CrovisierL. J. Díaz and A. Wilkinson, What is$\ldots$ a blender?, Notices Amer. Math. Soc., 63 (2016), 1175-1178.  doi: 10.1090/noti1438.

[4]

C. Bonatti and L. J. Díaz, Persistent nonhyperbolic transitive diffeomorphisms, Ann. of Math. (2), 143 (1996), 357–396. doi: 10.2307/2118647.

[5]

C. BonattiL. J. Díaz and S. Kiriki, Stabilization of heterodimensional cycles, Nonlinearity, 25 (2012), 931-960.  doi: 10.1088/0951-7715/25/4/931.

[6]

C. Bonatti, L. J. Díaz and M. Viana, Dynamics beyond Uniform Hyperbolicity. A global geometric and probabilistic perspective, vol. 102 of Encyclopaedia Math. Sci., Springer-Verlag, Berlin, 2005.

[7]

C. Bonatti and L. J. Díaz, Abundance of $C^1$-robust homoclinic tangencies, Trans. Amer. Math. Soc., 364 (2012), 5111-5148.  doi: 10.1090/S0002-9947-2012-05445-6.

[8]

L. J. DíazS. Kiriki and K. Shinohara, Blenders in centre unstable Hénon-like families: With an application to heterodimensional bifurcations, Nonlinearity, 27 (2014), 353-378.  doi: 10.1088/0951-7715/27/3/353.

[9]

L. J. Díaz and S. A. Pérez, Hénon-like families and blender-horseshoes at nontransverse heterodimensional cycles, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 29 (2019), 1930006. doi: 10.1142/S0218127419300064.

[10]

J. P. EnglandB. Krauskopf and H. M. Osinga, Computing one-dimensional stable manifolds and stable sets of planar maps without the inverse, SIAM J. Appl. Dyn. Syst., 3 (2004), 161-190.  doi: 10.1137/030600131.

[11]

A. C. Fowler and C. T. Sparrow, Bifocal homoclinic orbits in four dimensions, Nonlinearity, 4 (1991), 1159-1182.  doi: 10.1088/0951-7715/4/4/007.

[12]

R. Gilmore and M. Lefranc, The Topology of Chaos: Alice in Stretch and Squeezeland, Wiley-Interscience [John Wiley & Sons], New York, 2002.

[13]

C. GrebogiE. Ott and J. A. Yorke, Metamorphoses of basin boundaries in nonlinear dynamical systems, Phys. Rev. Lett., 56 (1986), 1011-1014.  doi: 10.1103/PhysRevLett.56.1011.

[14]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2.

[15]

A. Hammerlindl, B. Krauskopf, G. Mason and H. M. Osinga, Global manifold structure of a continuous-time heterodimensional cycle, arXiv: 1906.11438, 2019.

[16]

M. Hénon, A two-dimensional mapping with a strange attractor, Comm. Math. Phys., 50 (1976), 69-77.  doi: 10.1007/BF01608556.

[17]

S. Hittmeyer, B. Krauskopf, H. M. Osinga and K. Shinohara, Existence of blenders in a Hénon-like family: Geometric insights from invariant manifold computations, Nonlinearity, 31 (2018), R239–R267. doi: 10.1088/1361-6544/aacd66.

[18]

A. J. Homburg and J. S. W. Lamb, Symmetric homoclinic tangles in reversible systems, Ergodic Theory and Dynamical Systems, 26 (2006), 1769-1789.  doi: 10.1017/S0143385706000472.

[19]

A. J. Homburg and B. Sandstede, Homoclinic and heteroclinic bifurcations in vector fields, in H. Broer, F. Takens and B. Hasselblatt, Handbook of Dynamical Systems, vol. III Elsevier, Amsterdam, 2010,379–524.

[20]

B. Krauskopf and H. Osinga, Growing $1$D and quasi-$2$D unstable manifolds of maps, J. Comput. Phys., 146 (1998), 404-419.  doi: 10.1006/jcph.1998.6059.

[21]

B. Krauskopf and H. Osinga, Globalizing two-dimensional unstable manifolds of maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8 (1998), 483-503.  doi: 10.1142/S0218127498000310.

[22]

B. Krauskopf and H. M. Osinga, Investigating torus bifurcations in the forced Van der Pol oscillator, in Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems (Minneapolis, MN, 1997), 119, 199–208, IMA Vol. Math. Appl., Springer-Verlag, New York, 2000. doi: 10.1007/978-1-4612-1208-9_9.

