October  2020, 40(10): 6015-6041. doi: 10.3934/dcds.2020257

Nonuniformly hyperbolic systems arising from coupling of chaotic and gradient-like systems

1. 

Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA

2. 

Institute for Advanced Study, Princeton, New Jersey 08540, USA

* Corresponding author

Received  March 2020 Revised  May 2020 Published  June 2020

We investigate dynamical systems obtained by coupling two maps, one of which is chaotic and is exemplified by an Anosov diffeomorphism, and the other is of gradient type and is exemplified by a N-pole-to-S-pole map of the circle. Leveraging techniques from the geometric and ergodic theories of hyperbolic systems, we analyze three different ways of coupling together the two maps above. For weak coupling, we offer an addendum to existing theory showing that almost always the attractor has fractal-like geometry when it is not normally hyperbolic. Our main results are for stronger couplings in which the action of the Anosov diffeomorphism on the circle map has certain monotonicity properties. Under these conditions, we show that the coupled systems have invariant cones and possess SRB measures even though there are genuine obstructions to uniform hyperbolicity.

Citation: Matteo Tanzi, Lai-Sang Young. Nonuniformly hyperbolic systems arising from coupling of chaotic and gradient-like systems. Discrete & Continuous Dynamical Systems - A, 2020, 40 (10) : 6015-6041. doi: 10.3934/dcds.2020257
References:
[1]

F. AbdenurC. Bonatti and S. Crovisier, Global dominated splittings and the ${C}^1$ Newhouse phenomenon, Proc. Amer. Math. Soc., 134 (2006), 2229-2237.  doi: 10.1090/S0002-9939-06-08445-0.  Google Scholar

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J.-B. BardetG. Keller and and R. Zweimüller, Stochastically stable globally coupled maps with bistable thermodynamic limit, Comm. Math. Phys., 292 (2009), 237-270.  doi: 10.1007/s00220-009-0854-9.  Google Scholar

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M. Benedicks and L.-S. Young, Sina${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$-Bowen-Ruelle measures for certain Hénon maps, Invent. Math., 112 (1993), 541-576.  doi: 10.1007/BF01232446.  Google Scholar

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K. Bjerklöv, A note on circle maps driven by strongly expanding endomorphisms on $\Bbb{T}$, Dyn. Syst., 33 (2018), 361-368.  doi: 10.1080/14689367.2017.1386161.  Google Scholar

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S. BoccalettiV. LatoraY. MorenoM. Chavez and D.-U. Hwang, Complex networks: Structure and dynamics, Phys. Rep., 424 (2006), 175-308.  doi: 10.1016/j.physrep.2005.10.009.  Google Scholar

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C. Bonatti, L. J. Díaz and E. R. Pujals, A $C^1$-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources, Ann. of Math. (2), 158 (2003), 355–418. doi: 10.4007/annals.2003.158.355.  Google Scholar

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C. Bonatti and A. Wilkinson, Transitive partially hyperbolic diffeomorphisms on 3-manifolds, Topology, 44 (2005), 475-508.  doi: 10.1016/j.top.2004.10.009.  Google Scholar

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D. DolgopyatM. Viana and J. Yang, Geometric and measure-theoretical structures of maps with mostly contracting center, Comm. Math. Phys., 341 (2016), 991-1014.  doi: 10.1007/s00220-015-2554-y.  Google Scholar

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A. J. Homburg, Circle diffeomorphisms forced by expanding circle maps, Ergodic Theory Dynam. Systems, 32 (2012), 2011-2024.  doi: 10.1017/S014338571100068X.  Google Scholar

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K. Kaneko, Theory and Applications of Coupled Map Lattices, Nonlinear Science: Theory and Applications, John Wiley & Sons, Ltd., Chichester, 1993.  Google Scholar

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J. L. KaplanJ. Mallet-Paret and J. A. Yorke, The Lyapunov dimension of a nowhere differentiable attracting torus, Ergodic Theory Dynam. Systems, 4 (1984), 261-281.  doi: 10.1017/S0143385700002431.  Google Scholar

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G. Keller and C. Liverani, Uniqueness of the SRB measure for piecewise expanding weakly coupled map lattices in any dimension, Comm. Math. Phys., 262 (2006), 33-50.  doi: 10.1007/s00220-005-1474-7.  Google Scholar

