Abstract
Numerical discretization for the fractional differential equations is applied to the chaotic financial model described by the Caputo derivative. The graphical representations to support the numerical discretization are presented. We profit by analyzing the impact generated by the variations of the saving rate, the per investment cost, and the elasticity of demands in the dynamics of the solutions obtained with our numerical scheme. Notably, we use bifurcation diagrams to quantify the impact of the saving rate, the per investment cost, and the elasticity of demands, as well as the Lyapunov exponent to characterize the existence of chaos for the chosen value of the fractional order. The chaos observed depends strongly on these previously mentioned parameters. We finish by proposing a suitable control to synchronize the drive system and the response fractional financial model, using Lyapunov direct methods. The stability analysis of the equilibrium points of the chaotic financial model has been presented.
Highlights
Basic Fractional Calculus OperatorsWe address the fractional Caputo derivative, the fractional Riemann-Liouville derivative, and their associated integral
Following the literature, the savings can be considered as a source of instability for demand. e Keynesian and Neo-Keynesian and their macroeconomic models, Complexity contrary to the neoclassical ones, attribute a passive role to savings
We propose a new numerical scheme and depict the solutions according to this numerical scheme and analyze as well the impact of the saving rate, the per investment cost, and the elasticity of demands. ere exist investigations related to the fractional financial models
Summary
We address the fractional Caputo derivative, the fractional Riemann-Liouville derivative, and their associated integral. Consider the function x: [0, +∞[ ⟶ R; the fractional derivative in sense of Riemann-Liouville, of order α, is represented by the relationship. . .) is the gamma Euler function. Consider the function x: [0, +∞[ ⟶ R; the fractional derivative in sense of Caputo, of order α, is represented by the relationship. Definition 3 (see [27, 29]). E Riemann-Liouville fractional integral is represented as the following form for the function x: [0, +∞[ ⟶ R: Iαx(t). . .) represents the Gamma Euler function with the order α > 0 Γ(α) 0 where the function Γ(. . .) represents the Gamma Euler function with the order α > 0
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