# American Institute of Mathematical Sciences

April  2018, 38(4): 1745-1776. doi: 10.3934/dcds.2018072

## A formula for the boundary of chaos in the lexicographical scenario and applications to the bifurcation diagram of the standard two parameter family of quadratic increasing-increasing Lorenz maps

 1 Departamento de Matemáticas y Ciencia de la Computación, Universidad de Santiago de Chile, Santiago, Chile 2 Escuela de Matemática y Estadistíca, Universidad Central de Chile, Santiago, Chile

* Corresponding author: rafael.labarca@usach.cl

See section Acknowledgements

Received  May 2016 Revised  November 2017 Published  January 2018

The Geometric Lorenz Attractor has been a source of inspiration for many mathematical studies. Most of these studies deal with the two or one dimensional representation of its first return map. A one dimensional scenario (the increasing-increasing one's) can be modeled by the standard two parameter family of contracting Lorenz maps. The dynamics of any member of the standard family can be modeled by a subshift in the Lexicographical model of two symbols. These subshifts can be considered as the maximal invariant set for the shift map in some interval, in the Lexicographical model. For all of these subshifts, the lower extreme of the interval is a minimal sequence and the upper extreme is a maximal sequence. The Lexicographical world (LW) is precisely the set of sequences (lower extreme, upper extreme) of all of these subshifts. In this scenario the topological entropy is a map from LW onto the interval $[0, \log{2}]$. The boundary of chaos (that is the boundary of the set of $(a, b) ∈ LW$ such that $h_{top}(a, b)>0$) is given by a map $b = χ(a)$, which is defined by a recurrence formula. In the present paper we obtain an explicit formula for the value $χ(a)$ for $a$ in a dense set contained in the set of minimal sequences. Moreover, we apply this computation to determine regions of positive topological entropy for the standard quadratic family of contracting increasing-increasing Lorenz maps.