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T. PereiraS. van Strien and M. Tanzi, Heterogeneously coupled maps: Hub dynamics and emergence across connectivity layers, J. Eur. Math. Soc. (JEMS), 22 (2020), 2183-2252.  doi: 10.4171/JEMS/963.  Google Scholar

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show all references

References:
[1]

F. AbdenurC. Bonatti and S. Crovisier, Global dominated splittings and the ${C}^1$ Newhouse phenomenon, Proc. Amer. Math. Soc., 134 (2006), 2229-2237.  doi: 10.1090/S0002-9939-06-08445-0.  Google Scholar

[2]

V. Baladi, M. Degli Esposti, S. Isola, E. Järvenpää and A. Kupiainen, The spectrum of weakly coupled map lattices, J. Math. Pures Appl. (9), 77 (1998), 539–584. doi: 10.1016/S0021-7824(98)80138-4.  Google Scholar

[3]

P. BálintT. GilbertP. NándoriD. Szász and and I. P. Tóth, On the limiting Markov process of energy exchanges in a rarely interacting ball-piston gas, J. Stat. Phys., 166 (2017), 903-925.  doi: 10.1007/s10955-016-1598-5.  Google Scholar

[4]

J.-B. BardetG. Keller and and R. Zweimüller, Stochastically stable globally coupled maps with bistable thermodynamic limit, Comm. Math. Phys., 292 (2009), 237-270.  doi: 10.1007/s00220-009-0854-9.  Google Scholar

[5]

M. Benedicks and L. Carleson, The dynamics of the Hénon map, Ann. of Math. (2), 133 (1991), 73–169. doi: 10.2307/2944326.  Google Scholar

[6]

M. Benedicks and L.-S. Young, Sina${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$-Bowen-Ruelle measures for certain Hénon maps, Invent. Math., 112 (1993), 541-576.  doi: 10.1007/BF01232446.  Google Scholar

[7]

K. Bjerklöv, A note on circle maps driven by strongly expanding endomorphisms on $\Bbb{T}$, Dyn. Syst., 33 (2018), 361-368.  doi: 10.1080/14689367.2017.1386161.  Google Scholar

[8]

S. BoccalettiV. LatoraY. MorenoM. Chavez and D.-U. Hwang, Complex networks: Structure and dynamics, Phys. Rep., 424 (2006), 175-308.  doi: 10.1016/j.physrep.2005.10.009.  Google Scholar

[9]

J. Bochi and M. Viana, Lyapunov Exponents: How frequently are dynamical systems hyperbolic?, in Modern Dynamical Systems and Applications, Cambridge Univ. Press, Cambridge, 2004,271–297.  Google Scholar

[10]

C. Bonatti, L. J. Díaz and E. R. Pujals, A $C^1$-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources, Ann. of Math. (2), 158 (2003), 355–418. doi: 10.4007/annals.2003.158.355.  Google Scholar

[11]

C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel J. Math., 115 (2000), 157-193.  doi: 10.1007/BF02810585.  Google Scholar

[12]

C. Bonatti and A. Wilkinson, Transitive partially hyperbolic diffeomorphisms on 3-manifolds, Topology, 44 (2005), 475-508.  doi: 10.1016/j.top.2004.10.009.  Google Scholar

[13]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin-New York, 1975. doi: 10.1007/BFb0081279.  Google Scholar

[14]

J. Bricmont and A. Kupiainen, High temperature expansions and dynamical systems, Comm. Math. Phys., 178 (1996), 703-732.  doi: 10.1007/BF02108821.  Google Scholar

[15]

M. I. Brin and J. B. Pesin, Partially hyperbolic dynamical systems, Uspehi Mat. Nauk, 28 (1973), 169-170.   Google Scholar

[16]

L. BunimovichC. LiveraniA. Pellegrinotti and Y. Suhov, Ergodic systems of $n$ balls in a billiard table, Comm. Math. Phys., 146 (1992), 357-396.  doi: 10.1007/BF02102633.  Google Scholar

[17]

L. A. Bunimovich and Y. G. Sina${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$, Spacetime chaos in coupled map lattices, Nonlinearity, 1 (1988), 491-516.  doi: 10.1088/0951-7715/1/4/001.  Google Scholar

[18]