[23]

D. Li, Homoclinic bifurcations that give rise to heterodimensional cycles near a saddle-focus equilibrium, Nonlinearity, 30 (2017), 173-206.  doi: 10.1088/1361-6544/30/1/173.

[24]

J. J. Palis and W. de Melo, Geometric Theory of Dynamical Systems, Springer-Verlag, New York, 1982.

[25]

J. Palis and F. Takens, Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations, Cambridge Studies in Advanced Mathematics, 35. Cambridge University Press, Cambridge, 1993.

[26]

A. A. P. Rodrigues, Strange attractors and wandering domains near a homoclinic cycle to a bifocus, Journal of Differential Equations, 269 (2020), 3221-3258.  doi: 10.1016/j.jde.2020.02.027.

[27]

M. Shub, Global Stability of Dynamical Systems, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4757-1947-5.

[28]

M. Shub, What is $\ldots$ a horseshoe?, Notices Amer. Math. Soc., 52 (2005), 516-517. 

[29]

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817. 

[30]

W. ZhangB. Krauskopf and V. Kirk, How to find a codimension-one heteroclinic cycle between two periodic orbits, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 2825-2851.  doi: 10.3934/dcds.2012.32.2825.

show all references

References:
[1]

A. BackJ. GuckenheimerM. MyersF. Wicklin and P. Worfolk, DsTool: Computer assisted exploration of dynamical systems, Notices Amer. Math. Soc., 39 (1992), 303-309. 

[2]

P. G. BarrientosA. Raibekas and A. A. P. Rodrigues, Chaos near a reversible homoclinic bifocus, Dynamical Systems, 34 (2019), 504-516.  doi: 10.1080/14689367.2019.1569592.

[3]

C. BonattiS. CrovisierL. J. Díaz and A. Wilkinson, What is$\ldots$ a blender?, Notices Amer. Math. Soc., 63 (2016), 1175-1178.  doi: 10.1090/noti1438.

[4]

C. Bonatti and L. J. Díaz, Persistent nonhyperbolic transitive diffeomorphisms, Ann. of Math. (2), 143 (1996), 357–396. doi: 10.2307/2118647.

[5]

C. BonattiL. J. Díaz and S. Kiriki, Stabilization of heterodimensional cycles, Nonlinearity, 25 (2012), 931-960.  doi: 10.1088/0951-7715/25/4/931.

[6]

C. Bonatti, L. J. Díaz and M. Viana, Dynamics beyond Uniform Hyperbolicity. A global geometric and probabilistic perspective, vol. 102 of Encyclopaedia Math. Sci., Springer-Verlag, Berlin, 2005.

[7]

C. Bonatti and L. J. Díaz, Abundance of $C^1$-robust homoclinic tangencies, Trans. Amer. Math. Soc., 364 (2012), 5111-5148.  doi: 10.1090/S0002-9947-2012-05445-6.

[8]

L. J. DíazS. Kiriki and K. Shinohara, Blenders in centre unstable Hénon-like families: With an application to heterodimensional bifurcations, Nonlinearity, 27 (2014), 353-378.  doi: 10.1088/0951-7715/27/3/353.

[9]

L. J. Díaz and S. A. Pérez, Hénon-like families and blender-horseshoes at nontransverse heterodimensional cycles, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 29 (2019), 1930006. doi: 10.1142/S0218127419300064.

[10]

J. P. EnglandB. Krauskopf and H. M. Osinga, Computing one-dimensional stable manifolds and stable sets of planar maps without the inverse, SIAM J. Appl. Dyn. Syst., 3 (2004), 161-190.  doi: 10.1137/030600131.

[11]

A. C. Fowler and C. T. Sparrow, Bifocal homoclinic orbits in four dimensions, Nonlinearity, 4 (1991), 1159-1182.  doi: 10.1088/0951-7715/4/4/007.

[12]

R. Gilmore and M. Lefranc, The Topology of Chaos: Alice in Stretch and Squeezeland, Wiley-Interscience [John Wiley & Sons], New York, 2002.

[13]

C. GrebogiE. Ott and J. A. Yorke, Metamorphoses of basin boundaries in nonlinear dynamical systems, Phys. Rev. Lett., 56 (1986), 1011-1014.  doi: 10.1103/PhysRevLett.56.1011.

[14]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2.

[15]

A. Hammerlindl, B. Krauskopf, G. Mason and H. M. Osinga, Global manifold structure of a continuous-time heterodimensional cycle, arXiv: 1906.11438, 2019.

