doi: 10.3934/dcdss.2020277

Complicated dynamics and control of a hyperchaotic complex nonlinear autonomous Lü model with complex parameters

1. 

Department of Mathematics, Faculty of Science, Taif University, Taif, Saudi Arabia

2. 

Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt

3. 

Department of Mathematics, Umm Al-Qura University, P.O. Box 14949, Makkah, Saudi Arabia

4. 

Department of Mathematics, Turabah College, Taif University, Turabah, Saudi Arabia

* Corresponding author: Emad E. Mahmoud

Received  May 2019 Published  February 2020

In this work, we consider the dynamics and complicated properties of an independent Lü model with complex nonlinear conditions and parameters. We study the influence of complex parameters on the dynamics and behaviors of nonlinear hyperchaotic models. The complex parameters such as generalized Hamiltonian, symmetry, dispersal, equilibria and their stability, Lyapunov exponents, Lyapunov dimension, bifurcation graphs, and hyperchaotic satisfaction are considered. Furthermore, we analyze the stability of the trivial points and limit the conditions under which the complex nonlinear conditions with complex parameters have negative, zero, or positive Lyapunov exponents; we also focus on the chaos, hyperchaos, periodic, and quasiperiodic attractors for an extensive range of parameter values. Moreover, we verify the control of hyperchaotic arrangements of the autonomous Lü model with complex nonlinear conditions and complex parameters. We propose a method to transform the model from its hyperchaotic state to an unstable equilibrium point using a Lyapunov stability hypothesis. Finally, with the use of complex periodic driving, we demonstrate that the model can be transformed from hyperchaotic to quasiperiodic motions, thus resulting in a correct periodic arrangement in which the amplitude and frequency depend on the parameters of the model. Because the obtained arrangement is steady for an extensive range of parameter values, it may be used to control the model by entraining it with the connected periodic compelling term.

Citation: Emad E. Mahmoud, Kholod M. Abualnaja, Ohood A. Althagafi. Complicated dynamics and control of a hyperchaotic complex nonlinear autonomous Lü model with complex parameters. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020277
References:
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E. E. Mahmoud, An unusual kind of complex synchronizations and its applications in secure communications, Eur. Phys. J. Plus., 132 (2017), 1-14.  doi: 10.1140/epjp/i2017-11715-2.  Google Scholar

[32]

E. E. Mahmoud and M. A. AL-Adwani, Dynamical behaviors, control and synchronization of a new chaotic model with complex variables and cubic nonlinear terms, Results Phys., 7 (2017), 1346-1356.  doi: 10.1016/j.rinp.2017.02.039.  Google Scholar

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E. E. Mahmoud, Modified projective phase synchronization of chaotic complex nonlinear systems, Math. Comput. Simulat., 89 (2013), 69-85.  doi: 10.1016/j.matcom.2013.02.008.  Google Scholar

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E. E. Mahmoud, Dynamics and synchronization of new hyperchaotic complex Lorenz system, Math. Comput. Model., 55 (2012), 1951-1962.  doi: 10.1016/j.mcm.2011.11.053.  Google Scholar

[35]

A. Mohammadzadeh and S. Ghaemi, Synchronization of uncertain fractional-order hyperchaotic systems by using a new self-evolving non-singleton type-2 fuzzy neural network and its application to secure communication, Nonlinear Dyn., 88 (2017), 1-19.  doi: 10.1007/s11071-016-3227-x.  Google Scholar

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S. M. T. Nezhad, M. Nazari and E. A. Gharavol, A Novel DoS and DDoS attacks detection algorithm using ARIMA time series model and chaotic system in computer networks, IEEE Commun. Lett., 20 (2016), 700-703, http://dx.doi.org/10.1109/LCOMM.2016.2517622. Google Scholar

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N. S. Nise, "Stability" in Control Systems Engineering 6th Ed, JohnWiley & Sons, Inc., 2011. Google Scholar

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[40]

L. A. Quezada-TéllezS. Carrillo-MorenoO. Rosas-JaimesJ. J. Flores-Godoy and G. Fernández-Anaya, Dynamic analysis of a Lü model in six dimensions and its projections, Int. J. Nonlinear Sci. Numer. Simul., 18 (2017), 371-384.  doi: 10.1515/ijnsns-2016-0076.  Google Scholar

[41]

A. Shvets and A. Makaseyev, Deterministic chaos in pendulum systems with delay, Applied Mathematics and Nonlinear Sciences, 4 (2019), 1-8.  doi: 10.2478/AMNS.2019.1.00001.  Google Scholar

