| Citation: |
Ashish, Jinde Cao, Fawaz Alsaadi. CHAOTIC EVOLUTION OF DIFFERENCE EQUATIONS IN MANN ORBIT[J]. Journal of Applied Analysis & Computation, 2021, 11(6): 3063-3082. doi: 10.11948/20210164
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CHAOTIC EVOLUTION OF DIFFERENCE EQUATIONS IN MANN ORBIT
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Abstract
Because of the tremendous characteristics of discrete chaos and sensitivity to their initial parameters, discrete one-dimensional maps are extensively used in every branch of science and engineering such as a security system, cryptography and traffic control models. In this article, it is proposed to examine the chaotic characteristics of a logistic-type difference equation using Mann orbit. Due to the presence of an added parameter α the resulting orbit provides superior chaotic characteristics from those of the existing characteristics in chaotic maps. As compared to existing chaotic maps it provides more efficient and effective chaotic characteristics like better sensitivity, suitable maximum Lyapunov exponent value and superior stability behavior. The results are carried out mathematically as well as experimentally followed by theorems, a few counterexamples and some concluding remarks. Moreover, a superior discrete traffic flow model using macroscopic approach and Greenshield's model is also described.
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Keywords:
- Chaos /
- period-doubling /
- Lyapunov exponent /
- difference maps
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References
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Ashish, Jinde Cao, Fawaz Alsaadi. CHAOTIC EVOLUTION OF DIFFERENCE EQUATIONS IN MANN ORBIT[J]. Journal of Applied Analysis & Computation, 2021, 11(6): 3063-3082. doi: 10.11948/20210164
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Figure 1. (a) Fixed state plot for the system
$ S(\alpha, \mu, x) $ for$ \mu = 1.2 $ , (b) Periodic state plot for the system$ S(\alpha, \mu, x) $ for$ \mu = 1.5 $ , (c) Chaotic state plot for the system$ S(\alpha, \mu, x) $ for$ \mu = 3 $ , (d) Fixed state plot for the system$ S(\alpha, \mu, x) $ for$ \mu = 3.5 $ , (e) Chaotic state plot for the system$ S(\alpha, \mu, x) $ for$ \mu = 8.5 $ and (f) Chaotic state plot for the system$ S(\alpha, \mu, x) $ for$ \mu = 45 $ -
Figure 2. (a) Periodic plot for the system
$ S(\alpha, \mu, x) $ for$ 0\leq\mu \leq3.2 $ , (b) Chaotic regime for the system$ S(\alpha, \mu, x) $ for$ 2.32<\mu \leq3.2 $ , (c) Period three plot for the system$ S(\alpha, \mu, x) $ for$ 2.87\leq\mu \leq2.92 $ , (d) Periodic plot for the system$ S(\alpha, \mu, x) $ for$ 0\leq\mu \leq8.7 $ , (e) Periodic plot for the system$ S(\alpha, \mu, x) $ for$ 0\leq\mu \leq55 $ and (f) Comparative plot for the system$ S(\alpha, \mu, x) $ versus logistic map -
Figure 3. Functional plot for
$ S^3(\alpha, \mu, x) $ (a) when$ \mu = 2.9 $ , (b) when$ \mu = 2.75 $ , (c) when$ \mu = 7.82 $ , (d) when$ \mu = 7.55 $ , (e) when$ \mu = 50.2 $ and (f) when$ \mu = 48 $ -
Figure 4. Lyapunov
$ \gamma $ plot for the orbit$ \{S^n(\alpha, \mu, x_0)\} $ (a) when$ \alpha = 0.8 $ and$ 0\leq\mu\leq2.32 $ , (b) when$ \alpha = 0.8 $ and$ 2.32<\mu \leq 3.2 $ , (c) when$ \alpha = 0.5 $ and$ 0\leq\mu\leq6.34 $ , (d) when$ \alpha = 0.5 $ and$ 6.34<\mu \leq 8.7 $ , (e) when$ \alpha = 0.2 $ and$ 0\leq\mu\leq41 $ and (f) when$ \alpha = 0.2 $ and$ 41<\mu \leq 55 $ -
Figure 5. Lyapunov exponent
$ \eta $ plot for the system$ S(\alpha, \mu, x) $ (a) when$ \alpha = 0.8 $ and$ 0\leq \mu \leq3.2 $ , (b) when$ \alpha = 0.8 $ and$ 2.32\leq \mu \leq3.2 $ , (c) when$ \alpha = 0.5 $ and$ 0\leq \mu \leq8.7 $ , and (d) when$ \alpha = 0.5 $ and$ 6.34\leq \mu \leq8.7 $ -
Figure 6. Lyapunov exponent
$ \eta $ plot for the system$ S(\alpha, \mu, x) $ (a) when$ \alpha = 0.2 $ and$ 0\leq \mu \leq55 $ , (b) when$ \alpha = 0.2 $ and$ 0\leq \mu\leq41 $ , (c) when$ \alpha = 0.2 $ and$ 41<\mu\leq55 $ , and (d) versus$ \mu x(1-x) $ and$ 1-\mu x^2 $