May  2022, 27(5): 2833-2848. doi: 10.3934/dcdsb.2021162

On q-deformed logistic maps

Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, C/ Doctor Fleming sn, 30202, Cartagena, Spain

* Corresponding author: Jose S. Cánovas

Received  January 2021 Revised  March 2021 Published  May 2022 Early access  June 2021

Fund Project: This author has been partially supported by the Grant MTM2017-84079-P from Agencia Estatal de Investigación (AEI) y Fondo Europeo de Desarrollo Regional (FEDER)

We consider the logistic family $ f_{a} $ and a family of homeomorphisms $ \phi _{q} $. The $ q $-deformed system is given by the composition map $ f_{a}\circ \phi _{q} $. We study when this system has non zero fixed points which are LAS and GAS. We also give an alternative approach to study the dynamics of the $ q $-deformed system with special emphasis on the so-called Parrondo's paradox finding parameter values $ a $ for which $ f_{a} $ is simple while $ f_{a}\circ \phi _{q} $ is dynamically complicated. We explore the dynamics when several $ q $-deformations are applied.

Citation: Jose S. Cánovas. On q-deformed logistic maps. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2833-2848. doi: 10.3934/dcdsb.2021162
References:
[1]

R. L. AdlerA. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319.  doi: 10.1090/S0002-9947-1965-0175106-9.

[2]

L. Alsedá, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One, World Scientific Publishing, 1993. doi: 10.1142/1980.

[3]

F. Balibrea and V. Jiménez-López, The measure of scrambled sets: A survey, Acta Univ. M. Belii Ser. Math., 7 (1999), 3-11. 

[4]

S. Banerjee and R. Parthsarathy, A $q$-deformed logistic map and its implications, J. Phys. A, 44 (2011), 045104. doi: 10.1088/1751-8113/44/4/045104.

[5]

S. BehniaM. Yahyavi and R. Habibpourbisafar, Watermarking based on discrete wavelet transform and $q$-deformed chaotic map, Chaos Solitons & Fractals, 104 (2017), 6-17.  doi: 10.1016/j.chaos.2017.07.020.

[6]

F. BlanchardE. GlasnerS. Kolyada and A. Maass, On Li-Yorke pairs, J. Reine Angew. Math., 547 (2002), 51-68.  doi: 10.1515/crll.2002.053.

[7]

L. BlockJ. KeeslingS. H. Li and K. Peterson, An improved algorithm for computing topological entropy, J. Stat. Phys., 55 (1989), 929-939.  doi: 10.1007/BF01041072.

[8]

J. S. CánovasA. Linero and D. Peralta-Salas, Dynamic Parrondo's paradox, Phys. D, 218 (2006), 177-184.  doi: 10.1016/j.physd.2006.05.004.

[9]

J. S. Cánovas and M. Muñoz, Revisiting Parrondo's paradox for the logistic family, Fluct. Noise Lett., 12 (2013), 1350015. doi: 10.1142/S0219477513500156.

[10]

J. Cánovas and M. Muñoz, On the dynamics of the q-deformed logistic map, Phys. Lett. A, 383 (2019), 1742-1754.  doi: 10.1016/j.physleta.2019.03.003.

[11]

M. ChaichianA. P. Demichev and P. P. Kulish, Quasi-classical limit in $q$-deformed systems, non-commutativity and the $q$-path integral, Phys. Lett. A, 233 (1997), 251-260.  doi: 10.1016/S0375-9601(97)00513-6.

[12]

W. de Melo and S. van Strien, One Dimensional Dynamics, Springer Verlag, 1993. doi: 10.1007/978-3-642-78043-1.

[13]

S. N. Elaydi, Discrete Chaos. With Applications in Science and Engineering, Chapman & Hall CRC, Boca Raton, 2008.

[14]

J. GraczykD. Sands and G. Światek, Metric attractors for smooth unimodal maps, Ann. Math., 159 (2004), 725-740.  doi: 10.4007/annals.2004.159.725.

[15]

J. Guckenheimer, Sensitive dependence to initial conditions for one dimensional maps, Commun. Math. Phys., 70 (1979), 133-160.  doi: 10.1007/BF01982351.

[16]

R. Jaganathan and S. Sinha, A $q$-deformed nonlinear map, Phys. Lett. A, 338 (2005), 277-287.  doi: 10.1016/j.physleta.2005.02.042.