Citation: Rafael Labarca, Solange Aranzubia. A formula for the boundary of chaos in the lexicographical scenario and applications to the bifurcation diagram of the standard two parameter family of quadratic increasing-increasing Lorenz maps. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 1745-1776. doi: 10.3934/dcds.2018072
##### References:
 [1] V. S. Afraimovich, V. V. Bykov and L. P. Shil'nikov, The origin and structure of the Lorenz attractor, Dokl. Akad. Nauk SSSR, 234 (1977), 336-339. [2] V. S. Afraimovich, V. V. Bykov and L. P. Shil'nikov, On attracting structurally unstable limit sets of Lorenz attractor type, Trudy Moskov. Mat. Obshch., 44 (1982), 150-212. [3] L. Alseda, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One, 2 $^{nd}$ edition, Advenced Series in Nonlinear Dynamics, 5. World Scientific Publishing Co., Inc., River Edge, NJ, 2000. [4] S. Aranzubia, Hacia una demostración de la arco conexidad de la isentropa de entropía cero de la familia cuadrática de Lorenz lexicográfica: burbujas de entropía constante y cotas superiores para las burbujas de entropía cero., Ph. D thesis, Universidad de Santiago de Chile, 2015. [5] R. Bamón, R. Labarca, R. Mañé and M. J. Pacifico, The explosion of singular cycles, Inst. Hautes Études Sci. Publ. Math., 78 (1993), 207-232. [6] H. Bruin and S. Van Strien, Monotonicity of entropy for real multimodal maps, Journal of the American Mathematical Society, 28 (2015), 1-61. [7] W. De Melo and M. Martens, Universal Models for Lorenz maps, Ergodic Theory Dynam. Systems, 21 (2001), 833-860. [8] W. De Melo and S. Van Strien, One-dimensional Dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Springer-Verlag, Berlin, 1993. [9] P. Glendinning and C. Sparrow, Prime and renormalisable kneading invariants and the dynamics of expanding Lorenz maps. Homoclinic Chaos, Phys. D., 62 (1993), 22-50.  doi: 10.1016/0167-2789(93)90270-B. [10] J. Guckenheimer, A strange, strange attractor, in The Hopf Bifurcation and Its Applications, Springer, New York, (1976), 368–381. doi: 10.1007/978-1-4612-6374-6_25. [11] J. Guckenheimer and R. F. Williams, Structural Stability of Lorenz Attractors, Publ. Math. IHES, 50 (1979), 59-72. [12] R. Labarca, Bifurcation of Contracting Singular Cycles, Ann. Sci. École Norm. Sup.(4), 28 (1995), 705-745.  doi: 10.24033/asens.1731. [13] R. Labarca, A note on the topological classification of Lorenz maps on the interval, Topics in symbolic dynamics an applications, London Math. Soc. Lect. Not. Ser., 279 (2000), 229-245. [14] R. Labarca, Unfolding singular cycles, Notas Soc. Mat. Chile (N.S.), (2001), 38-71. [15] R. Labarca and C. Moreira, Bifurcations of the essential dynamics of Lorenz maps on the real line and the bifurcation scenario for the linear family, Sci. Ser. A Math. Sci. (N.S.), 7 (2001), 13-29. [16] R. Labarca and C. Moreira, Bifurcation of the essential dynamics of Lorenz maps and applications to Lorenz-like flows: contributions to the study of the Expanding Case, Bol. Soc. Brasil. Mat. (N.S.), 32 (2001), 107-144.  doi: 10.1007/BF01243862. [17] R. Labarca and C. Moreira, Essential dynamics for Lorenz maps on the real line and the lexicographical world, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 683-694.  doi: 10.1016/j.anihpc.2005.09.001. [18] R. Labarca and C. Moreira, Bifurcation of the essential dynamics of Lorenz Maps on the real line and the bifurcation scenario for Lorenz like flows: The contracting case, Proyecciones. Journal of Mathematics, 29 (2010), 241-289. [19] R. Labarca and S. Plaza, Bifurcation of discontinuous maps of the interval and palindromic numbers, Bol. Soc. Mat. Mexicana(3), 7 (2001), 99-116. [20] R. Labarca and B. San Martín, Prevalence of hyperbolicity for complex singular cycles, Bol. Soc. Brasil. Mat. (N.S.), 28 (1997), 343-362.  doi: 10.1007/BF01233397. [21] R. Labarca and L. Vásquez, On the characterization of the kneading sequences associated to injective Lorenz maps of the interval and to orientation preserving homeomorphisms of the circle, Bol. Soc. Mat. Mexicana(3), 16 (2010), 103-118. [22] R. Labarca and L. Vásquez, A characterization of the kneading sequences associated to Lorenz maps of the interval, Bull. Braz. Math. Soc. (N.S.), 43 (2012), 221-245.  doi: 10.1007/s00574-012-0011-5. [23] E. N. Lorenz, Deterministic non-periodic flow, J. Atmos. Sci., 20 (1963), 130-141. [24] N. Metropolis, M. L. Stein and P. R. Stein, Stable states of a non-linear transformation, Numer. Math., 10 (1967), 1-19.  doi: 10.1007/BF02165155. [25] N. Metropolis, M. L. Stein and P. R. Stein, On finite limit sets for transformations on the unit interval, J. Combinatorial Theory Ser. A, 15 (1973), 25-44.  doi: 10.1016/0097-3165(73)90033-2. [26] J. Milnor, Remarks on iterated cubic maps, Experiment. Math., 1 (1992), 5-24. [27] J. Milnor and W. Thurston, On iterated maps on the interval, Dynamical Systems (College Park, MD, 1986–7), 465–563, Lecture Notes in Math., 1342, Springer, Berlin, 1988. [28] M. Milnor and C. Tresser, On entropy and monotonicity for real cubic maps, Comm. Mat. Phys., No1 with an appendix by Adrien Douady and Pierrette Sentenac. 209 (2000), 123–178. doi: 10.1007/s002200050018. [29] C. Moreira, Maximal invariant sets for restriction of tent and unimodal maps, Qual. Theory Dyn. Syst., 2 (2001), 385-398.  doi: 10.1007/BF02969348. [30] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817.  doi: 10.1090/S0002-9904-1967-11798-1. [31] R. F. Williams, The structure of Lorenz Attractors, in Turbulence Seminar (Univ. Calif., Berkeley, Calif., 1976/1977). Lecture Notes in Math., Springer, Berlin, 615 (1977), 94–112. [32] M. A. Zaks, Scaling properties and renormalization invariants for the "homoclinic quasiperiodicity", Phys. D, 62 (1993), 300-316.  doi: 10.1016/0167-2789(93)90289-D.

