| Citation: |
H. I. Abdel-Gawad, B. Abdel-Aziz, M. Tantawy. EXTENDED CENTER MANIFOLD, GLOBAL BIFURCATION AND APPROXIMATE SOLUTIONS OF CHEN CHAOTIC DYNAMICAL SYSTEM[J]. Journal of Applied Analysis & Computation, 2024, 14(4): 2125-2139. doi: 10.11948/20230308
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EXTENDED CENTER MANIFOLD, GLOBAL BIFURCATION AND APPROXIMATE SOLUTIONS OF CHEN CHAOTIC DYNAMICAL SYSTEM
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Abstract
The study of the chaotic Chen dynamic System (CDS) has been a recent focus in the literature, with numerous works exploring its various chaotic features. However, the majority of these studies have relied primarily on numerical techniques to investigate nonlinear dynamic systems (NLDSs). In this context, our aim is to derive approximate analytical solutions for the CDS by developing an iterative scheme. We have proven the convergence theorem for this scheme, which ensures that our iterative process will converge to the exact solution. Additionally, we introduce a new method for constructing the extended center manifold, a critical component in the analysis of dynamical systems. The characteristics of the global bifurcation of the system components within the parameter space are explored. The error analysis of the iterated solutions demonstrates the efficiency of the present technique. We present both three-dimensional (3D) and two-dimensional (2D) phase portraits of the system. The 3D portrait reveals a feedback loop pattern, while the 2D portrait, which represents the interaction of the system components, exhibits multiple pools and cross pools. Furthermore, we illustrate the global bifurcation by visualizing the components of the CDS against the space-parameters. The sensitivity of CDS to infinitesimal variations in the initial conditions (ICs) is tested. It is found that even minor changes can lead to significant alterations in the system.
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References
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H. I. Abdel-Gawad, B. Abdel-Aziz, M. Tantawy. EXTENDED CENTER MANIFOLD, GLOBAL BIFURCATION AND APPROXIMATE SOLUTIONS OF CHEN CHAOTIC DYNAMICAL SYSTEM[J]. Journal of Applied Analysis & Computation, 2024, 14(4): 2125-2139. doi: 10.11948/20230308
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Figure 1.
(ⅰ)-(ⅴ). The phase portraits (2D-parametricplot) for each pair of
$ x^{(1)}, y^{(1)}, $ $ z^{(1)} $ $ a\text{=}2, \ b\text{=}0.1, \ x_{0}\text{=}0.1, \ y_{0}\text{=}0.2, \ z_{0}\text{=}0.8 $ -
Figure 2.
(ⅰ)-(ⅲ). Show the bifurcations of
$ x^{(1)}y^{(1)}, \boldsymbol{z}^{(1)} $ $ a $ $ b=0.1, \ x_{0}=2, \ y_{0}=-5, \ z_{0}=0.1 $ -
Figure 3.
(ⅰ), (ⅱ), show 3D and contour plot of Eq. (5.12), against
$ (t, \alpha $ -
Figure 4.
(ⅰ) - (ⅱ). When: (ⅰ)
$ b=1.5, \ x_{0}=2, \ y_{0}=1.2, \ z_{0}=3, $ $ a\text{=}0.5, $ $ \nu\text{=}-0.2, \ \mu\text{=}0.19, \ x_{0}=2.5, \ y_{0}=1.5, \ z_{0}=3 $ $ a $ $ b $ $ c<0 $ $ c>0 $ -
Figure 5.
(ⅰ). When
$ a=2, \ b=0.1, \ m=0.2, $