[17]

Y. A. Kuznetsov, Saddle-node bifurcation for maps, Scholarpedia 3 (2008), 4399. doi: 10.4249/scholarpedia.4399.

[18]

V. I. Man'koG. MarmoS. Solimeno and F. Zaccaria, Physical Nonlinear aspects of classical and quantum q-oscillators, Int. J. Mod. Phys. A, 8 (1993), 3577-3597.  doi: 10.1142/S0217751X93001454.

[19]

T. Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985-992.  doi: 10.1080/00029890.1975.11994008.

[20]

E. Liz, A global picture of the Gamma-Ricker map: A flexible discrete-time model with factors of positive and negative density dependence, Bull. Math. Biol., 80 (2018), 417-434.  doi: 10.1007/s11538-017-0382-2.

[21]

C. Luo, B.-Q. Liu and H.-S. Hou, Fractional chaotic maps with $q$-deformation, Appl. Math. Comput., 393 (2021), 125759. doi: 10.1016/j.amc.2020.125759.

[22]

J. Milnor, On the concept of attractor, Comm. Math. Phys., 99 (1985), 177-195.  doi: 10.1007/BF01212280.

[23]

J. Milnor and W. Thurston, On iterated maps of the interval, in Dynamical Systems, Lectures Notes in Mathematics, Springer-Verlag, 1342 1988,465–563. doi: 10.1007/BFb0082847.

[24]

M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math., 67 (1980), 45-63.  doi: 10.4064/sm-67-1-45-63.

[25]

V. PatidarG. Purohit and K. K. Sud, Dynamical behavior of $q$-deformed Henon map, Int. J. Bifurc. Chaos, 21 (2011), 1349-1356.  doi: 10.1142/S0218127411029215.

[26]

S. J. Schreiber, Allee effects, extinctions, and chaotic transients in simple population models, Theor. Popul. Biol., 64 (2003), 201-209.  doi: 10.1016/S0040-5809(03)00072-8.

[27]

M. D. Shrimali and S. Banerjee, Delayed $q$-deformed logistic map, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 3126-3133.  doi: 10.1016/j.cnsns.2013.03.017.

[28]

D. Singer, Stable orbits and bifurcations of maps on the interval, SIAM J. Appl. Math., 35 (1978), 260-267.  doi: 10.1137/0135020.

[29]

J. Smítal, Chaotic functions with zero topological entropy, Trans. Amer. Math. Soc., 297 (1986), 269-282.  doi: 10.1090/S0002-9947-1986-0849479-9.

[30]

C. Tresser, P. Coullet and E. de Faria, Period doubling, Scholarpedia, 9 (2014), 3958.

[31]

C. Tsallis, Nonextensive statistical mechanics: A brief review of its present status, An. Acad. Bras. Ci$\hat{\text{e}}$nc., 74 (2002), 393–414. doi: 10.1590/S0001-37652002000300003.

[32]

G.-C. Wu, M. N. Cankaya and S. Banerjee, Fractional q-deformed chaotic maps: A weight function approach, Chaos, 30 (2020), 121106. doi: 10.1063/5.0030973.

show all references

References:
[1]

R. L. AdlerA. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319.  doi: 10.1090/S0002-9947-1965-0175106-9.

[2]

L. Alsedá, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One, World Scientific Publishing, 1993. doi: 10.1142/1980.

[3]

F. Balibrea and V. Jiménez-López, The measure of scrambled sets: A survey, Acta Univ. M. Belii Ser. Math., 7 (1999), 3-11. 

[4]

S. Banerjee and R. Parthsarathy, A $q$-deformed logistic map and its implications, J. Phys. A, 44 (2011), 045104. doi: 10.1088/1751-8113/44/4/045104.

[5]

S. BehniaM. Yahyavi and R. Habibpourbisafar, Watermarking based on discrete wavelet transform and $q$-deformed chaotic map, Chaos Solitons & Fractals, 104 (2017), 6-17.  doi: 10.1016/j.chaos.2017.07.020.

[6]

F. BlanchardE. GlasnerS. Kolyada and A. Maass, On Li-Yorke pairs, J. Reine Angew. Math., 547 (2002), 51-68.  doi: 10.1515/crll.2002.053.

[7]

L. BlockJ. KeeslingS. H. Li and K. Peterson, An improved algorithm for computing topological entropy, J. Stat. Phys., 55 (1989), 929-939.  doi: 10.1007/BF01041072.

[8]

J. S. CánovasA. Linero and D. Peralta-Salas, Dynamic Parrondo's paradox, Phys. D, 218 (2006), 177-184.  doi: 10.1016/j.physd.2006.05.004.

[9]

J. S. Cánovas and M. Muñoz, Revisiting Parrondo's paradox for the logistic family, Fluct. Noise Lett., 12 (2013), 1350015. doi: 10.1142/S0219477513500156.