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##### References:
 [1] V. S. Afraimovich, V. V. Bykov and L. P. Shil'nikov, The origin and structure of the Lorenz attractor, Dokl. Akad. Nauk SSSR, 234 (1977), 336-339. [2] V. S. Afraimovich, V. V. Bykov and L. P. Shil'nikov, On attracting structurally unstable limit sets of Lorenz attractor type, Trudy Moskov. Mat. Obshch., 44 (1982), 150-212. [3] L. Alseda, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One, 2 $^{nd}$ edition, Advenced Series in Nonlinear Dynamics, 5. World Scientific Publishing Co., Inc., River Edge, NJ, 2000. [4] S. Aranzubia, Hacia una demostración de la arco conexidad de la isentropa de entropía cero de la familia cuadrática de Lorenz lexicográfica: burbujas de entropía constante y cotas superiores para las burbujas de entropía cero., Ph. D thesis, Universidad de Santiago de Chile, 2015. [5] R. Bamón, R. Labarca, R. Mañé and M. J. Pacifico, The explosion of singular cycles, Inst. Hautes Études Sci. Publ. Math., 78 (1993), 207-232. [6] H. Bruin and S. Van Strien, Monotonicity of entropy for real multimodal maps, Journal of the American Mathematical Society, 28 (2015), 1-61. [7] W. De Melo and M. Martens, Universal Models for Lorenz maps, Ergodic Theory Dynam. Systems, 21 (2001), 833-860. [8] W. De Melo and S. Van Strien, One-dimensional Dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Springer-Verlag, Berlin, 1993. [9] P. Glendinning and C. Sparrow, Prime and renormalisable kneading invariants and the dynamics of expanding Lorenz maps. Homoclinic Chaos, Phys. D., 62 (1993), 22-50.  doi: 10.1016/0167-2789(93)90270-B. [10] J. Guckenheimer, A strange, strange attractor, in The Hopf Bifurcation and Its Applications, Springer, New York, (1976), 368–381. doi: 10.1007/978-1-4612-6374-6_25. [11] J. Guckenheimer and R. F. Williams, Structural Stability of Lorenz Attractors, Publ. Math. IHES, 50 (1979), 59-72. [12] R. Labarca, Bifurcation of Contracting Singular Cycles, Ann. Sci. École Norm. Sup.(4), 28 (1995), 705-745.  doi: 10.24033/asens.1731. [13] R. Labarca, A note on the topological classification of Lorenz maps on the interval, Topics in symbolic dynamics an applications, London Math. Soc. Lect. Not. Ser., 279 (2000), 229-245. [14] R. Labarca, Unfolding singular cycles, Notas Soc. Mat. Chile (N.S.), (2001), 38-71. [15] R. Labarca and C. Moreira, Bifurcations of the essential dynamics of Lorenz maps on the real line and the bifurcation scenario for the linear family, Sci. Ser. A Math. Sci. (N.S.), 7 (2001), 13-29. [16] R. Labarca and C. Moreira, Bifurcation of the essential dynamics of Lorenz maps and applications to Lorenz-like flows: contributions to the study of the Expanding Case, Bol. Soc. Brasil. Mat. (N.S.), 32 (2001), 107-144.  doi: 10.1007/BF01243862. [17] R. Labarca and C. Moreira, Essential dynamics for Lorenz maps on the real line and the lexicographical world, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 683-694.  doi: 10.1016/j.anihpc.2005.09.001. [18] R. Labarca and C. Moreira, Bifurcation of the essential dynamics of Lorenz Maps on the real line and the bifurcation scenario for Lorenz like flows: The contracting case, Proyecciones. Journal of Mathematics, 29 (2010), 241-289. [19] R. Labarca and S. Plaza, Bifurcation of discontinuous maps of the interval and palindromic numbers, Bol. Soc. Mat. Mexicana(3), 7 (2001), 99-116. [20] R. Labarca and B. San Martín, Prevalence of hyperbolicity for complex singular cycles, Bol. Soc. Brasil. Mat. (N.S.), 28 (1997), 343-362.  