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July  2021, 41(7): 3367-3387. doi: 10.3934/dcds.2020409

A constructive approach to robust chaos using invariant manifolds and expanding cones

Paul A. Glendinning 1, and  David J. W. Simpson 2,,

1. 

Department of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL, UK
2. 

School of Fundamental Sciences, Massey University, Colombo Road, Palmerston North, 4410, New Zealand

*Corresponding author: D. J. W. Simpson

Received  July 2020 Revised  November 2020 Published  July 2021 Early access  December 2020

Fund Project: The authors were supported by Marsden Fund contract MAU1809, managed by Royal Society Te Apārangi

Chaotic attractors in the two-dimensional border-collision normal form (a piecewise-linear map) can persist throughout open regions of parameter space. Such robust chaos has been established rigorously in some parameter regimes. Here we provide formal results for robust chaos in the original parameter regime of [S. Banerjee, J.A. Yorke, C. Grebogi, Robust Chaos, Phys. Rev. Lett. 80(14):3049-3052, 1998]. We first construct a trapping region in phase space to prove the existence of a topological attractor. We then construct an invariant expanding cone in tangent space to prove that tangent vectors expand and so no invariant set can have only negative Lyapunov exponents. Under additional assumptions we characterise an attractor as the closure of the unstable manifold of a fixed point and prove that it satisfies Devaney's definition of chaos.

Keywords: Piecewise-linear, piecewise-smooth, border-collision bifurcation, Lyapunov exponent, robust chaos.
Mathematics Subject Classification: Primary: 37G35; Secondary: 39A28.
Citation: Paul A. Glendinning, David J. W. Simpson. A constructive approach to robust chaos using invariant manifolds and expanding cones. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3367-3387. doi: 10.3934/dcds.2020409
References:
[1]

G. Álvarez, F. Montoya, M. Romera and G. Pastor, Cryptanalysis of a discrete chaotic cryptosystem using external key, Phys. Lett. A, 319 (2003), 334-339.  doi: 10.1016/j.physleta.2003.10.044.

[2]

S. Banerjee and C. Grebogi, Border collision bifurcations in two-dimensional piecewise smooth maps, Phys. Rev. E, 59 (1999), 4052-4061.  doi: 10.1103/PhysRevE.59.4052.

[3]

S. Banerjee, J. A. Yorke and C. Grebogi, Robust chaos, Phys. Rev. Lett., 80 (1998), 3049-3052.  doi: 10.1103/PhysRevLett.80.3049.

[4]

J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacey, On Devaney's definition of chaos, Amer. Math. Monthly, 99 (1992), 332-334.  doi: 10.1080/00029890.1992.11995856.

[5]

L. Barreira and Y. Pesin, Nonuniform Hyperbolicity. Dynamics of Systems with Nonzero Lyapunov Exponents, volume 115 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2007. doi: 10.1017/CBO9781107326026.

[6]

M. Benedicks and L. Carleson, The dynamics of the Henon map, Ann. Math., 133 (1991), 73-169.  doi: 10.2307/2944326.

[7]

Y. Cao and Z. Liu, The geometric structure of strange attractors in the Lozi map, Commun. Nonlin. Sci. Numer. Simul., 3 (1998), 119-123.  doi: 10.1016/S1007-5704(98)90076-4.

[8]

Y. Cao and Z. Liu, Strange attractors in the orientation-preserving Lozi map, Chaos Solitons Fractals, 9 (1998), 1857–1863. doi: 10.1016/S0960-0779(97)00180-X.

[9]

P. Collet and Y. Levy, Ergodic properties of the Lozi mappings, Commun. Math. Phys., 93 (1984), 461-481.  doi: 10.1007/BF01212290.

[10]

E. Cornelis and M. Wojtkowski, A criterion for the positivity of the Liapunov characteristic exponent, Ergod. Th. & Dynam. Sys., 4 (1984), 527-539.  doi: 10.1017/S0143385700002625.

