Analysis of Fractional Order Chaotic Financial Model with Minimum Interest Rate Impact

: The main objective of this paper is to construct and test fractional order derivatives for the management and simulation of a fractional order disorderly ﬁnance system. In the developed system, we add the critical minimum interest rate d parameter in order to develop a new stable ﬁnancial model. The new emerging paradigm increases the demand for innovation, which is the gateway to the knowledge economy. The derivatives are characterized in the Caputo fractional order derivative and Atangana-Baleanu derivative. We prove the existence and uniqueness of the solutions with ﬁxed point theorem and an iterative scheme. The interest rate begins to rise according to initial conditions as investment demand and price exponent begin to fall, which shows the ﬁnancial system’s actual macroeconomic behavior. Speciﬁcally component of its application to the large scale and smaller scale forms, just as the utilization of speciﬁc strategies and instruments such fractal stochastic procedures and expectation.


Introduction
Different parts of the financial sector are investigated through mathematical models, this article is helpful in discussing advantages and drawbacks of mathematical models in the financial associations. Different solutions for improving mathematical models and obstructions in the application zone is also discussed. Mathematical modeling is the technique in which sensible and similar numerical expressions from vertical frameworks are made which is used in translating different issues including drawing systematic ideas. These systematic ideas are used in formulating strategy by choosing information and understanding problem [1]. Mathematical models help in different fields of science such as sociology and engineering. Mathematics have an important place in the field of finance. In the field of finance, account related theory on assessment of exercises on the money related administrators [2]. Different parts of financial market types and different scientific and numerical systems are drawn by using mathematical models [3].
Different types of genuine certified systems are derived through the device of fractional calculus [4]. We establish some writing with the help of the research work and construct the hypothesis of fractional calculus and exhibiting several utilizations. The recently presented Caputo Fabrizio fractional derivative is used to examine the partial model of an adjusted Kawahara condition by where Γ(.) refers to the function of Gamma. Laplace transform of the above derivative is obtained as [23,24]: Recently, Atangana and Baleanu proposed a fractional derivative with the Mittag-Leffler function as the kernel of differentiation. This kernel is non-singular and nonlocal and preserves the benefits of the above Liouville-Caputo derivative. The Atangana-Baleanu derivative has been defined as [25]: where κ , Z(κ) refers to the Z(0) = Z(1) = 1 and Eκ(.) refers to the equation Mittag-Leffler. Equation (3) Laplace is defined as follows: The fractional integral associated with the Atangana-Baleanu derivative with non-local kernel is defined as When κ is equivalent to zero, the initial function will be retrieved. κ = 1 will be retrieved from the classical ordinary integral.

Liouville-Caputo Sense
The strategy is an experimental system dependent on the blend of homotopy analysis technique and Laplace's transformation with polynomial homotopy [23,26]. The primary steps of this strategy are characterized as follows: Step 1. We should take a look at the following condition: is a continuous function. The boundary and initial conditions can be treated in a similar way.
Step 2. Applying the methodology proposed in [23,27], we get the following m-th order deformation equation: where the Laplace transform is implemented in Caputo sense (1) and P k is the homotopy polynomial described by Odibat in [28].
Step 3. Regarding homotopy polynomials, the nonlinear term ∧[h]g(h, t) is extended as Step 4. Expanding the nonlinear term in (6) as a progression of polynomials for homotopy, we can compute the diverse g m (h, t) for m > 1 and Equation solutions (5) can be written as The classical form of the model first studied in [29] and we modify the model by adding d as critical minimum interest rate. By using this methodology, a Liouville-Caputo fractional order derivative was utilized to solve the using time-fractional funding model: where x, y, and z are the state variables representing interest rate, investment demand, and price index, respectively, and we add the critical minimum interest rate d parameter in [30]. The parameter a is for savings, b is to cost per investment and c is the elasticity of market demand, although the parameters are non-negative constants i.e., a = 3, b = 0.1 and c = 1. From the above model (10)-(12) , we use the parameter d to modify the model, where d represents critical minimum interest rate with initial conditions x(0) = n 1 = 0.1, y(0) = n 2 = 4, z(0) = n 3 = 0.5.

Solution.
We also implemented the Laplace transform (2) to the system's first formula on (10): The initial conditions are taken and the above equation is simplified With inverse Laplace transform to Equation (14), getting For the other equations shown in Equations (11) and (12), we get with feature (e) = 0 where e is constant. Let's describe the following system as: The equation of so-called zero-order deformation is given by when p = 0 and p = 1, we have The deformation equations of the mth-order are given Transforming the inverse Laplace into Equations (24)- (26). We've got this where The mth-order deformation equation solution (24)-(26) is presented as: where Finally, the solutions of the Equations (10)-(12) are Through combining the Laplace transform (2) and its inverse, another model (10)-(12) solution can be obtained. The iterative scheme is given through where n 1 , n 2 , and n 3 are the initial conditions. If n tends to infinity, it is assumed that the solution is a limit Proof. We are going to assume the following. There are five positive constants D, E, and F can be found such that for all 0 ≤ t ≤ T ≤ ∞, Now, we consider a subset of L 2 ((e, f )(0, T)) defined as follows: We have Then, We obtain Additionally, if we find a non-null vector (x 1 , y 1 , z 1 ) using a certain routine as above, we get We conclude from the results of Equations (50) and (52) that the iterative method used is stable. Then, we obtain the same in [31]:

