Bifurcation and chaotic behavior of a discrete fractional order Lorenz system

Selvam, A. George Maria; Vianny, D. Abraham

doi:10.1063/1.5112237

Cited by 4 publications

(4 citation statements)

References 6 publications

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“…where C D α 0,t is a Caputo fractional derivative with 0 < α < 1 and f (x(t)) is potentially a nonlinear vector field. We show that the first order discretization given by El-Sayed and others [7][8][9][10][11][12][13] is incorrect. Further to this we correct the derivation of the difference equation resulting in an increasing order difference equation which captures the memory effect of the FDE.…”

Section: Introductionmentioning

confidence: 88%

“…Whilst the final result of the El-Sayed method [7][8][9][10][11][12][13] is incorrect, the initial approach that they undertake has some merit. Here we will consider the general Caputo initial value problem given in Eq.…”

Section: Piecewise Constant Integrablizationmentioning

confidence: 99%

“…In general the approach taken by El-Sayed and others [7][8][9][10][11][12][13] can be generalised easily for an equation in the form of Eq. ( 1).…”

Section: Comparison To El-sayed and Salmanmentioning

confidence: 99%

“…These models are often posed in the form of fractional differential equations (FDEs). Recently there have been attempts to construct a difference equation that captures the dynamical behaviour of a FDE by using a nonstandard piecewise approximation [7][8][9][10][11][12][13]. These attempts have resulted in the incorrect construction of first order difference equations that can not approximate the dynamics of FDEs.…”

Section: Introductionmentioning

confidence: 99%

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Discretization of fractional differential equations by a piecewise constant approximation

Angstmann

Henry

Jacobs

et al. 2017

Math. Model. Nat. Phenom.

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There has recently been considerable interest in using a nonstandard piecewise approximation to formulate fractional order differential equations as difference equations that describe the same dynamical behaviour and are more amenable to a dynamical systems analysis. Unfortunately, due to mistakes in the fundamental papers, the difference equations formulated through this process do not capture the dynamics of the fractional order equations. We show that the correct application of this nonstandard piecewise approximation leads to a one parameter family of fractional order differential equations that converges to the original equation as the parameter tends to zero. A closed formed solution exists for each member of this family and leads to the formulation of a difference equation that is of increasing order as time steps are taken. Whilst this does not lead to a simplified dynamical analysis it does lead to a numerical method for solving the fractional order differential equation. The method is shown to be equivalent to a quadrature based method, despite the fact that it has not been derived from a quadrature. The method can be implemented with non-uniform time steps. An example is provided showing that the difference equation can correctly capture the dynamics of the underlying fractional differential equation.

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