[11]

S. Das and J. A. Yorke, Multichaos from quasiperiodicity, SIAM J. Appl. Dyn. Syst., 16 (2017), 2196-2212.  doi: 10.1137/17M1113199.

[12]

R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989.

[13]

M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems. Theory and Applications, Springer-Verlag, London, Ltd., London, 2008.

[14]

J.-P. Duval, Génération d'une section des classes de conjugaison et arbre des mots de Lyndon de longueur bornée, Theoret. Comput. Sci., 60 (1988), 255–283. In French. doi: 10.1016/0304-3975(88)90113-2.

[15]

J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Mod. Phys., 57 (1985), 617-656.  doi: 10.1103/RevModPhys.57.617.

[16]

R. Edwards, J. J. McDonald and M. J. Tsatsomeros, On matrices with common invariant cones with applications in neural and gene networks, Linear Algebra Appl., 398 (2005), 37-67.  doi: 10.1016/j.laa.2004.04.005.

[17]

E. Glasner and B. Weiss, Sensitive dependence on initial conditions, Nonlinearity, 6 (1993), 1067-1075.  doi: 10.1088/0951-7715/6/6/014.

[18]

P. Glendinning, Bifurcation from stable fixed point to 2D attractor in the border collision normal form, IMA J. Appl. Math., 81 (2016), 699-710.  doi: 10.1093/imamat/hxw001.

[19]

P. Glendinning, Robust chaos revisited, Eur. Phys. J. Special Topics, 226 (2017), 1721-1738.  doi: 10.1140/epjst/e2017-70058-2.

[20]

B. R. Hunt, J. A. Kennedy, T.-Y. Li and H. E. Nusse, SLYRB measures: natural invariant measures for chaotic systems, Phys. D, 170 (2002), 50-71.  doi: 10.1016/S0167-2789(02)00445-1.

[21]

L. Kocarev and S. Lian, Chaos-Based Cryptography. Theory, Algorithms and Applications, Springer, New York, 2011. doi: 10.1007/978-3-642-20542-2.

[22]

P. Kowalczyk, Robust chaos and border-collision bifurcations in non-invertible piecewise-linear maps, Nonlinearity, 18 (2005), 485-504.  doi: 10.1088/0951-7715/18/2/002.

[23]

R. Lozi, Un attracteur étrange(?) du type attracteur de Hénon, J. Phys. (Paris), 39 (1978), 9–10. In French. doi: 10.1051/jphyscol:1978505.

[24]

R. S. MacKay, Renormalisation in Area-preserving Maps, World Scientific, World Scientific Publishing Co., Inc., River Edge, NJ, 1993. doi: 10.1142/9789814354462.

[25]

M. Misiurewicz, Strange attractors for the Lozi mappings, In R.G. Helleman, editor, Nonlinear Dynamics, Annals of the New York Academy of Sciences, New York, Wiley, 1980,348–358.

[26]

H. E. Nusse and J. A. Yorke, Border-collision bifurcations including "period two to period three" for piecewise smooth systems, Phys. D, 57 (1992), 39-57.  doi: 10.1016/0167-2789(92)90087-4.

[27]

V. Yu. Protasov, When do several linear operators share an invariant cone?, Linear Algebra Appl., 433 (2010), 781-789.  doi: 10.1016/j.laa.2010.04.006.

[28]

L. Rodman, H. Seyalioglu and I. M. Spitkovsky, On common invariant cones for families of matrices, Linear Algebra Appl., 432 (2010), 911-926.  doi: 10.1016/j.laa.2009.10.004.

[29]

M. Rychlik, Invariant measures and the variational principle for Lozi mappings, Springer, New York, 2004, pages 190–221. doi: 10.1007/978-0-387-21830-4_13.

