On fractional discrete financial system:Bifurcation,chaos,and control

The dynamic analysis of financial systems is a developing field that combines mathematics and economics to understand and explain fluctuations in financial markets.This paper introduces a new three-dimensional(3D)fractional financial map and we dissect its nonlinear dynamics system under commensurate and incommensurate orders.As such,we evaluate when the equilibrium points are stable or unstable at various fractional orders.We use many numerical methods,phase plots in 2D and 3D projections,bifurcation diagrams and the maximum Lyapunov exponent.These techniques reveal that financial maps exhibit chaotic attractor behavior.This study is grounded on the Caputo-like discrete operator,which is specifically influenced by the variance of the commensurate and incommensurate orders.Furthermore,we confirm the presence and measure the complexity of chaos in financial maps by the 0-1 test and the approximate entropy algorithm.Additionally,we offer nonlinear-type controllers to stabilize the fractional financial map.The numerical results of this study are obtained using MATLAB.

Composite Fractional Trapezoidal Rule with Romberg Integration

The aim of this research is to demonstrate a novel scheme for approximating the Riemann-Liouville fractional integral operator.This would be achieved by first establishing a fractional-order version of the 2-point Trapezoidal rule and then by proposing another fractional-order version of the(n+1)-composite Trapezoidal rule.In particular,the so-called divided-difference formula is typically employed to derive the 2-point Trapezoidal rule,which has accordingly been used to derive a more accurate fractional-order formula called the(n+1)-composite Trapezoidal rule.Additionally,in order to increase the accuracy of the proposed approximations by reducing the true errors,we incorporate the so-called Romberg integration,which is an extrapolation formula of the Trapezoidal rule for integration,into our proposed approaches.Several numerical examples are provided and compared with a modern definition of the Riemann-Liouville fractional integral operator to illustrate the efficacy of our scheme.

Hidden attractors in a new fractional-order discrete system: Chaos, complexity, entropy, and control

This paper studies the dynamics of a new fractional-order discrete system based on the Caputo-like difference operator.This is the first study to explore a three-dimensional fractional-order discrete chaotic system without equilibrium.Through phase portrait,bifurcation diagrams,and largest Lyapunov exponents,it is shown that the proposed fractional-order discrete system exhibits a range of different dynamical behaviors.Also,different tests are used to confirm the existence of chaos,such as 0-1 test and C0 complexity.In addition,the quantification of the level of chaos in the new fractional-order discrete system is measured by the approximate entropy technique.Furthermore,based on the fractional linearization method,a one-dimensional controller to stabilize the new system is proposed.Numerical results are presented to validate the findings of the paper.

Attractive Multistep Reproducing Kernel Approach for Solving Stiffness Differential Systems of Ordinary Differential Equations and Some Error Analysis

In this paper,an efficient multi-step scheme is presented based on reproducing kernel Hilbert space(RKHS)theory for solving ordinary stiff differential systems.The solution methodology depends on reproducing kernel functions to obtain analytic solutions in a uniform formfor a rapidly convergent series in the posed Sobolev space.Using the Gram-Schmidt orthogonality process,complete orthogonal essential functions are obtained in a compact field to encompass Fourier series expansion with the help of kernel properties reproduction.Consequently,by applying the standard RKHS method to each subinterval,approximate solutions that converge uniformly to the exact solutions are obtained.For this purpose,several numerical examples are tested to show proposed algorithm’s superiority,simplicity,and efficiency.The gained results indicate that themulti-step RKHSmethod is suitable for solving linear and nonlinear stiffness systems over an extensive duration and giving highly accurate outcomes.

An Efficient Approach for Solving One-Dimensional Fractional Heat Conduction Equation

Several researchers have dealt with the one-dimensional fractional heat conduction equation in the last decades,but as far as we know,no one has investigated such a problem from the perspective of developing suitable fractional-order methods.This has actually motivated us to address this problem by the way of establishing a proper fractional approach that involves employing a combination of a novel fractional difference formula to approximate the Caputo differentiator of orderαcoupled with the modified three-point fractional formula to approximate the Caputo differentiator of order 2α,where 0<α≤1.As a result,the fractional heat conduction equation is then reexpressed numerically using the aforementioned formulas,and by dividing the considered mesh into multiple nodes,a system is generated and algebraically solved with the aid of MATLAB.This would allow us to obtain the desired approximate solution for the problem at hand.

