Simulation and Lyapunov's Exponents Characterisation of Lorenz and Rosler Dynamics

Abstract

"This study investigated the characterisation of the dynamic responses of 3-dimensional Lorenz and Rӧsler models by Lyapunov's exponents using popular but laborious to implement Grahm Schmidt orthogonal rules over wider range of models driven parameters. The study also verifies a new proposed model for the validation of Lyapunov's spectrum when the requisite matrix depends on positions on the model attractor. Models and the corresponding Lyapunov's spectrums were simulated by appropriately effecting Grahm Schmidt orthogonal rules and using three different detailed constant step Runge-Kutta algorithms. The FORTRAN-90 coded algorithms were validated using literature results reported by Vladimir Golovko (2003). The stability of Lyapunov's exponents estimate variation was studied in the range of estimate reset period of 2≤τ≤16 .The Lorenz model was characterized at δ = 10, ƿ =28, and 1≤β≤2.8. This range covers both square and rectangular geometries. Similarly, Rӧsler model was characterised at a=ᵞ=0.2 and 2≤µ≤6. This range has potential to drive the model both periodically and chaotically depending on the choices of µ .The validation of the largest Lyapunov's exponents (λ)in Rosier model suffered the highest relative absolute percentage error of 14.29 while its absolute error is one of the lowest (0.01). The remaining five Lyapunov's exponents (three from Lorenz and two < from Rӧsier) suffered relative absolute percentage error of ≤ 2.00. Estimated Lyapunov's exponents stabilise for estimate reset period ≤ 10 .The most stable algorithms was found to be Butcher's modified fifth order followed respectively by fourth (RK4) and fifth (RK5) order. Estimation of Lyapunov's exponents' in Rӧsier model was found to be insensitive to algorithms due to its relative low degree of nonlinearity when compared with Lorenz model. It was established that the sum of Lyapunov's spectrum is the same as the average of trace of variation square matrix over large iteration regardless of dependence on position variable or not. This study demonstrated that the utility of Lyapunov's exponents as response characterising tool of dynamic systems driven by different parameters combination justify its laborious estimation by Grahm Schmidt method.

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