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Home  >  Transasctions of NAMP VOL2

30. Stability Analysis of Fractional Duffing Oscillator 1 by M.O. Oyesanya Transactions of NAMP Vol 2, (Nov., 2016), pp 325 – 342
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We solve fractional Duffing equation with two fractional derivatives namely

 With initial condition q (0) =1,  

Here λ is the damping coefficient, µ the stiffness coefficient, and ν the coefficient of nonlinearity with f(t) as the forcing function. Using complex analysis method we find that a Hankel contour ensue and we are unable to calculate residues because the location of the poles and branch points cannot be explicitly found except for special cases relating α and β in a simple way. Using the Laplace transform method in this first part we solve the linear homogenous Duffing equation in terms of Mittag-Leffler function. 

We then consider in the second part of the paper solving the nonlinear case using homotopy analysis method. We found that there are more interesting cases that were not evident in the integer calculus case. We found that for the case α = 2β with 0<β<1 there are 2n poles and branch points, where . We found that there are infinitely many solutions and as  implicating that the solution goes to a constant as β decreases acting as an asymptote just as was observed in the linear homogeneous case (observation (iii) above).

For the linear case (ν = 0) with f(t) =0 we observe that for µ < 0 and α = 2β

The solution is non-periodic

For β є (0,1) the solutions exhibit the characteristics of decaying exponential function

For fixed λ solutions with lower fractional derivatives are more damped than those with higher derivatives. The solution for β = 1/3 have some distinguishing property with faster damping.

We discuss the case for which α ≠ 2β. We found that for the case α = nβ we have a case and results similar to those discussed by [1]. For the case for which   we have a very non-trivial analysis for various f(t). Numerical consideration and simulations showed some chaotic nature as in [2]

Using some existing results in the literature we show that the fractional Duffing oscillator with two fractional derivatives can be seen as an appropriate model for earthquake prediction. We then have a comparative analysis of our results and the results known for the integer calculus case. We finally engage in the stability analysis of our problem.