June  2019, 24(6): 2941-2954. doi: 10.3934/dcdsb.2018293

Polynomial maps with hidden complex dynamics

1. 

Department of Mathematics, Shandong University, Weihai 264209, Shandong, China

2. 

Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, China

* Corresponding author

Received  November 2017 Revised  March 2018 Published  October 2018

Fund Project: This research is partially supported by the National Natural Science Foundation of China (Grant 11701328), Shandong Provincial Natural Science Foundation, China (Grant ZR2017QA006), Young Scholars Program of Shandong University, Weihai (Grant 2017WHWLJH09), and China Postdoctoral Science Foundation (Grant 2016M602126)

The dynamics of a class of one-dimensional polynomial maps are studied, and interesting dynamics are observed under certain conditions: the existence of periodic points with even periods except for one fixed point; the coexistence of two attractors, an attracting fixed point and a hidden attractor; the existence of a double period-doubling bifurcation, which is different from the classical period-doubling bifurcation of the Logistic map; the existence of Li-Yorke chaos. Furthermore, based on this one-dimensional map, the corresponding generalized Hénon map is investigated, and some interesting dynamics are found for certain parameter values: the coexistence of an attracting fixed point and a hidden attractor; the existence of Smale horseshoe for a subshift of finite type and also Li-Yorke chaos.

Citation: Xu Zhang, Guanrong Chen. Polynomial maps with hidden complex dynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2941-2954. doi: 10.3934/dcdsb.2018293
References:
[1]

G. Chen and T. Ueta, Yet another chaotic attractor, Int. J. Bifurcation Chaos, 9 (1999), 1465-1466. doi: 10.1142/S0218127499001024. Google Scholar

[2]

G. Chen, N. Kuznetsov, G. Leonov and T. Mokaev, Hidden attractors on one path: Glukhovsky-Dolzhansky, Lorenz, and Rabinovich systems, Int. J. Bifurcation Chaos, 27 (2017), 1750115 (9 pages). doi: 10.1142/S0218127417501152. Google Scholar

[3]

R. Devaney and Z. Nitecki, Shift automorphisms in the Hénon mapping, Commun. Math. Phys., 67 (1979), 137-146. doi: 10.1007/BF01221362. Google Scholar

[4]

H. Dullin and J. Meiss, Generalized Hénon maps: The cubic diffeomorphisms of the plane, Phys. D, 143 (2000), 262-289. doi: 10.1016/S0167-2789(00)00105-6. Google Scholar

[5]

S. Friedland and J. Milnor, Dynamical properties of plane polynomial automorphisms, Ergod. Th. & Dynam. Sys., 9 (1989), 67-99. doi: 10.1017/S014338570000482X. Google Scholar

[6]

S. GonchenkoM. Li and M. Malkin, Generalized Hénon maps and Smale horseshoes of new types, Int. J. Bifurcation Chaos, 18 (2008), 3029-3052. doi: 10.1142/S0218127408022238. Google Scholar

[7]

S. GonchenkoJ. Meiss and I. Ovsyannikov, Chaotic dynamics of three-dimensional Hénon maps that originate from a homoclinic bifurcation, Regular and Chaotic Dynamics, 11 (2006), 191-212. doi: 10.1070/RD2006v011n02ABEH000345. Google Scholar

[8]

S. GonchenkoI. OvsyannikovC. Simó and D. Turaev, Three-dimensional Hénon-like maps and wild Lorenz-like attractors, Int. J. Bifurcation Chaos, 15 (2005), 3493-3508. doi: 10.1142/S0218127405014180. Google Scholar

[9]

S. GonchenkoL. Shilnikov and D. Turaev, On global bifurcations in three-dimensional diffeomorphisms leading to wild Lorenz-like attractors, Regular and Chaotic Dynamics, 14 (2009), 137-147. doi: 10.1134/S1560354709010092. Google Scholar

[10]

M. Hénon, A two-dimensional mapping with a strange attractor, Commun. Math. Phys., 50 (1976), 69-77. doi: 10.1007/BF01608556. Google Scholar

