An Algorithm Based Levenberg Marquardt Method with Genetic Algorithm for Solving Continuation Problems

Hala A. Omar(1*), El-Sayed M. Zaky(2), Gamal Ibrahim(3), Ahmed Elsawy(4)

(1) Basic Engineering Sciences Department, Faculty of Engineering, Menoufia University, Egypt
(2) Basic Engineering Sciences Department, Faculty of Engineering, Menoufia University, Egypt
(3) Basic Engineering Sciences Department, Faculty of Engineering, Menoufia University, Egypt
(4) Dept. of Mathematics, Faculty of Science, Qassim University, Saudi Arabia
(*) Corresponding author

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In this paper, a new algorithm for bifurcation computation is proposed. The algorithm combines the Levenberg Marquardt method with the Genetic algorithm. With a set of random initial guesses, the Levenberg Marquardt algorithm is used to compute the static solutions of a bifurcation diagram. The Genetic algorithm is used to avoid getting trapped in local minima. The performance of the combined method is thoroughly investigated with the Lorenz Oscillator and the Co-oxidization models. The test results are compared to the corresponding results of the software package AUTO to verify the accuracy of the proposed algorithm. The results corroborate the suitability of the proposed algorithm for computational purposes in chemical applications. The performance of the Activated Sludge Reactor is finally tested as a typical example for chemical engineering systems. As revealed from the results the proposed algorithm gives accurate, fast, and versatile bifurcation results with one, two, or three bifurcation parameters.
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Bifurcations; Genetic Algorithm; Levenberg Marquardt Algorithm

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