[42]

O. I. TachaC. K. VolosI. M. KyprianidisI. N. StouboulosS. Vaidyanathan and V.-T. Pham, Analysis, adaptive control and circuit simulation of a novel nonlinear finance system, Appl. Math. Comput., 276 (2016), 200-217.  doi: 10.1016/j.amc.2015.12.015.  Google Scholar

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D. WilczakS. Serrano and R. Barrio, Coexistence and dynamical connections between hyperchaos and chaos in the 4D Rössler system: A computer-assisted proof, SIAM J. Appl. Dyn. Syst., 15 (2016), 356-390.  doi: 10.1137/15M1039201.  Google Scholar

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A. WolfJ. B. SwiftH. L. Swinney and J. A. Vastano, Determining Lyapunov exponents from a time series, Physica D, 16 (1985), 285-317.  doi: 10.1016/0167-2789(85)90011-9.  Google Scholar

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Q. Zhang and X. Wei, RGB color image encryption method based on Lorenz chaotic system and DNA computation, Iete. Tech. Rev., 30 (2014), 404-409, http://dx.doi.org/10.4103/0256-4602.123123. Google Scholar

show all references

References:
[1]

M. Arbid and A. Boukabou, Controlling unstable periodic orbits in complex dynamical networks with chaotic nodes, Optik, 128 (2017), 148-155.  doi: 10.1016/j.ijleo.2016.09.120.  Google Scholar

[2]

B. Bao, T. Jiang, G. Wang, P. Jin, H. Bao and M. Chen, Two-memristor-based Chua's hyperchaotic circuit with plane equilibrium and its extreme multistability, Nonlinear Dyn., 89 (2017), 1157-1171, http://dx.doi.org/10.1007/s10825-017-1013-8. doi: 10.1007/s11071-017-3507-0.  Google Scholar

[3]

V. A. BazhenovO. S. Pogorelova and T. G. Postnikova, Intermittent transition to chaos in vibroimpact system, Applied Mathematics and Nonlinear Sciences, 3 (2018), 475-485.  doi: 10.2478/AMNS.2018.2.00037.  Google Scholar

[4]

L. Cao, A four-dimensional hyperchaotic finance system and its control problems, Journal of Control Science and Engineering, 2018 (2018), Art. ID 4976380, 12 pp. doi: 10.1155/2018/4976380.  Google Scholar

[5]

T. CaraballoM. Herrera-Cobos and P. Marín-Rubio, An iterative method for non-autonomous nonlocal reaction-diffusion equations, Applied Mathematics and Nonlinear Sciences, 2 (2017), 73-82.  doi: 10.21042/AMNS.2017.1.00006.  Google Scholar

[6]

X. Chai, Z. Gan, K. Yang, Y. Chen and X. Liu, An image encryption algorithm based on the memristive hyperchaotic system, cellular automata and DNA sequence operations, Signal Process Image, 52 (2017), 6-19, http://dx.doi.org/10.1016/j.image.2016.12.007. Google Scholar

[7]

A. Chithra and I. R. Mohamed, Synchronization and chaotic communication in nonlinear circuits with nonlinear coupling, J. Comput. Electron., 16 (2017), 833-844.  doi: 10.1007/s10825-017-1013-8.  Google Scholar

[8]

C. Cruz-Hernandez, Synchronization of time-delay Chua's oscillator with application to secure communication, Nonlinear Dyn. Syst. Theory, 4 (2004), 1-13.   Google Scholar

[9]

N. A. A. FatafS. K. PalitS. MukherjeeM. R. M. SaidD. H. Son and S. Banerjee, Communication scheme using a hyperchaotic semiconductor laser model: Chaos shift key revisited, Eur. Phys. J. Plus., 132 (2017), 1-8.  doi: 10.1140/epjp/i2017-11786-y.  Google Scholar

[10]

A. C. FowlerJ. D. Gibbon and M. J. McGuinness, The complex Lorenz equations, Phys. D, 4 (1981/82), 139-163.  doi: 10.1016/0167-2789(82)90057-4.  Google Scholar

[11]

P. FredericksonJ. L. KaplanE. D. Yorke and J. A. Yorke, The Liapunov dimension of strange attractors, J. Differ. Equ., 49 (1983), 185-207.  doi: 10.1016/0022-0396(83)90011-6.  Google Scholar

[12]

