Narrow Results

This study investigates the chaotic behavior and stability of a nonlinear fractional-order financial system utilizing the Caputo fractional derivative. Initially, we formulate the nonlinear fractional-order financial model and establish the problem framework. Next, we prove the existence and uniqueness of solutions by applying the Banach and Schauder fixed-point theorems to the proposed system. Additionally, we analyze the Ulam–Hyers stability and discuss other significant findings related to the system’s stability. To simulate the proposed model, we develop numerical schemes based on fractional calculus, employing Lagrange polynomial interpolation. Finally, we present the numerical simulations to validate the theoretical results, highlighting the significant impact of fractional-order derivatives on the system’s behavior.

The main purpose of the present investigation is to find the solution of fractional coupled equations describing the romantic relationships using $q$-homotopy analysis transform method ($q$-HATM). The considered scheme is a unification of $q$-homotopy analysis technique with Laplace transform (LT). More preciously, we scrutinized the behavior of the obtained solution for the considered model with fractional-order, in order to elucidate the effectiveness of the proposed algorithm. Further, for the different fractional-order and parameters offered by the considered method, the physical natures have been apprehended. The obtained consequences evidence that the proposed method is very effective and highly methodical to study and examine the nature and its corresponding consequences of the system of fractional order differential equations describing the real word problems.

The emergent global economy depends on financial growth. Social and economic development is achieved by a stable, growing, and secure financial chain system. Financial instability poses huge financial inflation and economic decline. The financial crisis and the turbulence of the finance market also result in deterministic instability. This fluctuation during the financial operation may also critically affect the development of the economic system and other sociological and financial stabilities. For the chaotic behavior of the finance system, the mathematical formulation is developed along with controlling terms in this paper. Fitting the controlling parameters, the financial model can be made secure and safe from periodic behavior and will run in chaotic conditions. At the first attempt, the dynamical system and the controlling parameters are adjusted through assigned values and their ranges. Second, the impact of these parameters is studied along with the feasible techniques by graphical representations. The said financial dynamics are investigated in the sense of fractal–fractional derivatives. The concept of Ulam–Hyers stability is also developed for the considered model.