Analytic solutions of the generalized water wave dynamical equations based on time-space symmetric differential operator

It is well known that there is a deep connection between the symmetric and traveling wave solutions.It has been shown that all symmetric waves are traveling waves.In this paper,we establish new analytic solution collections of nonlinear conformable time-fractional water wave dynamical equation in a complex domain.For this purpose,we construct a new definition of a symmetric conformable differential operator(SCDO).The operator has a symmetric representation in the open unit disk.By using SCDO,we generalize a class of water wave dynamical equation type time-space fractional complex Ginzburg-Landau equation.The results show that the obtainable approaches are powerful,dependable and prepared to apply to all classes of complex differential equations.

An approach for approximate solution of fractional-order smoking model with relapse class

In this paper, we develop a fractional-order smoking model by considering relapse class. First, we formulate the model and find the unique positive solution for the proposed model. Then we apply the Griinwald-Letnikov approximation in the place of maintaining a general quadrature formula approach to the Riemann-Liouville integral definition of the fractional derivative. Building on this foundation avoids the need for domain trans-formations, contour integration or involved theory to compute accurate approximate solutions of fractional-order giving up smoking model A comparative study between Griinwald-Letnikov method and Runge-Kutta method is presented in the case of integer-order derivative. Finally, we present the obtained results graphically.

A Note on Rough Parametric Marcinkiewicz Functions

In this note, we obtain sharp Lp estimates of parametric Marcinkiewicz integral operators. Our result resolves a long standing open problem. Also, we present a class of parametric Marcinkiewicz integral operators that are bounded provided that their kernels belong to the sole space L^1(S^n-1).

Dynamics analysis of fractional-order Hopfield neural networks

This paper proposes fractional-order systems for Hopfield Neural Network(HNN).The so-called Predictor Corrector Adams Bashforth Moulton Method(PCABMM)has been implemented for solving such systems.Graphical comparisons between the PCABMM and the Runge-Kutla Method(RKM)solutions for the classical HNN reveal that the proposed technique is one of the powerful tools for handling these systems.To determine all Lyapunov exponents for them,the Benettin-Wolf algorithm has been involved in the PCABMM.leased on such algorithm,the Lyapunov exponents as a function of a given parameter and as another function of the fractional-order have been described,the intermittent chaos for these systems has been explored.A new result related to the Mittag-Leffler stability of some nonlinear Fractional-order Hopfield Neural Network(FoHNN)systems has been shown.Besides,the description and the dynamic analysis of those phenomena have been discussed and verified theoretically and numerically via illustrating the phase portraits and the Lyapunov exponents'diagrams.

Lie symmetry analysis,explicit solutions,and conservation laws of the time-fractional Fisher equation in two-dimensional space

In these analyses,we consider the time-fractional Fisher equation in two-dimensional space.Through the use of the Riemann-Liouville derivative approach,the well-known Lie point symmetries of the utilized equation are derived.Herein,we overturn the fractional fisher model to a fractional differential equation of nonlinear type by considering its Lie point symmetries.The diminutive equation’s derivative is in the Erdélyi-Kober sense,whilst we use the technique of the power series to conclude explicit solutions for the diminutive equations for the first time.The conservation laws for the dominant equation are built using a novel conservation theorem.Several graphical countenances were utilized to award a visual performance of the obtained solutions.Finally,some concluding remarks and future recommendations are utilized.

The Multi-Step Differential Transform Method and Its Application to Determine the Solutions of Non-Linear Oscillators

In this paper,a reliable algorithm based on an adaptation of the standard differential transform method is presented,which is the multi-step differential transform method(MSDTM).The solutions of non-linear oscillators were obtained by MSDTM.Figurative comparisons between the MSDTM and the classical fourthorder Runge-Kutta method(RK4)reveal that the proposed technique is a promising tool to solve non-linear oscillators.