H. Harraga and M. Yebdri, Attractors for a nonautonomous reaction-diffusion equation with delay, Applied Mathematics and Nonlinear Sciences, 3 (2018), 127-150.  doi: 10.21042/AMNS.2018.1.00010.  Google Scholar

[13]

A. K. Jagannatham and B. D. Rao, Cramer-Rao lower bound for constrained complex parameters, IEEE Signal Process. Lett., 11 (2004), 875-878.  doi: 10.1109/LSP.2004.836948.  Google Scholar

[14]

A. Khan and M. A. Bhat, Hyperchaotic analysis and adaptive projective synchronization of nonlinear dynamical system, Comput. Math. Model., 28 (2017), 517-530.  doi: 10.1007/s10598-017-9378-x.  Google Scholar

[15]

C. Y. KiyonoN. Pérez and E. C. N. Silva, Determination of full piezoelectric complex parameters using gradient-based optimization algorithm, Smart Mater. Struct., 25 (2016), 1-18.  doi: 10.1088/0964-1726/25/2/025019.  Google Scholar

[16]

G. C. Layek, Continuous Dynamical Systems in: An Introduction to Dynamical Systems and Chaos, Springer, India, 2015. Google Scholar

[17]

J. LiuK. X. Liu and S. T. Liu, Adaptive control for a class of nonlinear complex dynamical systems with uncertain complex parameters and perturbations, PLoS One, 12 (2017), 1-16.  doi: 10.1371/journal.pone.0175730.  Google Scholar

[18]

J. Liu and S. T. Liu, Complex modified function projective synchronization of complex chaotic systems with known and unknown complex parameters, Appl. Math. Model., 48 (2017), 440-450.  doi: 10.1016/j.apm.2017.04.001.  Google Scholar

[19]

J. LiuS. Liu and J. C. Sprott, Adaptive complex modified hybrid function projective synchronization of different dimensional complex chaos with uncertain complex parameters, Nonlinear Dyn., 83 (2017), 1109-1121.  doi: 10.1007/s11071-015-2391-8.  Google Scholar

[20]

J. LiuS. T. Liu and C. H. Yuan, Adaptive complex modified projective synchronization of complex chaotic (hyperchaotic) systems with uncertain complex parameters, Nonlinear Dyn., 79 (2015), 1035-1047.  doi: 10.1007/s11071-014-1721-6.  Google Scholar

[21]

E. N. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci., 20 (1963), 130-141.  doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.  Google Scholar

[22]

J. H. Lü and G. R. Chen, A new chaotic attractor coined, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 12 (2002), 659-661.  doi: 10.1142/S0218127402004620.  Google Scholar

[23]

E. E. N. Macau, Exploiting unstable periodic orbits of a chaotic invariant set for spacecraft control, Celest. Mech. Dyn. Astron., 87 (2003), 291-305.  doi: 10.1023/B:CELE.0000005725.77786.28.  Google Scholar

[24]

E. E. Mahmoud and S. M. Abo-Dahab, Dynamical properties and complex anti-synchronization with applications to secure communication for a novel chaotic complex nonlinear model, Chaos Solitons Fractals, 106 (2018), 273-284.  doi: 10.1016/j.chaos.2017.10.013.  Google Scholar

[25]

G. M. MahmoudT. Bountis and E. E. Mahmoud, Active control and global synchronization for complex Chen and Lü systems, Int. J. Bifurcation Chaos Appl. Sci. Eng., 17 (2007), 4295-4308.  doi: 10.1142/S0218127407019962.  Google Scholar

[26]

G. M. MahmoudE. E. Mahmoud and M. E. Ahmed, On the hyperchaotic complex Lü system, Nonlinear Dyn., 58 (2009), 725-738.  doi: 10.1007/s11071-009-9513-0.  Google Scholar

[27]

G. M. MahmoudM. E. Ahmed and E. E. Mahmoud, Analysis of hyperchaotic complex Lorenz systems, Int. J. Mod. Phys. C, 19 (2008), 1477-1494.  doi: 10.1142/S0129183108013151.  Google Scholar

[28]

G. M. MahmoudM. E. Ahmed and N. Sabor, On autonomous and nonautonomous modified hyperchaotic complex Lü systems, Int. J. Bifurcation Chaos Appl. Sci. Eng., 21 (2011), 1913-1926.  doi: 10.1142/S0218127411029525.  Google Scholar

[29]

