Abstract
A new four-dimensional hyperchaotic financial model is introduced. The novelties come from the fractional-order derivative and the use of the quadric function x 4 in modeling accurately the financial market. The existence and uniqueness of its solutions have been investigated to justify the physical adequacy of the model and the numerical scheme proposed in the resolution. We offer a numerical scheme of the new four-dimensional fractional hyperchaotic financial model. We have used the Caputo–Liouville fractional derivative. The problems addressed in this paper have much importance to approach the interest rate, the investment demand, the price exponent, and the average profit margin. The validation of the chaotic, hyperchaotic, and periodic behaviors of the proposed model, the bifurcation diagrams, the Lyapunov exponents, and the stability analysis has been analyzed in detail. The proposed numerical scheme for the hyperchaotic financial model is destined to help the agents decide in the financial market. The solutions of the 4D fractional hyperchaotic financial model have been analyzed, interpreted theoretically, and represented graphically in different contexts. The present paper is mathematical modeling and is a new tool in economics and finance. We also confirm, as announced in the literature, there exist hyperchaotic systems in the fractional context, which admit one positive Lyapunov exponent.
Highlights
Many behaviors of the dynamical systems are deterministic. e systems’ future behaviors follow the same evolutions and are explained by the initial conditions and the past of the systems
In terms of the characterization of the chaos, we will provide the existence of two positive Lyapunov exponents that are sufficient for the hyperchaotic behaviors in the integer version, but in the context of fractional-order derivative, the numbers of positive Lyapunov exponents are not an adequate definition to characterize the hyperchaotic dynamics because there exist systems which are hyperchaotic with the bifurcation diagrams but admits one positive Lyapunov exponent
We have mainly focused on the existence and the uniqueness of the solutions of the fractional 4D hyperchaotic financial model described by the generalized Caputo–Liouville derivative
Summary
Many behaviors of the dynamical systems are deterministic. e systems’ future behaviors follow the same evolutions and are explained by the initial conditions and the past of the systems. In [8], Kumar et al constructed a new finance model too, namely, the four-dimensional chaotic financial model and used the Lyapunov direct method to study the stability of the equilibrium points and proposed the numerical simulations of the new model. In terms of the characterization of the chaos, we will provide the existence of two positive Lyapunov exponents that are sufficient for the hyperchaotic behaviors in the integer version, but in the context of fractional-order derivative, the numbers of positive Lyapunov exponents are not an adequate definition to characterize the hyperchaotic dynamics because there exist systems which are hyperchaotic with the bifurcation diagrams but admits one positive Lyapunov exponent.
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