doi: 10.1007/BF01233397. [21] R. Labarca and L. Vásquez, On the characterization of the kneading sequences associated to injective Lorenz maps of the interval and to orientation preserving homeomorphisms of the circle, Bol. Soc. Mat. Mexicana(3), 16 (2010), 103-118. [22] R. Labarca and L. Vásquez, A characterization of the kneading sequences associated to Lorenz maps of the interval, Bull. Braz. Math. Soc. (N.S.), 43 (2012), 221-245.  doi: 10.1007/s00574-012-0011-5. [23] E. N. Lorenz, Deterministic non-periodic flow, J. Atmos. Sci., 20 (1963), 130-141. [24] N. Metropolis, M. L. Stein and P. R. Stein, Stable states of a non-linear transformation, Numer. Math., 10 (1967), 1-19.  doi: 10.1007/BF02165155. [25] N. Metropolis, M. L. Stein and P. R. Stein, On finite limit sets for transformations on the unit interval, J. Combinatorial Theory Ser. A, 15 (1973), 25-44.  doi: 10.1016/0097-3165(73)90033-2. [26] J. Milnor, Remarks on iterated cubic maps, Experiment. Math., 1 (1992), 5-24. [27] J. Milnor and W. Thurston, On iterated maps on the interval, Dynamical Systems (College Park, MD, 1986–7), 465–563, Lecture Notes in Math., 1342, Springer, Berlin, 1988. [28] M. Milnor and C. Tresser, On entropy and monotonicity for real cubic maps, Comm. Mat. Phys., No1 with an appendix by Adrien Douady and Pierrette Sentenac. 209 (2000), 123–178. doi: 10.1007/s002200050018. [29] C. Moreira, Maximal invariant sets for restriction of tent and unimodal maps, Qual. Theory Dyn. Syst., 2 (2001), 385-398.  doi: 10.1007/BF02969348. [30] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817.  doi: 10.1090/S0002-9904-1967-11798-1. [31] R. F. Williams, The structure of Lorenz Attractors, in Turbulence Seminar (Univ. Calif., Berkeley, Calif., 1976/1977). Lecture Notes in Math., Springer, Berlin, 615 (1977), 94–112. [32] M. A. Zaks, Scaling properties and renormalization invariants for the "homoclinic quasiperiodicity", Phys. D, 62 (1993), 300-316.  doi: 10.1016/0167-2789(93)90289-D.
Picture of the Topological Entropy
Bubble $B(\underline{0}, \underline{1})$
Region $B_1(\underline{0}, 00\underline{10}, 11\underline{01}, \underline{1})$
Region $B_2(00\underline{10}, \underline{01}, 11\underline{01}, \underline{1})$
Region $B_3(\underline{0}, 00\underline{10}, \underline{10}, 11\underline{01})$
Region $B_1 \cup B_2 \cup B_3$
Region $C_1 \cup C_2 \cup C_3$
Region B(a)
Graph of the equation $-\mu = y(\nu)$
Graph of the equation $\nu = x(\mu)$
Transversal intersection at $(2, 2)$
Quadratic family for $\mu = \nu = 2$
$F_{\mu, \nu}$ for $\nu = \sqrt{\mu}$ and $\mu = \dfrac{1+\sqrt{1+4\nu}}{2}$
Intersection of the curves $\nu = \sqrt{\mu}$ and $\mu = \dfrac{1+\sqrt{1+4\nu}}{2}$
Graph of map $F_{\mu_1, \nu_1}$
Graph of map $F_{(\mu_n, \nu_n)}$
Graph of the map $F_{(\mu, \nu)}$ for $-\mu < -\dfrac{1+ \sqrt{1+4 \nu}}{2}$
Graph of the map $F_{ \mu(t), 0}(x)$
Graph of map $F_{\mu(t), \nu}, \, 0 \leq t < \nu^2$
Graph of map $F_{(\mu(t), \sqrt{t})}$
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