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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

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  • 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.

    Mathematics Subject Classification: Primary: 37B10, 37B40; Secondary: 37E05.


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  • Figure 1.  Picture of the Topological Entropy

    Figure 2.  Bubble $B(\underline{0}, \underline{1})$

    Figure 3.  Region $B_1(\underline{0}, 00\underline{10}, 11\underline{01}, \underline{1})$

    Figure 4.  Region $B_2(00\underline{10}, \underline{01}, 11\underline{01}, \underline{1})$

    Figure 5.  Region $B_3(\underline{0}, 00\underline{10}, \underline{10}, 11\underline{01})$

    Figure 6.  Region $B_1 \cup B_2 \cup B_3$

    Figure 7.  Region $C_1 \cup C_2 \cup C_3$

    Figure 8.  Region B(a)

    Figure 9.  The standard quadratic family

    Figure 10.  Graph of the equation $-\mu = y(\nu)$

    Figure 11.  Graph of the equation $\nu = x(\mu)$

    Figure 12.  Transversal intersection at $(2, 2)$

    Figure 13.  Quadratic family for $ \mu = \nu = 2$

    Figure 14.  $F_{\mu, \nu}$ for $ \nu = \sqrt{\mu}$ and $ \mu = \dfrac{1+\sqrt{1+4\nu}}{2}$

    Figure 15.  Intersection of the curves $\nu = \sqrt{\mu}$ and $ \mu = \dfrac{1+\sqrt{1+4\nu}}{2}$

    Figure 16.  Graph of map $F_{\mu_1, \nu_1}$

    Figure 17.  Graph of map $F_{(\mu_n, \nu_n)}$

    Figure 18.  Graph of the map $F_{(\mu, \nu)}$ for $-\mu < -\dfrac{1+ \sqrt{1+4 \nu}}{2} $

    Figure 19.  Graph of the map $ F_{ \mu(t), 0}(x)$

    Figure 20.  Graph of map $F_{\mu(t), \nu}, \, 0 \leq t < \nu^2$

    Figure 21.  Graph of map $F_{(\mu(t), \sqrt{t})}$

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