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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, 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.

[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.

[3]

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

[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.

[5]

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

[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.

[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.

[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.

[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.

[10]

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

[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.

[12]

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

[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.

[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.

[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.

[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.

[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.

[18]

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

[19]

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

[20]

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

[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.

[22]

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

[23]

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

[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.

[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.

[26]

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

[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.

[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.

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.

[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.

[3]

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

[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.

[5]

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

[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.

[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.

[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.

[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.

[10]

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

[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.

[12]

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

[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.

[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.

[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.

[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.

[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.

[18]

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

[19]

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

[20]

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

[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.

[22]

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

[23]

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

[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.

[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.

[26]

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

[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.

[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.

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 $m = 2$.
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