[16]

M. Hénon, A two-dimensional mapping with a strange attractor, Comm. Math. Phys., 50 (1976), 69-77.  doi: 10.1007/BF01608556.

[17]

S. Hittmeyer, B. Krauskopf, H. M. Osinga and K. Shinohara, Existence of blenders in a Hénon-like family: Geometric insights from invariant manifold computations, Nonlinearity, 31 (2018), R239–R267. doi: 10.1088/1361-6544/aacd66.

[18]

A. J. Homburg and J. S. W. Lamb, Symmetric homoclinic tangles in reversible systems, Ergodic Theory and Dynamical Systems, 26 (2006), 1769-1789.  doi: 10.1017/S0143385706000472.

[19]

A. J. Homburg and B. Sandstede, Homoclinic and heteroclinic bifurcations in vector fields, in H. Broer, F. Takens and B. Hasselblatt, Handbook of Dynamical Systems, vol. III Elsevier, Amsterdam, 2010,379–524.

[20]

B. Krauskopf and H. Osinga, Growing $1$D and quasi-$2$D unstable manifolds of maps, J. Comput. Phys., 146 (1998), 404-419.  doi: 10.1006/jcph.1998.6059.

[21]

B. Krauskopf and H. Osinga, Globalizing two-dimensional unstable manifolds of maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8 (1998), 483-503.  doi: 10.1142/S0218127498000310.

[22]

B. Krauskopf and H. M. Osinga, Investigating torus bifurcations in the forced Van der Pol oscillator, in Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems (Minneapolis, MN, 1997), 119, 199–208, IMA Vol. Math. Appl., Springer-Verlag, New York, 2000. doi: 10.1007/978-1-4612-1208-9_9.

[23]

D. Li, Homoclinic bifurcations that give rise to heterodimensional cycles near a saddle-focus equilibrium, Nonlinearity, 30 (2017), 173-206.  doi: 10.1088/1361-6544/30/1/173.

[24]

J. J. Palis and W. de Melo, Geometric Theory of Dynamical Systems, Springer-Verlag, New York, 1982.

[25]

J. Palis and F. Takens, Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations, Cambridge Studies in Advanced Mathematics, 35. Cambridge University Press, Cambridge, 1993.

[26]

A. A. P. Rodrigues, Strange attractors and wandering domains near a homoclinic cycle to a bifocus, Journal of Differential Equations, 269 (2020), 3221-3258.  doi: 10.1016/j.jde.2020.02.027.

[27]

M. Shub, Global Stability of Dynamical Systems, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4757-1947-5.

[28]

M. Shub, What is $\ldots$ a horseshoe?, Notices Amer. Math. Soc., 52 (2005), 516-517. 

[29]

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817. 

[30]

W. ZhangB. Krauskopf and V. Kirk, How to find a codimension-one heteroclinic cycle between two periodic orbits, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 2825-2851.  doi: 10.3934/dcds.2012.32.2825.

