# American Institute of Mathematical Sciences

## 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
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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.
Results of model (3) for ${a}^{i} = 1, $${b}^{r} = 22, {b}^{i} = 1, c = 5 vary {a}^{r }with under a similar quantity as in Fig. 1. (a) Hyperchaotic trajectories besides two PLEs, {a} ^{r} = 41 in (z, x^{i} , y^{i}) , (b) Chaotic trajectories with one PLE, { a}^{r} = 45 in (z, y^{r}, x^{r} ) , (c) Quasiperiodic solution, {a}^{r} = 60 in (y^{r}, y^{i}, x^{r} ) , (d) Periodic solutions, {a}^{r} = 95 in (x^{r}, x^{i}, y^{i} ). Numerical solution of model (24) (before and after control) for the case a = a^{r} +ja^{i} = 41+j, b = b^{r} +jb^{i} = 22+j, c = 5 .(a) State space in { (t, y}^{i}) plane (before control). (b) Hyperchaotic attractor in { (x}^{i}, y^{i}) plane (before control). (c) Hyperchaotic attractor in { (x}^{i}, y^{i}, z) space (before control). (d) State space in (t, y^{i}) plane (after control). (e) Hyperchaotic attractor in (x^{i} , y^{i}) plane (after control). (f) Hyperchaotic attractor in (x^{i} , y^{i}, z) space (after control). Time evolution of the controller: (a) (\Psi _{11}, t) graph. (b) (\Psi _{12}, t) graph. (c) (\Psi _{21} , t) graph. (d) (\Psi _{22} , t) graph. (e) (\Psi _{31}, t) graph. The projection of the motion for model (30) with initial conditions for { a = 41+j}, { b = 22+j}, c = 5,$$ 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.
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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