[11]

S. Jafari, V.-T. Pham, S. Moghtadaei and S. Kingni, The relationship between chaotic maps and some chaotic systems with hidden attractors, Int. J. Bifurcation Chaos, 26 (2016), 1650211 (8 pages). doi: 10.1142/S0218127416502114. Google Scholar

[12]

S. JafariJ. Sprott and F. Nazarimehr, Recent new examples of hidden attractors, Eur. Phys. J. Special Topics, 224 (2015), 1469-1476. Google Scholar

[13]

H. JiangY. LiuZ. Wei and L. Zhang, Hidden chaotic attractors in a class of two-dimensional maps, Nonlinear Dyn., 85 (2016), 2719-2727. doi: 10.1007/s11071-016-2857-3. Google Scholar

[14]

N. KuznetsovG. LeonovT. MokaevA. Prasad and M. Shrimali, Finite-time Lyapunov dimension and hidden attractor of the Rabinovich system, Nonlinear Dyn., 92 (2018), 267-285. doi: 10.1007/s11071-018-4054-z. Google Scholar

[15]

G. Leonov and N. Kuznetsov, Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits, Int. J. Bifurcation Chaos, 23 (2013), 1330002 (69 pages). doi: 10.1142/S0218127413300024. Google Scholar

[16]

G. Leonov and N. Kuznetsov, On differences and similarities in the analysis of Lorenz, Chen, and Lu systems, Appl. Math. Comput., 256 (2015), 334-343. doi: 10.1016/j.amc.2014.12.132. Google Scholar

[17]

G. LeonovN. Kuznetsov and V. Vagaitsev, Localization of hidden Chua's attractors, Phys. Lett. A., 375 (2011), 2230-2233. doi: 10.1016/j.physleta.2011.04.037. Google Scholar

[18]

T. Li and J. Yorke, Period three implies chaos, Am. Math. Mon., 82 (1975), 985-992. doi: 10.1080/00029890.1975.11994008. Google Scholar

[19]

E. Lorenz, Deterministic non-periodic flow, J. Atmos. Sci., 20 (1963), 130-141. Google Scholar

[20]

R. May, Simple mathematical models with very complicated dynamics, Nature, 261 (1976), 45-67. Google Scholar

[21]

M. Molaie, S Jafari, J. Sprott and M. Golpayegani, Simple chaotic flows with one stable equilibrium, Int. J. Bifurcation Chaos, 23 (2013), 1350188 (7 pages). doi: 10.1142/S0218127413501885. Google Scholar

[22]

C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics and Chaos, CRC Press, Florida, 1999. Google Scholar

[23]

J. Sprott, Some simple chaotic flows, Phys. Rev. E, 50 (1994), R647-R650. doi: 10.1103/PhysRevE.50.R647. Google Scholar

[24]

X. Wang and G. Chen, Constructing a chaotic system with any number of equilibria, Nonlinear Dyn., 71 (2013), 429-436. doi: 10.1007/s11071-012-0669-7. Google Scholar

[25]

X. Zhang, Hyperbolic invariant sets of the real generalized Hénon maps, Chaos, Solitons, Fractals, 43 (2010), 31-41. doi: 10.1016/j.chaos.2010.07.003. Google Scholar

[26]

X. Zhang, Chaotic polynomial maps, Int. J. Bifurcation Chaos, 26 (2016), 1650131 (37 pages). doi: 10.1142/S0218127416501315. Google Scholar

[27]

X. Zhang and Y. Shi, Coupled-expanding maps for irreducible transition matrices, Int. J. Bifurcation Chaos, 20 (2010), 3769-3783. doi: 10.1142/S0218127410028094. Google Scholar

[28]

X. ZhangY. Shi and G. Chen, Some properties of coupled-expanding maps in compact sets, Proc. Amer. Math. Soc., 141 (2013), 585-595. doi: 10.1090/S0002-9939-2012-11339-5. Google Scholar

show all references

References:
[1]