Figure 1.  Illustration of the hyperbolic set $ \Lambda_h $ (black dots) as the closure of the intersection between the manifolds $ W^s(p_h^\pm) $ (blue curves) and $ W^u(p_h^\pm) $ (red curves) of the saddle fixed points $ p_h^\pm $ (green crosses); panels (a) and (b) show two views of the $ (x, y) $-plane, and panel (c) shows the Poincaré disk in the $ (\bar{x}, \bar{y}) $-plane
Figure 2.  The hyperbolic set $ \Lambda $ (black dots) of $ H $ with $ \xi = 1.2 $, determined as the intersection set of $ W^s(p^-) $ (dark blue) and $ W^s(p^+) $ (light blue) with $ W^u(p^-) $ (red surface), shown in $ (\bar{x}, \bar{y}, \bar{z}) $-space (a) and in projection onto the $ (\bar{x}, \bar{z}) $-plane (b). Panels (c) and (d) illustrate $ \Lambda $ and its tangent space $ T^s(\Lambda) $ (green lines) in $ (x, y, z) $-space and in projection onto the $ (x, z) $-plane, respectively; four regions are highlighted with different shades of green
Figure 3.  The hyperbolic set $ \Lambda $ (black dots) of $ H $ with $ \xi = 2.0 $, determined as the intersection set of $ W^s(p^-) $ (dark blue) and $ W^s(p^+) $ (light blue) with $ W^u(p^-) $ (red surface), shown in $ (\bar{x}, \bar{y}, \bar{z}) $-space (a) and in projection onto the $ (\bar{x}, \bar{z}) $-plane (b). Panels (c) and (d) illustrate $ \Lambda $ and its tangent space $ T^s(\Lambda) $ (green lines) in $ (x, y, z) $-space and in projection onto the $ (x, z) $-plane, respectively; four different regions are highlighted with different shades of green
Figure 4.  The hyperbolic set $ \Lambda $ of $ H $ with $ \xi = 0.8 $, determined as the intersection set of $ W^u(p^-) $ (red curves) and $ W^u(p^+) $ (magenta curves) with $ W^s(p^-) $ (blue surface), shown in $ (\bar{x}, \bar{y}, \bar{z}) $-space (a) and in projection onto the $ (\bar{y}, \bar{z}) $-plane (b). Panels (c) and (d) illustrate $ \Lambda $ and its tangent space $ T^u(\Lambda) $ (green lines) in $ (x, y, z) $-space and in projection onto the $ (y, z) $-plane, respectively; four regions are highlighted with different shades of green
Figure 5.  The hyperbolic set $ \Lambda $ of $ H $ with $ \xi = 0.45 $, determined as the intersection set of $ W^u(p^-) $ (red curves) and $ W^u(p^+) $ (magenta curves) with $ W^s(p^-) $ (blue surface), shown in $ (\bar{x}, \bar{y}, \bar{z}) $-space (a) and in projection onto the $ (\bar{y}, \bar{z}) $-plane (b). Panels (c) and (d) illustrate $ \Lambda $ and its tangent space $ T^u(\Lambda) $ (green lines) in $ (x, y, z) $-space and in projection onto the $ (y, z) $-plane, respectively; four regions are highlighted with different shades of green
Figure 6.  The intersection set for $ \xi = 1.2 $ of the stable manifolds $ W^s(p^-) $ (dark blue) and $ W^s(p^+) $ (light blue) with the section $ \Sigma $ (gray plane) defined by $ \bar{x} = \bar{y} $. Panel (a) shows the intersection points in $ \Sigma $ and panel (b) shows how $ W^s(p^\pm) $ intersect $ \Sigma $ in $ (\bar{x}, \bar{y}, \bar{z}) $-space
Figure 7.  The intersection set for $ \xi = 2.0 $ of the stable manifold $ W^s(p^-) $ (dark blue) and $ W^s(p^+) $ (light blue) with the section $ \Sigma $ (gray plane) defined by $ \bar{x} = \bar{y} $. Panel (a) shows the intersection points in $ \Sigma $ and panel (b) shows how $ W^s(p^\pm) $ intersect $ \Sigma $ in $ (\bar{x}, \bar{y}, \bar{z}) $-space
Figure 8.  The five largest $ \bar{z} $-gaps $ \Delta^i $, for $ i = 1, \dots, 5 $, of $ W^s(p^-) $ in $ \Sigma $ as a function of the arclength, represented by the exponent $ k $, for $ \xi = 1.2 $ (a1) and for $ \xi = 2.0 $ (a2). Panel (b) shows the largest gap $ \Delta^1 $ versus $ \xi $ for $ k = 6 $ (red) when $ 0 < \xi <1 $ and for $ k = 7 $ (blue) when $ 1 < \xi $. Panel (c) shows the associated $ \bar{z} $-values of $ p^\pm $ (green), $ W^s(p^-) \cap \Sigma $ (blue) and $ W^u(p^-) \cap \Sigma $ (red), respectively
Figure 9.  Self-similar structure of the intersection set $ W^s(p^-) \cap \Sigma $ for $ \xi = 1.2 $. Panel (a1) shows a part of $ W^s(p^-) \cap \Sigma $ in a color coding according to the $ \bar{x} $-values, and panel (a2) is an enlargement. Panels (b1) and (b2) show $ \bar{x}_{n+1} $ versus $ \bar{x}_{n} $ and $ \bar{z}_{n+1} $ versus $ \bar{z}_{n} $, respectively, of successive points of $ W^s(p^-) \cap \Sigma $
Figure 10.  Self-similar structure of the intersection set $ W^s(p^-) \cap \Sigma $ for $ \xi = 2.0 $. Panel (a1) shows a part of $ W^s(p^-) \cap \Sigma $ in a color coding according to the $ \bar{x} $-values, and panel (a2) is an enlargement. Panels (b1) and (b2) show $ \bar{x}_{n+1} $ versus $ \bar{x}_{n} $ and $ \bar{z}_{n+1} $ versus $ \bar{z}_{n} $, respectively, of successive points of $ W^s(p^-) \cap \Sigma $
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