| Citation: |
Ashish, Jinde Cao, Muhammad Aslam Noor. STABILIZATION OF FIXED POINTS IN CHAOTIC MAPS USING NOOR ORBIT WITH APPLICATIONS IN CARDIAC ARRHYTHMIA[J]. Journal of Applied Analysis & Computation, 2023, 13(5): 2452-2470. doi: 10.11948/20220350
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STABILIZATION OF FIXED POINTS IN CHAOTIC MAPS USING NOOR ORBIT WITH APPLICATIONS IN CARDIAC ARRHYTHMIA
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Abstract
Controlling chaos through stability in fixed and periodic states is used in various engineering problems such as heat convection, reduction control, spine-wave instability, traffic control models, cardiac arrhythmia, chemical chaos, etc. Traditionally, this process is done in the coordination of chaos and stability in fixed and periodic points by using fixed point iterative procedures. Therefore, this article deals with a novel alliance between stabilization in one-dimensional discrete maps and Noor fixed point iterative procedure. The procedure contains $\alpha$, $\beta$, $\gamma$ and $r$, as its four new control parameters due to which the stability rate increases more rapidly than the other existing procedures. The stability theorems and a few time varying examples for fixed and periodic points are studied using Noor control system. Further, the Lyapunov exponent property is also described and the maximum Lyapunov value is determined to examine the stability in fixed and periodic points. Moreover, an improved control-based cardiac arrhythmia model is discussed in the Noor control system. Surprisingly, it is noted that the added new parameters $\alpha$, $\beta$, and $\gamma$ may increase the stability in chaotic arrhythmia rapidly.
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Keywords:
- Chaos /
- stability /
- lterative procedure /
- difference maps /
- Lyapunov exponent
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References
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Ashish, Jinde Cao, Muhammad Aslam Noor. STABILIZATION OF FIXED POINTS IN CHAOTIC MAPS USING NOOR ORBIT WITH APPLICATIONS IN CARDIAC ARRHYTHMIA[J]. Journal of Applied Analysis & Computation, 2023, 13(5): 2452-2470. doi: 10.11948/20220350
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Figure 1.
(a) Unstable fixed point plot for an original map
$ \phi_4(v)=4v(1-v) $ $ \phi_4(v)=4v(1-v) $ $ N=50 $ $ \alpha = 0.4, \beta = 0.3 $ $ \gamma = 0.3 $ $ \alpha = 0.4, \beta = 0.3 $ $ \gamma = 0.3 $ -
Figure 2.
(a) Unstable periodic point plot for an original map
$ \phi_4(v)=4v(1-v) $ $ \phi^2_4(v) $ $ N=50 $ $ \alpha = 0.3, \beta = 0.2 $ $ \gamma = 0.2 $ $ \alpha = 0.3, \beta = 0.2 $ $ \gamma = 0.2 $ -
Figure 3.
(a) Bifurcation plot for the Cardiac Arrhythmia model
$ I_{n} = r I_{n-1}(1-I_{n-1}) $ $ \alpha = 0.4 $ $ \beta = 0.3 $ $ \gamma = 0.3 $ $ r\in[1, 4] $ $ \alpha = 0.3 $ $ \beta = 0.2 $ $ \gamma = 0.2 $ $ r\in[3, 4] $