G. Chen and T. Ueta, Yet another chaotic attractor, Int. J. Bifurcation Chaos, 9 (1999), 1465-1466. doi: 10.1142/S0218127499001024. Google Scholar

[2]

G. Chen, N. Kuznetsov, G. Leonov and T. Mokaev, Hidden attractors on one path: Glukhovsky-Dolzhansky, Lorenz, and Rabinovich systems, Int. J. Bifurcation Chaos, 27 (2017), 1750115 (9 pages). doi: 10.1142/S0218127417501152. Google Scholar

[3]

R. Devaney and Z. Nitecki, Shift automorphisms in the Hénon mapping, Commun. Math. Phys., 67 (1979), 137-146. doi: 10.1007/BF01221362. Google Scholar

[4]

H. Dullin and J. Meiss, Generalized Hénon maps: The cubic diffeomorphisms of the plane, Phys. D, 143 (2000), 262-289. doi: 10.1016/S0167-2789(00)00105-6. Google Scholar

[5]

S. Friedland and J. Milnor, Dynamical properties of plane polynomial automorphisms, Ergod. Th. & Dynam. Sys., 9 (1989), 67-99. doi: 10.1017/S014338570000482X. Google Scholar

[6]

S. GonchenkoM. Li and M. Malkin, Generalized Hénon maps and Smale horseshoes of new types, Int. J. Bifurcation Chaos, 18 (2008), 3029-3052. doi: 10.1142/S0218127408022238. Google Scholar

[7]

S. GonchenkoJ. Meiss and I. Ovsyannikov, Chaotic dynamics of three-dimensional Hénon maps that originate from a homoclinic bifurcation, Regular and Chaotic Dynamics, 11 (2006), 191-212. doi: 10.1070/RD2006v011n02ABEH000345. Google Scholar

[8]

S. GonchenkoI. OvsyannikovC. Simó and D. Turaev, Three-dimensional Hénon-like maps and wild Lorenz-like attractors, Int. J. Bifurcation Chaos, 15 (2005), 3493-3508. doi: 10.1142/S0218127405014180. Google Scholar

[9]

S. GonchenkoL. Shilnikov and D. Turaev, On global bifurcations in three-dimensional diffeomorphisms leading to wild Lorenz-like attractors, Regular and Chaotic Dynamics, 14 (2009), 137-147. doi: 10.1134/S1560354709010092. Google Scholar

[10]

M. Hénon, A two-dimensional mapping with a strange attractor, Commun. Math. Phys., 50 (1976), 69-77. doi: 10.1007/BF01608556. Google Scholar

[11]

S. Jafari, V.-T. Pham, S. Moghtadaei and S. Kingni, The relationship between chaotic maps and some chaotic systems with hidden attractors, Int. J. Bifurcation Chaos, 26 (2016), 1650211 (8 pages). doi: 10.1142/S0218127416502114. Google Scholar

[12]

S. JafariJ. Sprott and F. Nazarimehr, Recent new examples of hidden attractors, Eur. Phys. J. Special Topics, 224 (2015), 1469-1476. Google Scholar

[13]

H. JiangY. LiuZ. Wei and L. Zhang, Hidden chaotic attractors in a class of two-dimensional maps, Nonlinear Dyn., 85 (2016), 2719-2727. doi: 10.1007/s11071-016-2857-3. Google Scholar

[14]

N. KuznetsovG. LeonovT. MokaevA. Prasad and M. Shrimali, Finite-time Lyapunov dimension and hidden attractor of the Rabinovich system, Nonlinear Dyn., 92 (2018), 267-285. doi: 10.1007/s11071-018-4054-z. Google Scholar

[15]

G. Leonov and N. Kuznetsov, Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits, Int. J. Bifurcation Chaos, 23 (2013), 1330002 (69 pages). doi: 10.1142/S0218127413300024. Google Scholar

[16]

G. Leonov and N. Kuznetsov, On differences and similarities in the analysis of Lorenz, Chen, and Lu systems, Appl. Math. Comput., 256 (2015), 334-343. doi: 10.1016/j.amc.2014.12.132. Google Scholar

