On q-deformed logistic maps

Figure 1.  (a) The region $ R^+ $ where the fixed point $ x_0^+ $ exists. (b) The region $ R^- $ where the fixed point $ x_0^- $ exists

Figure 2.  (a) The region $ L^0 $ where the fixed point $ 0 $ is LAS. (b) The region $ L^+ $ where the fixed point $ x_0^+ $ is LAS

Figure 3.  (a) The region $ G^0 $ where the fixed point $ 0 $ is GAS. (b) The region $ G^+ $ where the fixed point $ x_0^+ $ is GAS

Figure 4.  For $ k = 2 $, (a) The region $ G^0(q,2) $ where the fixed point $ 0 $ is GAS. (b) The region $ G^+(q,2) $ where the fixed point $ x_0^+ $ is GAS. For $ k = 5 $, (c) The region $ G^0(q,5) $ where the fixed point $ 0 $ is GAS. (d) The region $ G^+(q,5) $ where the fixed point $ x_0^+ $ is GAS

Figure 5.  We fix $ q = 0 $. Bifurcation diagrams when the $ q $-deformation $ \phi _{q} $ is applied twice. We compute 10000 points of each orbit, with initial conditions $ 0.75 $ (black) and $ 0.05 $ (green), and draw the last 200. Black color overwrites green when the attractor is the same, covering completely when $ 0 $ is GAS. The parameter $ a $ ranges $ (0,4] $ with step size $ 0.005 $. The dashed red line represents the unstable fixed point $ x_0^- $ when it exists, which acts as a separatrix between the basins of attraction of the two attractors. In (b) we depict the region where the existence of two different attractors is possible when we apply the same $ q $-deformation twice

Figure 6.  With accuracy $ 10^{-4} $, topological entropy of the $ q $-deformed logistic map for $ a\in [3.5,4] $ and $ q\in [-25,2) $ (a) and associated level curves (b). Note that the darker regions are, the smaller the topological entropy is. (c) and (d) show the same computations for $ a\in [3.5,4] $ and $ q\in [0.7,1.2] $. (e) and (f) depict the region where the Lyaounov exponents are positive, and hence chaos is physically observable

Figure 7.  Level curves of the topological entropy of the $ q $-deformed logistic maps (a) $ \Phi _{q,q,a} $, (b) $ \Phi _{q,q,q,a} $, (c) $ \Phi _{q,q,q,q,a} $ and (d) $ \Phi _{q,q,q,q,q,a} $ for $ a\in [3.5,4] $ and $ q\in [0.7,1.2] $. Note that the darker regions are, the smaller the topological entropy is

Figure 8.  For $ q = 0.95 $: (a)-(b) Bifurcation diagrams of the $ q $-deformed logistic map $ \Phi _{q,q,a} $. In (a) we cannot see any chaotic behavior, but a zoom in (b) shows that it exists. The parameter $ a $ ranges from $ 3.569 $ to $ 3.57 $ with step size $ 10^{-6} $. (c)-(d) The same for $ \Phi _{q,q,q,a} $

Figure 9.  For $ a = 3.5697 $ and $ q_1,q_2\in [0.7,1.2] $. Region of positive Lyapunov exponents of the $ q $-deformed logistic maps (a) $ \Phi _{q_2,q_1,q_1,a} $ and (b) $ \Phi _{q_2,q_1,q_2,q_1,a} $. (c) and (d) level curves of the topological entropy of these maps with accuracy $ 10^{-4} $