J.-R. Chazottes and B. Fernandez, Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems, Lecture Notes in Physics, 671, Springer, Berlin, 2005. doi: 10.1007/b103930.  Google Scholar

[19]

H. de Jong, Modeling and simulation of genetic regulatory systems: A literature review, J. Comput. Biol., 9 (2002), 67-103.  doi: 10.1089/10665270252833208.  Google Scholar

[20]

L. J. DíazE. R. Pujals and R. Ures, Partial hyperbolicity and robust transitivity, Acta Math., 183 (1999), 1-43.  doi: 10.1007/BF02392945.  Google Scholar

[21]

D. Dolgopyat, On dynamics of mostly contracting diffeomorphisms, Comm. Math. Phys., 213 (2000), 181-201.  doi: 10.1007/s002200000238.  Google Scholar

[22]

D. DolgopyatM. Viana and J. Yang, Geometric and measure-theoretical structures of maps with mostly contracting center, Comm. Math. Phys., 341 (2016), 991-1014.  doi: 10.1007/s00220-015-2554-y.  Google Scholar

[23]

J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, in The Theory of Chaotic Attractors, Springer, New York, 1985,273–312. doi: 10.1007/978-0-387-21830-4_17.  Google Scholar

[24]

J.-P. Eckmann and L.-S. Young, Nonequilibrium energy profiles for a class of 1-D models, Comm. Math. Phys., 262 (2006), 237-267.  doi: 10.1007/s00220-005-1462-y.  Google Scholar

[25]

B. Fernandez, Computer-assisted proof of loss of ergodicity by symmetry breaking in expanding coupled maps, Ann. Henri Poincaré, 21 (2020), 649–674. doi: 10.1007/s00023-019-00876-2.  Google Scholar

[26]

T. Fischer and H. H. Rugh, Transfer operators for coupled analytic maps, Ergodic Theory Dynam. Systems, 20 (2000), 109-143.  doi: 10.1017/S0143385700000079.  Google Scholar

[27]

D. HadjiloucasM. J. Nicol and C. P. Walkden, Regularity of invariant graphs over hyperbolic systems, Ergodic Theory Dynam. Systems, 22 (2002), 469-482.  doi: 10.1017/S0143385702000226.  Google Scholar

[28]

B. Hasselblatt and A. Katok, Handbook of Dynamical Systems, North-Holland, Amsterdam, 2002.  Google Scholar

[29]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, 1977. doi: 10.1007/BFb0092042.  Google Scholar

[30]

A. J. Homburg, Circle diffeomorphisms forced by expanding circle maps, Ergodic Theory Dynam. Systems, 32 (2012), 2011-2024.  doi: 10.1017/S014338571100068X.  Google Scholar

[31]

K. Kaneko, Theory and Applications of Coupled Map Lattices, Nonlinear Science: Theory and Applications, John Wiley & Sons, Ltd., Chichester, 1993.  Google Scholar

[32]

J. L. KaplanJ. Mallet-Paret and J. A. Yorke, The Lyapunov dimension of a nowhere differentiable attracting torus, Ergodic Theory Dynam. Systems, 4 (1984), 261-281.  doi: 10.1017/S0143385700002431.  Google Scholar

[33]

G. Keller and M. Künzle, Transfer operators for coupled map lattices, Ergodic Theory Dynam. Systems, 12 (1992), 297-318.  doi: 10.1017/S0143385700006763.  Google Scholar

[34]

G. Keller and C. Liverani, Uniqueness of the SRB measure for piecewise expanding weakly coupled map lattices in any dimension, Comm. Math. Phys., 262 (2006), 33-50.  doi: 10.1007/s00220-005-1474-7.  Google Scholar

[35]

G. Keller and C. Liverani, Map lattices coupled by collisions, Comm. Math. Phys., 291 (2009), 591-597.  doi: 10.1007/s00220-009-0835-z.  Google Scholar

[36]

J. Koiller and L.-S. Young, Coupled map networks, Nonlinearity, 23 (2010), 1121-1141.  doi: 10.1088/0951-7715/23/5/006.  Google Scholar

[37]

Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, in International Symposium on Mathematical Problems in Theoretical Physics, Lecture Notes in Phys., 39, Springer, Berlin, 1975,420–422. doi: 10.1007/BFb0013365.  Google Scholar

[38]