Atangana-Baleanu-Caputo Sense
Considering the system with an ABC fractional order derivative according to the methodology mentioned in [23,26]: with initial conditions x(0) = n 1 ≥ 0, y(0) = n 2 ≥ 0, z(0) = n 3 ≥ 0 Solution: We apply the Laplace transformation (4) to Equation (56), we have Simplifying the above equation with taking initial conditions Then, we have Similarly to Equations (57) and (58), we have Here, we select a operator that is of linear type as Next, we describe the model below: This is the so-called zeroth-order deformation is presented by: The equations of the mth-order deformation are presented by Use the inverse Laplace to transform the Equations (69)-(71), and we obtain where The mth-order deformation of the system is specified as where [ d m dp m N[(pφ 1 (t; p))(pφ 2 (t; p))]] p=0 (81) Finally, the solutions of Equations (56)-(58) are given as Models (56)-(58) solution can be obtained with Equation (4). Systems (56)-(58) are similar to the Volterra form in the Atangana-Baleanu sense. With the iterative scheme, we get Theorem 2. We prove the existence and uniqueness of the solution using a Picard-Lindelof approach.
Proof. The following operator is considered: Let where Considering the Picards operator, we have defined as follows: where Now, we presume that all solutions are bound in a certain amount of time Here, we request that ϑ < k ξ Then, we obtain with ω less than 1.
Since Ω is a contraction, we obtain θω < 1, so the specified ϑ operator is also a contraction. The Atangana-Baleanu fractional integral numerical approximation [27] using the Adams-Moulton rule is given by where b κ k = (k + 1) 1−κ − (k) 1−κ . Hence, it shows that existence and uniqueness of the solution for the dynamical finance system. We have the following generalized solution with the iterative method:

Numerical Results and Discussion
The ABC derivative has been used to present the theoretical solution of the fractional-order model consisting of a nonlinear system of the fractional differential equation. In this model, we represent x(t), y(t) and z(t) are interest rate, investment demand, and price exponent with initial conditions x(0) = 0.1, y(0) = 4 and z(0) = 0.5, while the parameter a is for savings, b is to cost per investment, and c is the commercial markets demand elasticity with a = 3, b = 0.1, and c = 1 are given in [30,32]. By utilizing Caputo and ABC fractional derivative, the numerical results of interest rate, investment demand, and price exponent for various fractional estimations of η are acquired. Figures 1-3 refer to the graphical solution of the finance system with the Caputo derivative of the finance system. Within this figure, we noticed that interest rate, investment demand, and the price exponent have more degree of freedom as contrasted with ordinary derivatives. From Figures 4-6, we use ABC fractional-order derivative of the financial system, we effectively have seen that interest rate, investment demand, and price exponent rates are the better estimations compared with ordinary derivatives. Figures 7-10 present the comparison of Caputo derivative and ABC derivative for the finance system. It should be observed that the behavior of the finance system is almost the same but ABC derivative presents more convenient and comfortable behavior in a system for closed-loop design. Caputo and ABC fractional derivatives are increasing or decreasing in the relationship between these variables. From Figures 1-10, remarkable responses are obtained from the developed model for compartments by taking non-integer fractional parameter values. Numerical results show that the system keeps the η chaotic motion. The interest rate begins to rise according to initial conditions as investment demand and price exponent begin to fall, which shows the financial system's actual macroeconomic behavior. It is observed here that a complex chaotic fractional system provides more appropriate and reliable results compared to time integer parameters for non-integer time-fractional parameters. In this system, we add parameter d to develop the new financial stable model which is the critical minimum interest rate. In order to observe the impact of factors on the mechanics of the fractional-order model, different numerical ways can be observed in Figures 11-13. These simulations reveal a change in the value of a critical minimum interest rate of the model. We see that decreasing the critical minimum interest rate decreases the price of the exponent and the investment demand becomes high. Due to increasing the investment demand, our economy will become stronger. Interest Rate x(t)

Complex choatic System with ABC Derivative
Interest Rate x(t) at κ=1 Interest Rate x(t) at κ=0.8 Interest Rate x(t) at κ=0.9 Interest Rate x(t) at κ=0.7

Complex Finance System
Interest Rate at κ=0.9 Investment Demand at κ=0.

impact of critical minimum interest rate with x(t)
ABC derivative at d=0.9 ABC derivative at d=0.7 ABC derivative at d=0.5 ABC derivative at d=0.1 Figure 11. Impact of critical minimum interest rate with x(t).

impact of critical minimum interest rate with z(t)
ABC derivative at d = 0.9 ABC derivative at d = 0.8 ABC derivative at d = 0.6 ABC derivative at d = 0.7 Figure 13. Impact of critical minimum interest rate with z(t).

Conclusions
This paper uses a dynamic chaotic fractional order model with an ABC derivative to conduct the economic system. The basis of this fractional model consists of exponentially decreasing non-singular kernels that appear in the derivation of the ABC. The financial model is presented with theoretical and numerical investigation. This demonstrates the regulation of the economic system's critical minimum interest rate. In order to control the economic system, we are discussing a fractional order financing model. The modified model with ABC derivative shows a good financial system control agreement. The model offers the effect of evaluating numerical results on a critical minimum interest rate. Graphical representation shows the impact on the amount of critical minimum interest rate for variables with time. We can observe κ = 1 revealing more absorbing characteristics by numerical simulation using ABC non-integer order derivative. For interest rate, investment demand and price exponent, the concept of this research provides important results. Therefore, we conclude that the ABC derivative is useful to control and maintain the finance system to overcome the risk factors. The interest rate begins to rise according to initial conditions as investment demand and price exponent begin to fall, which shows the actual macroeconomic behavior of the financial system. It is observed here that the complex chaotic fractional system provides more appropriate and reliable results as compared to time integer parameters for non-integer time-fractional parameters.