Abstract
Four discrete models, using the exact spectral derivative discretization finite difference (ESDDFD) method, are proposed for a chaotic five-dimensional, conformable fractional derivative financial system incorporating ethics and market confidence. Since the system considered was recently studied using the conformable Euler finite difference (CEFD) method and found to be hyperchaotic, and the CEFD method was recently shown to be valid only at fractional index α=1, the source of the hyperchaos is in question. Through numerical experiments, illustration is presented that the hyperchaos previously detected is, in part, an artifact of the CEFD method, as it is absent from the ESDDFD models.
Highlights
Hyperchaotic systems [1,2]—typically defined as systems with at least two positiveLyapunov exponents [3,4,5]—of a fractional-order have been investigated in many contexts, such as systems of Rössler [6] or Lorenz [7] type, those with flux controlled memristors [8]or realized in circuits [9,10,11], those arising from cellular neural networks [12], and financial systems [13]
As recounted in [13], a nonlinear financial system depicting the relationship among interest rates, investments, prices, and savings was first introduced by Huang and Li [14]
The average profit margin was added as a variable in Yu et al [19], while investment incentive and market confidence were introduced in Xin et al [20,21]
Summary
Hyperchaotic systems [1,2]—typically defined as systems with at least two positive. Lyapunov exponents [3,4,5]—of a fractional-order have been investigated in many contexts, such as systems of Rössler [6] or Lorenz [7] type, those with flux controlled memristors [8]. Methodology is a novel extension, developed in the context of advection–reaction–diffusion equations [51], of the NSFD method to non-integer derivatives [52] It is, natural to ask whether some of the hyperchaotic behavior detected in the fractional financial system is an artifact of the method and whether ESDDFD models can be constructed to eliminate such induced hyperchaos. The purpose of the present study is to investigate this question—in particular, the effects of the discretization of the derivative and that of non-linear terms To this end, the following four discrete models using the ESDDFD method are constructed for the system (1) and the bifurcation experiments of [13].
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