Dynamical behaviour of fractional-order finance system
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Dynamical behaviour of fractional-order finance system MUHAMMAD FARMAN1 , ALI AKGÜL2 ,∗ , MUHAMMAD UMER SALEEM3 , SUMAIYAH IMTIAZ1 and AQEEL AHMAD1 1 Department
of Mathematics and Statistics, University of Lahore, Lahore 54590, Pakistan of Mathematics, Art and Science Faculty, Siirt University, 56100 Siirt, Turkey 3 Division of Science and Technology, Department of Mathematics, University of Education, Lahore, Pakistan ∗ Corresponding author. E-mail: [email protected] 2 Department
MS received 10 March 2020; revised 18 August 2020; accepted 8 September 2020 Abstract. In this paper, we developed the fractional-order finance system transmission model. The main objective of this paper is to construct and evaluate a fractional derivative to track the shape of the dynamic chaotic financial system of fractional order. The numerical solution for fractional-order financial system is determined using the Atangana–Baleanu–Caputo (ABC) and Caputo derivatives. Picard–Lindelof’s method shows the existence and uniqueness of the solution. Numerical techniques show that ABC derivative strategy can be used effectively to overcome the risk of investment. An active control strategy for controlling chaos is used in this system. The stabilisation of equilibrium is obtained by both theoretical analysis and simulation results. Keywords. Finance system; fractional derivative; Picard–Lindelof; stability analysis; price index. PACS nos. 02.30.Hq; 02.60.Cb; 02.30.Gp
1. Introduction Among various fields of natural science, researchers are attracted more to nonlinear chaotic systems. Such systems are dynamic and sensitive to initial conditions. Although the chaotic phenomenon first became evident in 1985, the influence of Western science has been affected by the fact that in an economic system the chaotic phenomenon itself makes the macroeconomic feature infinite. The efficiency of the intervention is very small, while the government can take macrocontrol measures like finance policies or monetary policies to intervene. The precise economy is significantly restricted due to the instability and the difficulty and also the rational prediction behaviour. The internal structures of the economy have shown discrepancies and inconsistencies, and in the fields of finance, supply and social economics, extremely difficult phenomena are occurring because of interaction between non-linear variables, increasingly complicated with all kinds of economic problems, and processes developing at a low to a higher level. The monitoring and stabilisation of unstable, regular or stationary solutions for accurate economic forecasts is thus becoming increasingly important [1,2].
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The fractional calculus has grown significantly during the last forty years as an excellent modelling technique. This is a strong tool in designing most of the physical processes with a memory effect which cannot be expressed well by integer and differential equations. It has been applied to many fields of science and technology. For
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