[17]

G. LeonovN. Kuznetsov and V. Vagaitsev, Localization of hidden Chua's attractors, Phys. Lett. A., 375 (2011), 2230-2233. doi: 10.1016/j.physleta.2011.04.037. Google Scholar

[18]

T. Li and J. Yorke, Period three implies chaos, Am. Math. Mon., 82 (1975), 985-992. doi: 10.1080/00029890.1975.11994008. Google Scholar

[19]

E. Lorenz, Deterministic non-periodic flow, J. Atmos. Sci., 20 (1963), 130-141. Google Scholar

[20]

R. May, Simple mathematical models with very complicated dynamics, Nature, 261 (1976), 45-67. Google Scholar

[21]

M. Molaie, S Jafari, J. Sprott and M. Golpayegani, Simple chaotic flows with one stable equilibrium, Int. J. Bifurcation Chaos, 23 (2013), 1350188 (7 pages). doi: 10.1142/S0218127413501885. Google Scholar

[22]

C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics and Chaos, CRC Press, Florida, 1999. Google Scholar

[23]

J. Sprott, Some simple chaotic flows, Phys. Rev. E, 50 (1994), R647-R650. doi: 10.1103/PhysRevE.50.R647. Google Scholar

[24]

X. Wang and G. Chen, Constructing a chaotic system with any number of equilibria, Nonlinear Dyn., 71 (2013), 429-436. doi: 10.1007/s11071-012-0669-7. Google Scholar

[25]

X. Zhang, Hyperbolic invariant sets of the real generalized Hénon maps, Chaos, Solitons, Fractals, 43 (2010), 31-41. doi: 10.1016/j.chaos.2010.07.003. Google Scholar

[26]

X. Zhang, Chaotic polynomial maps, Int. J. Bifurcation Chaos, 26 (2016), 1650131 (37 pages). doi: 10.1142/S0218127416501315. Google Scholar

[27]

X. Zhang and Y. Shi, Coupled-expanding maps for irreducible transition matrices, Int. J. Bifurcation Chaos, 20 (2010), 3769-3783. doi: 10.1142/S0218127410028094. Google Scholar

[28]

X. ZhangY. Shi and G. Chen, Some properties of coupled-expanding maps in compact sets, Proc. Amer. Math. Soc., 141 (2013), 585-595. doi: 10.1090/S0002-9939-2012-11339-5. Google Scholar

Figure 1.  Illustration diagram of the function $P(x) = ax^m(x+b)(x-b)$ with $a>0$, $x\in[-b,b]$, and odd $m$.
Figure 2.  Illustration diagram of the function $P(x) = ax^m(x+b)(x-b)$ with $a>0$, $x\in[-b,b]$, and even $m$.
Figure 3.  Bifurcation diagram of $P(x) = ax^2(x+1)(x-1)$ for $3\leq a\leq4$, where the initial value is $0.6$.
Figure 4.  Bifurcation diagram of $P(x) = ax^3(x+1)(x-1)$ for $4.6\leq a\leq5$, where the initial value is $0.6$.
Figure 5.  Illustration diagram of the horseshoe for a subshift of finite type for the matrix $A$.
Figure 6.  Illustration diagram for the map (4) with $a = 6$, $b = c = d = 1$, and $m = 3$.
Figure 7.  Simulation of the map (4) with $a = 5$, $b = 1$, $m = 3$, $d = 1$, and $c = 0.005$, where the initial value is $(0.8,0.8)$.
Figure 8.  Illustration diagram for the map (4) with $a = 5$, $b = c = d = 1$, and $m = 2$.
[1]

Ghassen Askri. Li-Yorke chaos for dendrite maps with zero topological entropy and ω-limit sets. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 2957-2976. doi: 10.3934/dcds.2017127

[2]

Jakub Šotola. Relationship between Li-Yorke chaos and positive topological sequence entropy in nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5119-5128. doi: 10.3934/dcds.2018225

[3]