[30]

D. J. W. Simpson, Border-collision bifurcations in $\mathbb{R}^n$, SIAM Rev., 58 (2016), 177-226.  doi: 10.1137/15M1006982.

[31]

D. J. W. Simpson, The stability of fixed points on switching manifolds of piecewise-smooth continuous maps, J. Dyn. Diff. Equat., 32 (2020), 1527-1552.  doi: 10.1007/s10884-019-09803-9.

[32]

D. J. W. Simpson and J. D. Meiss, Neimark-Sacker bifurcations in planar, piecewise-smooth, continuous maps, SIAM J. Appl. Dyn. Sys., 7 (2008), 795-824.  doi: 10.1137/070704241.

[33]

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817.  doi: 10.1090/S0002-9904-1967-11798-1.

[34]

I. Sushko and L. Gardini, Center bifurcation for two-dimensional border-collision normal form, Int. J. Bifurcation Chaos, 18 (2008), 1029-1050.  doi: 10.1142/S0218127408020823.

[35]

S. van Strien, One-parameter families of smooth interval maps: Density of hyperbolicity and robust chaos, Proc. Amer. Math. Soc., 138 (2010), 4443-4446.  doi: 10.1090/S0002-9939-2010-10446-X.

[36]

M. Viana, Lectures on Lyapunov Exponents., volume 145 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2014. doi: 10.1017/CBO9781139976602.

[37]

M. Wojtkowski, Invariant families of cones and Lyapunov exponents, Ergod. Th. & Dynam. Sys., 5 (1985), 145-161.  doi: 10.1017/S0143385700002807.

[38]

L.-S. Young, Bowen-Ruelle measures for certain piecewise hyperbolic maps, Trans. Amer. Math. Soc., 287 (1985), 41-48.  doi: 10.1090/S0002-9947-1985-0766205-1.

[39]

Z. T. Zhusubaliyev, E. Mosekilde, S. Maity, S. Mohanan and S. Banerjee, Border collision route to quasiperiodicity: Numerical investigation and experimental confirmation, Chaos, 16 (2006), 023122, 11 pp. doi: 10.1063/1.2208565.

show all references

References:
[1]

G. Álvarez, F. Montoya, M. Romera and G. Pastor, Cryptanalysis of a discrete chaotic cryptosystem using external key, Phys. Lett. A, 319 (2003), 334-339.  doi: 10.1016/j.physleta.2003.10.044.

[2]

S. Banerjee and C. Grebogi, Border collision bifurcations in two-dimensional piecewise smooth maps, Phys. Rev. E, 59 (1999), 4052-4061.  doi: 10.1103/PhysRevE.59.4052.

[3]

S. Banerjee, J. A. Yorke and C. Grebogi, Robust chaos, Phys. Rev. Lett., 80 (1998), 3049-3052.  doi: 10.1103/PhysRevLett.80.3049.

[4]

J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacey, On Devaney's definition of chaos, Amer. Math. Monthly, 99 (1992), 332-334.  doi: 10.1080/00029890.1992.11995856.

[5]

L. Barreira and Y. Pesin, Nonuniform Hyperbolicity. Dynamics of Systems with Nonzero Lyapunov Exponents, volume 115 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2007. doi: 10.1017/CBO9781107326026.

[6]

M. Benedicks and L. Carleson, The dynamics of the Henon map, Ann. Math., 133 (1991), 73-169.  doi: 10.2307/2944326.

[7]

Y. Cao and Z. Liu, The geometric structure of strange attractors in the Lozi map, Commun. Nonlin. Sci. Numer. Simul., 3 (1998), 119-123.  doi: 10.1016/S1007-5704(98)90076-4.

[8]

Y. Cao and Z. Liu, Strange attractors in the orientation-preserving Lozi map, Chaos Solitons Fractals, 9 (1998), 1857–1863. doi: 10.1016/S0960-0779(97)00180-X.