F. Ledrappier, Propriétés ergodiques des mesures de Sinaï, Inst. Hautes Études Sci. Publ. Math., 59 (1984), 163–188.  Google Scholar

[39]

F. Ledrappier and J.-M. Strelcyn, A proof of the estimation from below in Pesin entropy formula, Ergodic Theory Dynam. Systems, 2 (1982), 203-219.  doi: 10.1017/S0143385700001528.  Google Scholar

[40]

F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin's entropy formula, Ann. of Math. (2), 122 (1985), 509–539. doi: 10.2307/1971328.  Google Scholar

[41]

K. Lu, Q. Wang and L.-S. Young, Strange attractors for periodically forced parabolic equations, Mem. Amer. Math. Soc., 224 (2013). doi: 10.1090/S0065-9266-2012-00669-1.  Google Scholar

[42]

T. PereiraS. van Strien and M. Tanzi, Heterogeneously coupled maps: Hub dynamics and emergence across connectivity layers, J. Eur. Math. Soc. (JEMS), 22 (2020), 2183-2252.  doi: 10.4171/JEMS/963.  Google Scholar

[43]

Y. B. Pesin and Y. G. Sinai, Gibbs measures for partially hyperbolic attractors, Ergodic Theory Dynam. Systems, 2 (1982), 417-438.  doi: 10.1017/S014338570000170X.  Google Scholar

[44]

A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization. A Universal Concept in Nonlinear Sciences, Cambridge Nonlinear Science Series, 12, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511755743.  Google Scholar

[45]

C. Pugh and M. Shub, Ergodic attractors, Trans. Amer. Math. Soc., 312 (1989), 1-54.  doi: 10.1090/S0002-9947-1989-0983869-1.  Google Scholar

[46]

E. R. Pujals and M. Sambarino, On the dynamics of dominated splitting, Ann. of Math. (2), 169 (2009), 675–739. doi: 10.4007/annals.2009.169.675.  Google Scholar

[47]

C. Robinson, Structural stability of $C^1$ diffeomorphisms, J. Differential Equations, 22 (1976), 28-73.  doi: 10.1016/0022-0396(76)90004-8.  Google Scholar

[48]

D. Ruelle, A measure associated with axiom-A attractors, Amer. J. Math., 98 (1976), 619-654.  doi: 10.2307/2373810.  Google Scholar

[49]

D. Ruelle and A. Wilkinson, Absolutely singular dynamical foliations, Comm. Math. Phys., 219 (2001), 481-487.  doi: 10.1007/s002200100420.  Google Scholar

[50]

F. Sélley and P. Bálint, Mean-field coupling of identical expanding circle maps, J. Stat. Phys., 164 (2016), 858-889.  doi: 10.1007/s10955-016-1568-y.  Google Scholar

[51]

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Figure 1.  (Courtesy of Bastien Fernandez) Illustration of a coupled system having an attractor with wild geometry. The {plotted} points represent the intersection of the attractor $ \Lambda $ with a $ W^{cs}- $manifold through one of the points in the attractor. The horizontal and vertical axes indicate approximate directions of $ E^s $ and $ E^c $
Figure 2.  Top: Evolution of a curve in $ W^{cu} $. Here we show the action of the map on a curve $ \gamma_0 $, $ \gamma_0(J) \subset \{z = \mbox{constant}\} $, such that $ \pi_{xy}\gamma_0 $ is contained in a $ W^u_A- $curve. The pictures on the top show the effect of $ \tilde f_1 $, $ \tilde f_2 $, and $ \tilde f_3 $ on $ \gamma_0 $, where $ \tilde f_1 $, $ \tilde f_2 $, and $ \tilde f_3 $ are the lifts of $ f_1 $, $ f_2 $, and $ f_3 $ respectively. Here we assume that $ r(x) = x $. Bottom: action of $ \tilde F^6 $ on a piece of curve $ \gamma_0 $, where $ \tilde F $ is the lift of the map $ F $. These plots highlight the effect of the monotonicity of the coupling
Figure 3.  The pictures above illustrate assumptions (B2) and (B3). On the left is the graph of an example of $ r $: $ r = 0 $ outside of $ I_\epsilon $ and is very steep when $ r(x) $ is in between $ d $ and $ 1-d $. On the right is the graph of an example of $ g $. The highlighted intervals are $ g(I_+)\backslash I_+ $; each component has diameter less than $ d $
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