A new financial chaotic model in Atangana-Baleanu stochastic fractional differential equations

Abstract We formulate and analyze a new financial chaotic model in fractional stochastic differential equation in Atangana-Baleanu operator. The model is constructed initially in integer case and then the application of fractional and stochastic approach are used for the model generalization. We study with care some of the mathematical results for the model in integer case by identifying the suitable values of parameters and initial values for the state variables for which the model behaves chaotic. Further, using the concept of stochastic differential equation, the model is further extended to the stochastic fractional differential equations. The equilibriums possibly the model posses are shown and analyzed. Lyapanove exponent are investigated for the model. We give a new method that is introduced recently in literature to find the model solution numerically for the case of fractional stochastic differential equation. We obtained many interesting graphical results for the model by considering different scenarios.

[1]  Abdon Atangana,et al.  Modeling the dynamics of novel coronavirus (2019-nCov) with fractional derivative , 2020, Alexandria Engineering Journal.

[2]  Yong Zhou,et al.  Nonlocal Cauchy problem for fractional evolution equations , 2010 .

[3]  Yong Zhou,et al.  Existence of mild solutions for fractional neutral evolution equations , 2010, Comput. Math. Appl..

[4]  Abdon Atangana,et al.  Can transfer function and Bode diagram be obtained from Sumudu transform , 2020 .

[5]  A. Akgül.  A novel method for a fractional derivative with non-local and non-singular kernel , 2018, Chaos, Solitons & Fractals.

[6]  Analysis of New Trends of Fractional Differential Equations , 2020 .

[7]  D. G. Prakasha,et al.  A fractional model for propagation of classical optical solitons by using nonsingular derivative , 2020 .

[8]  I. Podlubny.  Fractional differential equations , 1998 .

[9]  B. Ghanbari,et al.  Turing-Hopf bifurcation in a diffusive mussel-algae model with time-fractional-order derivative , 2020 .

[10]  Yong Zhou,et al.  EXISTENCE AND UNIQUENESS FOR FRACTIONAL NEUTRAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY , 2009 .

[11]  C. F. Li,et al.  Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations , 2010, Comput. Math. Appl..

[12]  J. F. Gómez‐Aguilar,et al.  Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena , 2018 .

[13]  B. Ghanbari,et al.  Dynamical behavior of two predators–one prey model with generalized functional response and time-fractional derivative , 2021, Advances in Difference Equations.

[14]  Dumitru Baleanu,et al.  Two analytical methods for time-fractional nonlinear coupled Boussinesq–Burger’s equations arise in propagation of shallow water waves , 2016 .

[15]  Salih Djilali,et al.  A Heroin Epidemic Model: Very General Non Linear Incidence, Treat-Age, and Global Stability , 2017 .

[16]  M. Khan,et al.  Modeling the dynamics of hepatitis E via the Caputo–Fabrizio derivative , 2019, Mathematical Modelling of Natural Phenomena.

[17]  B. Ghanbari,et al.  Mathematical and numerical analysis of a three‐species predator‐prey model with herd behavior and time fractional‐order derivative , 2019, Mathematical Methods in the Applied Sciences.

[18]  Seda İğret Araz,et al.  Nonlinear equations with global differential and integral operators: Existence, uniqueness with application to epidemiology , 2020 .

[19]  Abdon Atangana,et al.  Analysis of fractal fractional differential equations , 2020 .

[20]  A. Tridane,et al.  Age-Structured Modeling of COVID-19 Epidemic in the USA, UAE and Algeria , 2020, Alexandria Engineering Journal.

[21]  Sunil Kumar,et al.  New analytical method for gas dynamics equation arising in shock fronts , 2014, Comput. Phys. Commun..

[22]  Behzad Ghanbari,et al.  Mathematical analysis of a fractional-order predator-prey model with prey social behavior and infection developed in predator population , 2020 .

[23]  Abdon Atangana,et al.  Modelling and analysis of fractal-fractional partial differential equations: Application to reaction-diffusion model , 2020 .

[24]  Saif Ullah,et al.  A fractional model for the dynamics of TB virus , 2018, Chaos, Solitons & Fractals.