[9]

P. Collet and Y. Levy, Ergodic properties of the Lozi mappings, Commun. Math. Phys., 93 (1984), 461-481.  doi: 10.1007/BF01212290.

[10]

E. Cornelis and M. Wojtkowski, A criterion for the positivity of the Liapunov characteristic exponent, Ergod. Th. & Dynam. Sys., 4 (1984), 527-539.  doi: 10.1017/S0143385700002625.

[11]

S. Das and J. A. Yorke, Multichaos from quasiperiodicity, SIAM J. Appl. Dyn. Syst., 16 (2017), 2196-2212.  doi: 10.1137/17M1113199.

[12]

R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989.

[13]

M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems. Theory and Applications, Springer-Verlag, London, Ltd., London, 2008.

[14]

J.-P. Duval, Génération d'une section des classes de conjugaison et arbre des mots de Lyndon de longueur bornée, Theoret. Comput. Sci., 60 (1988), 255–283. In French. doi: 10.1016/0304-3975(88)90113-2.

[15]

J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Mod. Phys., 57 (1985), 617-656.  doi: 10.1103/RevModPhys.57.617.

[16]

R. Edwards, J. J. McDonald and M. J. Tsatsomeros, On matrices with common invariant cones with applications in neural and gene networks, Linear Algebra Appl., 398 (2005), 37-67.  doi: 10.1016/j.laa.2004.04.005.

[17]

E. Glasner and B. Weiss, Sensitive dependence on initial conditions, Nonlinearity, 6 (1993), 1067-1075.  doi: 10.1088/0951-7715/6/6/014.

[18]

P. Glendinning, Bifurcation from stable fixed point to 2D attractor in the border collision normal form, IMA J. Appl. Math., 81 (2016), 699-710.  doi: 10.1093/imamat/hxw001.

[19]

P. Glendinning, Robust chaos revisited, Eur. Phys. J. Special Topics, 226 (2017), 1721-1738.  doi: 10.1140/epjst/e2017-70058-2.

[20]

B. R. Hunt, J. A. Kennedy, T.-Y. Li and H. E. Nusse, SLYRB measures: natural invariant measures for chaotic systems, Phys. D, 170 (2002), 50-71.  doi: 10.1016/S0167-2789(02)00445-1.

[21]

L. Kocarev and S. Lian, Chaos-Based Cryptography. Theory, Algorithms and Applications, Springer, New York, 2011. doi: 10.1007/978-3-642-20542-2.

[22]

P. Kowalczyk, Robust chaos and border-collision bifurcations in non-invertible piecewise-linear maps, Nonlinearity, 18 (2005), 485-504.  doi: 10.1088/0951-7715/18/2/002.

[23]

R. Lozi, Un attracteur étrange(?) du type attracteur de Hénon, J. Phys. (Paris), 39 (1978), 9–10. In French. doi: 10.1051/jphyscol:1978505.

[24]

R. S. MacKay, Renormalisation in Area-preserving Maps, World Scientific, World Scientific Publishing Co., Inc., River Edge, NJ, 1993. doi: 10.1142/9789814354462.

[25]

M. Misiurewicz, Strange attractors for the Lozi mappings, In R.G. Helleman, editor, Nonlinear Dynamics, Annals of the New York Academy of Sciences, New York, Wiley, 1980,348–358.

[26]

H. E. Nusse and J. A. Yorke, Border-collision bifurcations including "period two to period three" for piecewise smooth systems, Phys. D, 57 (1992), 39-57.  doi: 10.1016/0167-2789(92)90087-4.

[27]

V. Yu. Protasov, When do several linear operators share an invariant cone?, Linear Algebra Appl., 433 (2010), 781-789.  doi: 10.1016/j.laa.2010.04.006.

[28]

L. Rodman, H. Seyalioglu and I. M. Spitkovsky, On common invariant cones for families of matrices, Linear Algebra Appl., 432 (2010), 911-926.  doi: 10.1016/j.laa.2009.10.004.

