| Citation: |
Ashish, Jinde Cao. DYNAMICAL INTERPRETATIONS OF A GENERALIZED CUBIC SYSTEM[J]. Journal of Applied Analysis & Computation, 2022, 12(6): 2314-2329. doi: 10.11948/20210455
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DYNAMICAL INTERPRETATIONS OF A GENERALIZED CUBIC SYSTEM
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Abstract
Chaotic map is a typical route in mathematics to describe the dynamical interpretations in various applications of science and engineering. However, the dynamics in the traditional logistic chaotic map $ r\vartheta(1-\vartheta) $ depends on the single control parameter $ r $. In this article, a generalized cubic chaotic map with three changeable parameters $ a $, $ b $ and $ r $ is introduced and its dynamical properties are studied. The added new control parameter increases the flexibility in the system due to which it can fit in various applications. A few cases are discussed showing the effectiveness of the changeable parameters in various properties such as stationary and periodic states, stability in stationary states, Lyapunov exponent property, bifurcation interpretation, and the minimum entropy control. Further, the developments are illustrated mathematically as well as experimentally followed by period-doubling bifurcation and Lyapunov exponent diagrams. Moreover, it is noticed that as compared to traditional chaotic systems, a bifurcation self-similarity is seen along the x-axis in all the cases of the cubic map. Moreover, a brief summary on Shannon minimum entropy is also given to control the unstable stationary and periodic states in chaotic regime.
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Keywords:
- Bifurcation plot /
- cubic map /
- Lyapunov exponent /
- entropy
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References
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Ashish, Jinde Cao. DYNAMICAL INTERPRETATIONS OF A GENERALIZED CUBIC SYSTEM[J]. Journal of Applied Analysis & Computation, 2022, 12(6): 2314-2329. doi: 10.11948/20210455
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Figure 1. (a) Bifurcation plot for the cubic map
$ r \vartheta(1- \vartheta^2) $ when$ \vartheta \in [-1, 1] $ , (b) Magnified period-doubling regime for the cubic map$ r \vartheta(1- \vartheta^2) $ when$ \vartheta \in [-1, 1] $ , (c) Magnified chaotic regime for the cubic map$ r \vartheta(1- \vartheta^2) $ when$ \vartheta \in [-1, 1] $ , (d) Stabilization in fixed and periodic states for the cubic map$ r \vartheta(1- \vartheta^2) $ -
Figure 2. (a) Bifurcation plot for the cubic map
$ r \vartheta(2- \vartheta^2) $ when$ \vartheta \in [-1.42, 1.42] $ , (b) Bifurcation plot for the cubic map$ r \vartheta(4- \vartheta^2) $ when$ \vartheta \in [-2, 2] $ , (c) Bifurcation plot for the cubic map$ r \vartheta(6- \vartheta^2) $ when$ \vartheta \in [-2.5, 2.5] $ , (d) Comparative bifurcation plot for the map$ r \vartheta_n(a-\vartheta_n^2) $ when$ a = 2, 4 $ and$ 6 $ -
Figure 3. (a) Bifurcation plots for the map
$ r \vartheta_n(1-b\vartheta_n^2) $ when$ b = 2, 4 $ and$ 6 $ , (b) Bifurcation plots for the map$ r \vartheta_n(a-b\vartheta_n^2) $ when$ a, b = 2, 4 $ and$ 6 $ -
Figure 4. (a) Lyapunov exponent plot for the map
$ r \vartheta_n(1-\vartheta_n^2) $ when$ 0\leq r\leq 2.59 $ , (b) LE versus bifurcation plot for the map$ r \vartheta_n(1-\vartheta_n^2) $ when$ 0\leq r\leq 2.59 $ , (c) Lyapunov exponent plot for the map$ r \vartheta_n(a-\vartheta_n^2) $ when$ a=2, 4 $ and$ 6 $ , (d) LE versus bifurcation plot for the map$ r \vartheta_n(a-\vartheta_n^2) $ when$ a=2, 4 $ and$ 6 $ , (e) Lyapunov exponent plot for the map$ r \vartheta_n(1-b \vartheta_n^2) $ when$ b=2, 4 $ and$ 6 $ , (f) LE versus bifurcation plot for the map$ r \vartheta_n(1-b \vartheta_n^2) $ when$ a=2, 4 $ and$ 6 $ -
Figure 5. (a) Lyapunov exponent plot for the map
$ r \vartheta_n(a-b \vartheta_n^2) $ when$ a, b=2, 4 $ and$ 6 $ , (b) LE versus bifurcation plot for the map$ r \vartheta_n(a-b \vartheta_n^2) $ when$ a, b =2, 4 $ and$ 6 $