[10]

J. Cánovas and M. Muñoz, On the dynamics of the q-deformed logistic map, Phys. Lett. A, 383 (2019), 1742-1754.  doi: 10.1016/j.physleta.2019.03.003.

[11]

M. ChaichianA. P. Demichev and P. P. Kulish, Quasi-classical limit in $q$-deformed systems, non-commutativity and the $q$-path integral, Phys. Lett. A, 233 (1997), 251-260.  doi: 10.1016/S0375-9601(97)00513-6.

[12]

W. de Melo and S. van Strien, One Dimensional Dynamics, Springer Verlag, 1993. doi: 10.1007/978-3-642-78043-1.

[13]

S. N. Elaydi, Discrete Chaos. With Applications in Science and Engineering, Chapman & Hall CRC, Boca Raton, 2008.

[14]

J. GraczykD. Sands and G. Światek, Metric attractors for smooth unimodal maps, Ann. Math., 159 (2004), 725-740.  doi: 10.4007/annals.2004.159.725.

[15]

J. Guckenheimer, Sensitive dependence to initial conditions for one dimensional maps, Commun. Math. Phys., 70 (1979), 133-160.  doi: 10.1007/BF01982351.

[16]

R. Jaganathan and S. Sinha, A $q$-deformed nonlinear map, Phys. Lett. A, 338 (2005), 277-287.  doi: 10.1016/j.physleta.2005.02.042.

[17]

Y. A. Kuznetsov, Saddle-node bifurcation for maps, Scholarpedia 3 (2008), 4399. doi: 10.4249/scholarpedia.4399.

[18]

V. I. Man'koG. MarmoS. Solimeno and F. Zaccaria, Physical Nonlinear aspects of classical and quantum q-oscillators, Int. J. Mod. Phys. A, 8 (1993), 3577-3597.  doi: 10.1142/S0217751X93001454.

[19]

T. Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985-992.  doi: 10.1080/00029890.1975.11994008.

[20]

E. Liz, A global picture of the Gamma-Ricker map: A flexible discrete-time model with factors of positive and negative density dependence, Bull. Math. Biol., 80 (2018), 417-434.  doi: 10.1007/s11538-017-0382-2.

[21]

C. Luo, B.-Q. Liu and H.-S. Hou, Fractional chaotic maps with $q$-deformation, Appl. Math. Comput., 393 (2021), 125759. doi: 10.1016/j.amc.2020.125759.

[22]

J. Milnor, On the concept of attractor, Comm. Math. Phys., 99 (1985), 177-195.  doi: 10.1007/BF01212280.

[23]

J. Milnor and W. Thurston, On iterated maps of the interval, in Dynamical Systems, Lectures Notes in Mathematics, Springer-Verlag, 1342 1988,465–563. doi: 10.1007/BFb0082847.

[24]

M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math., 67 (1980), 45-63.  doi: 10.4064/sm-67-1-45-63.

[25]

V. PatidarG. Purohit and K. K. Sud, Dynamical behavior of $q$-deformed Henon map, Int. J. Bifurc. Chaos, 21 (2011), 1349-1356.  doi: 10.1142/S0218127411029215.

[26]

S. J. Schreiber, Allee effects, extinctions, and chaotic transients in simple population models, Theor. Popul. Biol., 64 (2003), 201-209.  doi: 10.1016/S0040-5809(03)00072-8.

[27]

M. D. Shrimali and S. Banerjee, Delayed $q$-deformed logistic map, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 3126-3133.  doi: 10.1016/j.cnsns.2013.03.017.

[28]

D. Singer, Stable orbits and bifurcations of maps on the interval, SIAM J. Appl. Math., 35 (1978), 260-267.  doi: 10.1137/0135020.

[29]

J. Smítal, Chaotic functions with zero topological entropy, Trans. Amer. Math. Soc., 297 (1986), 269-282.  doi: 10.1090/S0002-9947-1986-0849479-9.

[30]

C. Tresser, P. Coullet and E. de Faria, Period doubling, Scholarpedia, 9 (2014), 3958.

[31]

C. Tsallis, Nonextensive statistical mechanics: A brief review of its present status, An. Acad. Bras. Ci$\hat{\text{e}}$nc., 74 (2002), 393–414. doi: 10.1590/S0001-37652002000300003.

[32]

G.-C. Wu, M. N. Cankaya and S. Banerjee, Fractional q-deformed chaotic maps: A weight function approach, Chaos, 30 (2020), 121106. doi: 10.1063/5.0030973.

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} $
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