Partial chaos suppression in a fractional order macroeconomic model

This work investigates the possibility of suppressing chaos in a fractional-nonlinear macroeconomic dynamic model. The system generalizes a model recently reported in the literature in which chaos is strongly present. This description involves the inclusion of the public sector deficit and its coupling with other variables. The system is simulated for integer and non-integer orders that produce a complex dynamics. The time histories and the phase diagrams are presented. The main contribution of this work refers to the adoption of the largest Lyapunov exponent (LLE) criteria based on Wolf's algorithm. This approach improves the response of the system, suppressing, at least partially, the strong presence of chaos reported in previous studies.

[1]  G. Hondroyiannis,et al.  Estimation of Parameters in the Presence of Model Misspecification and Measurement Error , 2010 .

[2]  M. Rivero,et al.  Fractional calculus: A survey of useful formulas , 2013, The European Physical Journal Special Topics.

[3]  Enrico Scalas,et al.  The application of continuous-time random walks in finance and economics , 2006 .

[4]  M. Rosenstein,et al.  A practical method for calculating largest Lyapunov exponents from small data sets , 1993 .

[5]  S. A. David,et al.  Fractional order calculus: historical apologia, basic concepts and some applications , 2011 .

[6]  J.A. Tenreiro Machado,et al.  Analysis of World Economic Variables Using Multidimensional Scaling , 2014, PloS one.

[7]  J. A. Tenreiro Machado,et al.  Pseudo Phase Plane and Fractional Calculus modeling of western global economic downturn , 2015, Commun. Nonlinear Sci. Numer. Simul..

[8]  Ma Junhai,et al.  Study for the bifurcation topological structure and the global complicated character of a kind of nonlinear finance system (I) , 2001 .

[9]  J. Machado,et al.  A Review of Definitions for Fractional Derivatives and Integral , 2014 .

[10]  K. B. Oldham,et al.  The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order , 1974 .

[11]  R. Gorenflo,et al.  Fractional calculus and continuous-time finance II: the waiting-time distribution , 2000, cond-mat/0006454.

[12]  E. Panas,et al.  Long memory and chaotic models of prices on the London Metal Exchange , 2001 .

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

[14]  William A. Barnett,et al.  A single-blind controlled competition among tests for nonlinearity and chaos , 1997 .

[15]  A. Wolf,et al.  Determining Lyapunov exponents from a time series , 1985 .

[16]  Alfredo Medio,et al.  On lags and chaos in economic dynamic models , 1991 .

[17]  N. Laskin.  Fractional market dynamics , 2000 .

[18]  Wenjuan Wu,et al.  Complex nonlinear dynamics and controlling chaos in a Cournot duopoly economic model , 2015 .

[19]  R. Gorenflo,et al.  Fractional calculus and continuous-time finance , 2000, cond-mat/0001120.

[20]  Changpin Li,et al.  Chaos in Chen's system with a fractional order , 2004 .

[21]  Liang Chen,et al.  Controlling chaos in an economic model , 2007 .

[22]  A. Wolf,et al.  13. Quantifying chaos with Lyapunov exponents , 1986 .

[23]  Abraham C.-L. Chian,et al.  NONLINEAR DYNAMICS AND CHAOS IN MACROECONOMICS , 2000 .

[24]  S. A. David,et al.  Fractional-order in a macroeconomic dynamic model , 2013 .

[25]  Ramazan Gençay,et al.  Testing chaotic dynamics via Lyapunov exponents , 1998 .

[26]  David Hsieh.  Chaos and Nonlinear Dynamics: Application to Financial Markets , 1991 .

[27]  F. Mainardi,et al.  Recent history of fractional calculus , 2011 .

[28]  J. A. Tenreiro Machado,et al.  A fractional perspective to the bond graph modelling of world economies , 2015 .

[29]  Enrico Scalas,et al.  Coupled continuous time random walks in finance , 2006 .

[30]  Enrico Scalas,et al.  Fractional Calculus and Continuous-Time Finance III : the Diffusion Limit , 2001 .

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

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

[33]  Bruce J. West,et al.  Fractional Langevin model of memory in financial time series. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.