Frédéric Faure. Prequantum chaos: Resonances of the prequantum cat map. Journal of Modern Dynamics, 2007, 1 (2) : 255-285. doi: 10.3934/jmd.2007.1.255

[4]

C. Bonanno, G. Menconi. Computational information for the logistic map at the chaos threshold. Discrete & Continuous Dynamical Systems - B, 2002, 2 (3) : 415-431. doi: 10.3934/dcdsb.2002.2.415

[5]

Olivier P. Le Maître, Lionel Mathelin, Omar M. Knio, M. Yousuff Hussaini. Asynchronous time integration for polynomial chaos expansion of uncertain periodic dynamics. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 199-226. doi: 10.3934/dcds.2010.28.199

[6]

Jianhe Shen, Maoan Han. Bifurcations of canard limit cycles in several singularly perturbed generalized polynomial Liénard systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3085-3108. doi: 10.3934/dcds.2013.33.3085

[7]

Victor Kozyakin. Polynomial reformulation of the Kuo criteria for v- sufficiency of map-germs. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 587-602. doi: 10.3934/dcdsb.2010.14.587

[8]

Ming Zhao, Cuiping Li, Jinliang Wang, Zhaosheng Feng. Bifurcation analysis of the three-dimensional Hénon map. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 625-645. doi: 10.3934/dcdss.2017031

[9]

Kuan-Ju Huang, Yi-Jung Lee, Tzung-Shin Yeh. Classification of bifurcation curves of positive solutions for a nonpositone problem with a quartic polynomial. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1497-1514. doi: 10.3934/cpaa.2016.15.1497

[10]

Thai Son Doan, Martin Rasmussen, Peter E. Kloeden. The mean-square dichotomy spectrum and a bifurcation to a mean-square attractor. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 875-887. doi: 10.3934/dcdsb.2015.20.875

[11]

Tian Ma, Shouhong Wang. Attractor bifurcation theory and its applications to Rayleigh-Bénard convection. Communications on Pure & Applied Analysis, 2003, 2 (4) : 591-599. doi: 10.3934/cpaa.2003.2.591

[12]

José Miguel Pasini, Tuhin Sahai. Polynomial chaos based uncertainty quantification in Hamiltonian, multi-time scale, and chaotic systems. Journal of Computational Dynamics, 2014, 1 (2) : 357-375. doi: 10.3934/jcd.2014.1.357

[13]

Tijana Levajković, Hermann Mena, Amjad Tuffaha. The stochastic linear quadratic optimal control problem in Hilbert spaces: A polynomial chaos approach. Evolution Equations & Control Theory, 2016, 5 (1) : 105-134. doi: 10.3934/eect.2016.5.105

[14]

André de Carvalho, Toby Hall. Decoration invariants for horseshoe braids. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 863-906. doi: 10.3934/dcds.2010.27.863

[15]

Marcus Fontaine, William D. Kalies, Vincent Naudot. A reinjected cuspidal horseshoe. Conference Publications, 2013, 2013 (special) : 227-236. doi: 10.3934/proc.2013.2013.227

[16]

Xingwu Chen, Jaume Llibre, Weinian Zhang. Averaging approach to cyclicity of hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3953-3965. doi: 10.3934/dcdsb.2017203

[17]

Masoud Yari. Attractor bifurcation and final patterns of the n-dimensional and generalized Swift-Hohenberg equations. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 441-456. doi: 10.3934/dcdsb.2007.7.441

[18]

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 & Continuous Dynamical Systems - A, 2018, 38 (4) : 1745-1776. doi: 10.3934/dcds.2018072

[19]

Mats Gyllenberg, Yan Ping. The generalized Liénard systems. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 1043-1057. doi: 10.3934/dcds.2002.8.1043

[20]

Teresa Faria, Eduardo Liz, José J. Oliveira, Sergei Trofimchuk. On a generalized Yorke condition for scalar delayed population models. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 481-500. doi: 10.3934/dcds.2005.12.481

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (66)
  • HTML views (526)
  • Cited by (0)

Other articles
by authors

[Back to Top]