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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

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.

Citation: Paul A. Glendinning, David J. W. Simpson. A constructive approach to robust chaos using invariant manifolds and expanding cones. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3367-3387. doi: 10.3934/dcds.2020409
References:
[1]

G. ÁlvarezF. MontoyaM. 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.  Google Scholar

[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.  Google Scholar

[3]

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

[4]

J. BanksJ. BrooksG. CairnsG. Davis and P. Stacey, On Devaney's definition of chaos, Amer. Math. Monthly, 99 (1992), 332-334.  doi: 10.1080/00029890.1992.11995856.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[11]

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

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

R. EdwardsJ. 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.  Google Scholar

[17]

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

[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.  Google Scholar

[19]

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

[20]

B. R. HuntJ. A. KennedyT.-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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

L. RodmanH. 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.  Google Scholar

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[37]

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

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

G. ÁlvarezF. MontoyaM. 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.  Google Scholar

[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.  Google Scholar

[3]

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

[4]

J. BanksJ. BrooksG. CairnsG. Davis and P. Stacey, On Devaney's definition of chaos, Amer. Math. Monthly, 99 (1992), 332-334.  doi: 10.1080/00029890.1992.11995856.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[11]

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

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

R. EdwardsJ. 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.  Google Scholar

[17]

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

[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.  Google Scholar

[19]

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

[20]

B. R. HuntJ. A. KennedyT.-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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

L. RodmanH. 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.  Google Scholar

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[37]

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

[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.  Google Scholar

[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.  Google Scholar

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
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 $.)
Figure 3.  Initial portions of the stable and unstable manifolds of the fixed point $ X $
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">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
Figure 6.  The trapping region $ \Omega_{\rm trap} $
Figure 5.  The forward invariant region $ \Omega $ and its image $ f(\Omega) $
Figure 7.  The functions $ p $ (27), $ q $ (23), and $ r $ (24) for $ \tau > \delta + 1 $ and a fixed value of $ \delta \in (0, 1) $
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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