E. E. Mahmoud and F. S. Abood, A new nonlinear chaotic complex model and its complex antilag synchronization, Complexity, 2017 (2017), Art. ID 3848953, 13 pp. doi: 10.1155/2017/3848953.  Google Scholar

[30]

G. M. MahmoudT. BountisM. A. Al-Kashif and S. A. Aly, Dynamical properties and synchronization of complex non-linear equations for detuned lasers, Dynamical Syst., 24 (2009), 63-79.  doi: 10.1080/14689360802438298.  Google Scholar

[31]

E. E. Mahmoud, An unusual kind of complex synchronizations and its applications in secure communications, Eur. Phys. J. Plus., 132 (2017), 1-14.  doi: 10.1140/epjp/i2017-11715-2.  Google Scholar

[32]

E. E. Mahmoud and M. A. AL-Adwani, Dynamical behaviors, control and synchronization of a new chaotic model with complex variables and cubic nonlinear terms, Results Phys., 7 (2017), 1346-1356.  doi: 10.1016/j.rinp.2017.02.039.  Google Scholar

[33]

E. E. Mahmoud, Modified projective phase synchronization of chaotic complex nonlinear systems, Math. Comput. Simulat., 89 (2013), 69-85.  doi: 10.1016/j.matcom.2013.02.008.  Google Scholar

[34]

E. E. Mahmoud, Dynamics and synchronization of new hyperchaotic complex Lorenz system, Math. Comput. Model., 55 (2012), 1951-1962.  doi: 10.1016/j.mcm.2011.11.053.  Google Scholar

[35]

A. Mohammadzadeh and S. Ghaemi, Synchronization of uncertain fractional-order hyperchaotic systems by using a new self-evolving non-singleton type-2 fuzzy neural network and its application to secure communication, Nonlinear Dyn., 88 (2017), 1-19.  doi: 10.1007/s11071-016-3227-x.  Google Scholar

[36]

A. H. Nayfeh and B. Balachandran, Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods, John Wiley & Sons, Inc., New York, 1995. doi: 10.1002/9783527617548.  Google Scholar

[37]

S. M. T. Nezhad, M. Nazari and E. A. Gharavol, A Novel DoS and DDoS attacks detection algorithm using ARIMA time series model and chaotic system in computer networks, IEEE Commun. Lett., 20 (2016), 700-703, http://dx.doi.org/10.1109/LCOMM.2016.2517622. Google Scholar

[38]

N. S. Nise, "Stability" in Control Systems Engineering 6th Ed, JohnWiley & Sons, Inc., 2011. Google Scholar

[39]

E. OttC. Grebogi and J. A. Yorke, Controlling chaos, Phys. Rev. Lett., 64 (1990), 1196-1199.  doi: 10.1103/PhysRevLett.64.1196.  Google Scholar

[40]

L. A. Quezada-TéllezS. Carrillo-MorenoO. Rosas-JaimesJ. J. Flores-Godoy and G. Fernández-Anaya, Dynamic analysis of a Lü model in six dimensions and its projections, Int. J. Nonlinear Sci. Numer. Simul., 18 (2017), 371-384.  doi: 10.1515/ijnsns-2016-0076.  Google Scholar

[41]

A. Shvets and A. Makaseyev, Deterministic chaos in pendulum systems with delay, Applied Mathematics and Nonlinear Sciences, 4 (2019), 1-8.  doi: 10.2478/AMNS.2019.1.00001.  Google Scholar

[42]

O. I. TachaC. K. VolosI. M. KyprianidisI. N. StouboulosS. Vaidyanathan and V.-T. Pham, Analysis, adaptive control and circuit simulation of a novel nonlinear finance system, Appl. Math. Comput., 276 (2016), 200-217.  doi: 10.1016/j.amc.2015.12.015.  Google Scholar

[43]

D. WilczakS. Serrano and R. Barrio, Coexistence and dynamical connections between hyperchaos and chaos in the 4D Rössler system: A computer-assisted proof, SIAM J. Appl. Dyn. Syst., 15 (2016), 356-390.  doi: 10.1137/15M1039201.  Google Scholar

[44]

A. WolfJ. B. SwiftH. L. Swinney and J. A. Vastano, Determining Lyapunov exponents from a time series, Physica D, 16 (1985), 285-317.  doi: 10.1016/0167-2789(85)90011-9.  Google Scholar

[45]

Q. Zhang and X. Wei, RGB color image encryption method based on Lorenz chaotic system and DNA computation, Iete. Tech. Rev., 30 (2014), 404-409, http://dx.doi.org/10.4103/0256-4602.123123. Google Scholar

