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arXiv2015

A Modified Quadratic Lorenz attractor

This study introduces a modified quadratic Lorenz attractor. The properties of this new chaotic system are analysed and discussed in detail, by determining the equilibria points, the eigenvalues of the Jacobian, and the Lyapunov exponents. The numerical simulations, the time series analysis, and the projections to the $xy$-plane, $xz$-plane, and $yz$-plane are conducted to highlight the chaotic behaviour. The multiplicative form of the new system is also presented and the simulations are conducted using multiplicative Runge-Kutta methods.

math.DS↗

arXiv2015

The Runge-Kutta Method in Geometric Multiplicative Calculus

This paper illuminates the derivation, the applicability, and the efficiency of the Multiplicative Runge-Kutta Method, derived in the frame- work of geometric multiplicative calculus. The removal of the restrictions of geometric multiplicative calculus to positive-valued functions of real variable and the fact that the multiplicative derivative does not exist at the roots of the function, is presented explicitly to ensure that the proposed method is uni- versally applicable. The error analysis is also carried out in the framework of geometric multiplicative calculus explicitly. The presented method is applied to various problems and the results are compared to the ones obtained from the Ordinary Runge-Kutta Method. Moreover, for one example, a comparison of the computation time vs. relative error, is worked out, to illustrate the general advantage of the proposed method.

math.NA↗