[25]  Phatiphat Thounthong,et al.  A comparison study of bank data in fractional calculus , 2019, Chaos, Solitons & Fractals.

[26]  Abdon Atangana,et al.  Fractional derivatives with no-index law property: Application to chaos and statistics , 2018, Chaos, Solitons & Fractals.

[27]  Lakshmi Narayan Guin,et al.  Impact of predation in the spread of an infectious disease with time fractional derivative and social behavior , 2021, Int. J. Model. Simul. Sci. Comput..

[28]  Abraham C.-L. Chian,et al.  Attractor merging crisis in chaotic business cycles , 2005 .

[29]  Luciano Fanti,et al.  Chaotic business cycles and fiscal policy: An IS-LM model with distributed tax collection lags , 2007 .

[30]  Mario Sportelli,et al.  A dynamic IS-LM model with delayed taxation revenues , 2005 .

[31]  A. Atangana,et al.  On solutions of fractal fractional differential equations , 2021, Discrete & Continuous Dynamical Systems - S.

[32]  Esra Karatas Akgül,et al.  Solutions of the linear and nonlinear differential equations within the generalized fractional derivatives. , 2019, Chaos.

[33]  Kazuyuki Sasakura,et al.  On the dynamic behavior of Schinasi's business cycle model , 1994 .

[34]  Abraham C.-L. Chian,et al.  Complex economic dynamics: Chaotic saddle, crisis and intermittency , 2006 .

[35]  F. Mainardi.  Fractional Relaxation-Oscillation and Fractional Diffusion-Wave Phenomena , 1996 .

[36]  M. Khan,et al.  Fractional investigations of zoonotic visceral leishmaniasis disease with singular and non-singular kernel , 2019, The European Physical Journal Plus.

[37]  A. Atangana.  Extension of rate of change concept: From local to nonlocal operators with applications , 2020 .

[38]  Sara Dadras,et al.  Control of a fractional-order economical system via sliding mode , 2010 .

[39]  R. Agarwal,et al.  A study of fractional Lotka‐Volterra population model using Haar wavelet and Adams‐Bashforth‐Moulton methods , 2020, Mathematical Methods in the Applied Sciences.

[40]  B. Ghanbari,et al.  The influence of an infectious disease on a prey-predator model equipped with a fractional-order derivative , 2021 .

[41]  Yuming Chen,et al.  Global Dynamics of an SEIR Model with Two Age Structures and a Nonlinear Incidence , 2020, Acta Applicandae Mathematicae.

[42]  Wei-Ching Chen,et al.  Nonlinear dynamics and chaos in a fractional-order financial system , 2008 .

[43]  M. Khan,et al.  Modeling the transmission of dengue infection through fractional derivatives , 2019, Chaos, Solitons & Fractals.

[44]  E. Lorenz.  Deterministic nonperiodic flow , 1963 .

[45]  Yong Zhou,et al.  Existence and uniqueness for p-type fractional neutral differential equations☆ , 2009 .

[46]  B. Samet,et al.  A wavelet based numerical scheme for fractional order SEIR epidemic of measles by using Genocchi polynomials , 2020, Numerical Methods for Partial Differential Equations.

[47]  B. Ghanbari,et al.  Backward, Hopf bifurcation in a heroin epidemic model with treat age⋆ , 2020, Int. J. Model. Simul. Sci. Comput..

[48]  Muhammad Altaf Khan,et al.  The dynamics of a new chaotic system through the Caputo–Fabrizio and Atanagan–Baleanu fractional operators , 2019, Advances in Mechanical Engineering.

[49]  M. Jleli,et al.  A fractional model for population dynamics of two interacting species by using spectral and Hermite wavelets methods , 2020, Numerical Methods for Partial Differential Equations.

[50]  K. M. Owolabi,et al.  Modeling and simulation of nonlinear dynamical system in the frame of nonlocal and non-singular derivatives , 2019, Chaos, Solitons & Fractals.

[51]  Yong Zhou,et al.  A class of fractional evolution equations and optimal controls , 2011 .