[29]

M. Rychlik, Invariant measures and the variational principle for Lozi mappings, Springer, New York, 2004, pages 190–221. doi: 10.1007/978-0-387-21830-4_13.

[30]

D. J. W. Simpson, Border-collision bifurcations in $\mathbb{R}^n$, SIAM Rev., 58 (2016), 177-226.  doi: 10.1137/15M1006982.

[31]

D. J. W. Simpson, The stability of fixed points on switching manifolds of piecewise-smooth continuous maps, J. Dyn. Diff. Equat., 32 (2020), 1527-1552.  doi: 10.1007/s10884-019-09803-9.

[32]

D. J. W. Simpson and J. D. Meiss, Neimark-Sacker bifurcations in planar, piecewise-smooth, continuous maps, SIAM J. Appl. Dyn. Sys., 7 (2008), 795-824.  doi: 10.1137/070704241.

[33]

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817.  doi: 10.1090/S0002-9904-1967-11798-1.

[34]

I. Sushko and L. Gardini, Center bifurcation for two-dimensional border-collision normal form, Int. J. Bifurcation Chaos, 18 (2008), 1029-1050.  doi: 10.1142/S0218127408020823.

[35]

S. van Strien, One-parameter families of smooth interval maps: Density of hyperbolicity and robust chaos, Proc. Amer. Math. Soc., 138 (2010), 4443-4446.  doi: 10.1090/S0002-9939-2010-10446-X.

[36]

M. Viana, Lectures on Lyapunov Exponents., volume 145 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2014. doi: 10.1017/CBO9781139976602.

[37]

M. Wojtkowski, Invariant families of cones and Lyapunov exponents, Ergod. Th. & Dynam. Sys., 5 (1985), 145-161.  doi: 10.1017/S0143385700002807.

[38]

L.-S. Young, Bowen-Ruelle measures for certain piecewise hyperbolic maps, Trans. Amer. Math. Soc., 287 (1985), 41-48.  doi: 10.1090/S0002-9947-1985-0766205-1.

[39]

Z. T. Zhusubaliyev, E. Mosekilde, S. Maity, S. Mohanan and S. Banerjee, Border collision route to quasiperiodicity: Numerical investigation and experimental confirmation, Chaos, 16 (2006), 023122, 11 pp. doi: 10.1063/1.2208565.

Figure 1.  Initial portions of the stable and unstable manifolds of the fixed point $ Y $. Throughout this paper stable and unstable manifolds are coloured blue and red respectively
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Figure 2.  The parameter region $ {\mathcal{R}} $: (3) and $ \phi > 0 $, where $ \phi $ is given by (10). The striped region indicates parameter values valid for Theorem 2.3. (This figure was created using $ \delta_L = 0.2 $ and $ \delta_R = 0.4 $.)
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Figure 3.  Initial portions of the stable and unstable manifolds of the fixed point $ X $
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Figure 4.  A phase portrait of (1) using the parameter values (15). This shows all periodic solutions (except $ Y $) up to period $ 20 $ (as black dots). These were computed via a brute-force search and the algorithm of [14] to generate all possible symbolic itineraries. The unstable manifold $ W^u(X) $ (coloured red but mostly obscured by the periodic solutions) was computed numerically by following it outwards from $ X $ until no further growth could be discerned
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Figure 6.  The trapping region $ \Omega_{\rm trap} $
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Figure 5.  The forward invariant region $ \Omega $ and its image $ f(\Omega) $
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Figure 7.  The functions $ p $ (27), $ q $ (23), and $ r $ (24) for $ \tau > \delta + 1 $ and a fixed value of $ \delta \in (0, 1) $
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Figure 8.  The slope maps (32). $ G_L(m) $ and $ G_R(m) $ are the slopes of $ A_L v $ and $ A_R v $, respectively, where $ v $ has slope $ m $
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