Figure 1.  When $ {a}^{i}= 1, $ $ {b}^{r}= 22, $ $ {b}^{i} = 1, $ $ {c = 5} $ and $ {a}^{r} $ is changed under the underlying conditions $ {x}^{r}= 1, $ $ {x}^{i}= 2, $ $ {y}^{r}= 3, { y}^{i} = 4, $ $ {z = 5} $ and $ { t = 0.} $ (a) Lyapunov types of system (3): $ {L}_{1}, $ $ {L}_{2} $ and $ {L}_{3} $ , (b) Lyapunov types of system (3): $ {L}_{4} $ and $ {L}_{5} $ , (c) Bifurcation graphs in $ (a^{r}, x^{r} ) $ plane.
Figure 2.  Results of model (3) for ${a}^{i} = 1, $${b}^{r} = 22, $ ${b}^{i} = 1, $ $ c = 5 $ vary ${a}^{r }$with under a similar quantity as in Fig. 1. (a) Hyperchaotic trajectories besides two PLEs, ${a} ^{r} = 41 $ in $ (z, x^{i} , y^{i}) $, (b) Chaotic trajectories with one PLE, ${ a}^{r} = 45 $ in $ (z, y^{r}, x^{r} ) $, (c) Quasiperiodic solution, ${a}^{r} = 60 $ in $(y^{r}, y^{i}, x^{r} ) $, (d) Periodic solutions, ${a}^{r} = 95 $ in $ (x^{r}, x^{i}, y^{i} ). $
Figure 3.  Numerical solution of model (24) (before and after control) for the case $a = a^{r} +ja^{i} = 41+j, $ $ b = b^{r} +jb^{i} = 22+j, $ $ c = 5 $.(a) State space in ${ (t, y}^{i}) $ plane (before control). (b) Hyperchaotic attractor in ${ (x}^{i}, y^{i}) $ plane (before control). (c) Hyperchaotic attractor in ${ (x}^{i}, y^{i}, z) $ space (before control). (d) State space in $ (t, y^{i}) $ plane (after control). (e) Hyperchaotic attractor in $ (x^{i} , y^{i}) $ plane (after control). (f) Hyperchaotic attractor in $ (x^{i} , y^{i}, z) $ space (after control).
Figure 4.  Time evolution of the controller: (a)$ (\Psi _{11}, t) $ graph. (b) $ (\Psi _{12}, t) $graph. (c) $ (\Psi _{21} , t) $ graph. (d) $ (\Psi _{22} , t) $ graph. (e) $ (\Psi _{31}, t) $ graph.
Figure 5.  The projection of the motion for model (30) with initial conditions for ${ a = 41+j}, $ ${ b = 22+j}, $ $ c = 5, $$ K = 60 $ and $\omega = 3 $. (a) Quasiperiodic motion on the $ (x^{r}, x^{i}, y^{i} ) $ space. (b) Quasiperiodic motion on the $(x^{r}, z, y^{i}) $ space.
Table 1.  The solutions of model (3)
$ L_{1} $ $ L_{2} $ $ L_{3} $ $ L_{4} $ $ L_{5} $ $ \textbf{Attractors} $
$ 0 $ $ - $ $ - $ $ - $ $ - $ $ \text{Periodic solutions} $
$ 0 $ $ 0 $ $ - $ $ - $ $ - $ $ \text{Quasi-periodic solution} $
$ + $ $ 0 $ $ - $ $ - $ $ - $ $ \text{Chaotic } $trajectories
$ + $ $ 0 $ $ 0 $ $ - $ $ - $ $ \text{Chaotic } $trajectories
$ + $ $ + $ $ 0 $ $ - $ $ - $ Hyperchaotic trajectories
$ L_{1} $ $ L_{2} $ $ L_{3} $ $ L_{4} $ $ L_{5} $ $ \textbf{Attractors} $
$ 0 $ $ - $ $ - $ $ - $ $ - $ $ \text{Periodic solutions} $
$ 0 $ $ 0 $ $ - $ $ - $ $ - $ $ \text{Quasi-periodic solution} $
$ + $ $ 0 $ $ - $ $ - $ $ - $ $ \text{Chaotic } $trajectories
$ + $ $ 0 $ $ 0 $ $ - $ $ - $ $ \text{Chaotic } $trajectories
$ + $ $ + $ $ 0 $ $ - $ $ - $ Hyperchaotic trajectories
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