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The Role of the Poincaré Map is Indicating a New Direction in the Analysis of the Genetic Network

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In this study, a modern technique dealing with a specific 4-dimensional genetic network has been introduced. Theoretically, the analysis has been performed and validated geometrically under the ambient hypercube. Both synchronized and asynchronous switching dynamics have been investigated for discrete time steps. By presenting its dynamics on a hypercube, we have obtained cycles that is sharing the same vertex. It is known as nothing but a weak condition of chaotic dynamics which is proven for the proposed problem of this article. Moreover, due to no self-input condition, only one gene can reach the threshold. For the rest genes, to terminate the integration, a value of 6.7 is placed; this will expedite the process of starting the new integration for the next approach of the threshold. Eigenvalues with negative real parts are obtained, which means the system is stable. Based on that,  its behavioral dynamics are predicted. Finally, the results have been investigated in Matlab and express its behavior through figures. From that figures, the stability of its dynamics has been concluded. The numerical calculation of Poincaré maps is used to address new evidence of dynamics, actually pointing to a new research direction. To the best of our knowledge, such numerical results were not studied before.
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Piecewise Linear Equations; Chaotic Dynamics; Genetic Networks; Synchronous and Asynchronous Logical Networks

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