Chemical reactions form the basis of life and understanding the different patterns and unpredictable changes in the reactions are noteworthy in real life situations. The article aims at constructing a mathematical model of two step reversible chemical reactions with a Caputo fractional difference operator. The reversible reaction involves the breakdown of an ester compound in the presence of water followed by the formation of fatty acid salts from the reaction of a free fatty acid with alkali hydroxide, such as NaOH. Using bifurcation diagrams, the chaotic response exhibited by the system is illustrated for state variables with identical fractional order and variables with non-identical fractional orders. The changes in periodic states of the system are investigated for each state variables with time varying plots and maximum Lyapunov exponents using the Jacobian matrix method are presented in support of the bifurcation diagrams. The synchronization of the subsystems of the proposed system is achieved with nonlinear control functions. Numerical simulations are presented to provide comparison of commensurate and incommensurate order models. Understanding the nature of these processes has significant applications in the production of bio-fuels from vegetable oils and animal fats by a transesterification reaction.
Our review focused on nanomaterials-based toxicity evaluation and its exposure to the human and aquatic animals when it was leached and contaminated in the environment. Ecotoxicological assessment and its mechanism mainly affect the skin covering layers and its preventive barriers that protect the foreign particles' skin. Nanoscale materials are essential in the medical field, especially in biomedical and commercial applications such as nanomedicine and drug delivery, mainly in therapeutic treatments. However, various commercial formulations of pharmaceutical drugs are manufactured through a series of clinical trials. The role of such drugs and their metabolites has not met the requirement of an individual's need at the early stage of the treatments except few drugs and medicines with minimal or no side effects. Therefore, biology and medicines are taken up the advantages of nano scaled drugs and formulations for the treatment of various diseases. The present study identifies and analyses the different nanoparticles and their chemical components on the skin and their effects due to penetration. There are advantageous factors available to facilitate positive and negative contact between dermal layers. It creates a new agenda for an established application that is mainly based on skin diseases.