×

zbMATH — the first resource for mathematics

Analysis of nonlinear dynamics and chaos in a fractional order financial system with time delay. (English) Zbl 1228.35253
Summary: A delayed fractional order financial system is proposed and the complex dynamical behaviors of such a system are discussed by numerical simulations. A great variety of interesting dynamical behaviors of such a system including single-periodic, multiple-periodic, and chaotic motions are displayed. In particular, the effect of time delay on the chaotic behavior is investigated, it is found that an approximate time delay can enhance or suppress the emergence of chaos. Meanwhile, corresponding to different values of delay, the lowest orders for chaos to exist in the delayed fractional order financial systems are determined, respectively.

MSC:
35Q91 PDEs in connection with game theory, economics, social and behavioral sciences
35R11 Fractional partial differential equations
91G80 Financial applications of other theories
37N40 Dynamical systems in optimization and economics
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Shone, R., Economic dynamics, (2002), Cambridge: Cambridge University Press · Zbl 1066.91072
[2] Chian, A.L.; Zorotto, F.A.; Rempel, E.L.; Rogers, C., Attractor merging crisis in chaotic business cycles, Chaos, solitons and fractals, 24, 869-875, (2005) · Zbl 1081.37058
[3] Chian, A.L.; Rempel, E.L.; Rogers, C., Complex economic dynamics: chaotic saddle, crisis and intermittency, Chaos, solitons and fractals, 29, 1194-1218, (2006) · Zbl 1142.91652
[4] Sasakura, K., On the dynamic behavior of schinas’s business cycle model, Journal of macroeconomics, 16, 3, 423-424, (1994)
[5] Cesare, L.D.; Sportelli, M., A dynamic IS-LM model with delayed taxation revenues, Chaos, solitons and fractals, 25, 233-244, (2005) · Zbl 1110.91020
[6] Fanti, L.; Manfredi, P., Chaotic business cycles and fiscal policy: an IS-LM model with distributed tax collection lags, Chaos, solitons and fractals, 32, 736-744, (2007) · Zbl 1133.91482
[7] Lorenz, H.W., Nonlinear economic dynamics and chaotic motion, (1993), Springer New York
[8] Lorenz, H.W.; Nusse, H.E., Chaotic attractors, chaotic saddles, and fractal basin boundaries: goodwin’s nonlinear accelerator model reconsidered, Chaos, solitons and fractals, 13, 957-965, (2002) · Zbl 1016.37052
[9] Zhou, Y.; Jiao, F.; Li, J., Existence and uniqueness for fractional neutral differential equations with infinite delay, Nonlinear analysis, 71, 3249-3256, (2009) · Zbl 1177.34084
[10] Zhou, Y.; Jiao, F.; Li, J., Existence and uniqueness for p-type fractional neutral differential equations, Nonlinear analysis, 71, 2724-2733, (2009) · Zbl 1175.34082
[11] Zhou, Y.; Jiao, F., Existence of mild solutions for fractional neutral evolution equations, Computers and mathematics with applications, 59, 1063-1077, (2010) · Zbl 1189.34154
[12] Li, C.F.; Luo, X.N.; Zhou, Y., Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations, Computers and mathematics with applications, 59, 1363-1375, (2010) · Zbl 1189.34014
[13] Zhou, Y.; Jiao, F., Nonlocal Cauchy problem for fractional evolution equations, Nonlinear analysis: real world applications, 11, 4465-4475, (2010) · Zbl 1260.34017
[14] Wang, J.R.; Zhou, Y., A class of fractional evolution equations and optimal controls, Nonlinear analysis: real world applications, 12, 262-272, (2011) · Zbl 1214.34010
[15] Podlubny, I., Fractional differential equations, (1999), Academic New York · Zbl 0918.34010
[16] Hilfer, R., Applications of fractional calculus in physics, (2001), World Scientific Hackensack, NJ · Zbl 0998.26002
[17] Butzer, P.L.; Westphal, U., An introduction to fractional calculus, (2000), World Scientific Singapore · Zbl 0987.26005
[18] Mainardi, F., Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos, solitons and fractals, 7, 1461-1477, (1996) · Zbl 1080.26505
[19] Chen, W.C., Nonlinear dynamics and chaos in a fractional-order financial system, Chaos, solitons and fractals, 36, 1305-1314, (2008)
[20] Dadras, S.; Momeni, H.R., Control of a fractional-order economical system via sliding mode, Physica A, 389, 2434-2442, (2010)
[21] M. Salah, N. Hamri, J. Wang, Chaos control of a fractional-order financial system, Mathematical Problem in Engineering (2010), doi:10.1155/2010/270646. · Zbl 1195.91185
[22] Kalecki, M., A macrodynamic theory of business cycles, Econometrica, 3, 327-344, (1935) · JFM 61.1326.06
[23] Cesare, L.D.; Sportelli, M., A dynamic IS-LM model with delayed taxation revenues, Chaos, solitons and fractals, 25, 233-244, (2005) · Zbl 1110.91020
[24] Zhou, L.J.; Li, Y.Q., A generalized dynamic IS-LM model with delayed time in investment processes, Applied mathematics and computation, 196, 774-781, (2008) · Zbl 1136.91572
[25] Kaddar, A.; Alaoui, H.T., On the dynamic behavior of a delayed IS-LM business cycle model, Applied mathematical sciences, 2, 1529-1539, (2008) · Zbl 1154.91561
[26] Matsumotoa, A.; Szidarovszky, F., Delayed dynamics in heterogeneous competition with product differentiation, Nonlinear analysis: realworld applications, 11, 601-611, (2010) · Zbl 1187.91081
[27] Takeuchi, Y.; Yamamura, T., Stability analysis of the kaldor model with time delays: monetary policy and government budget constraint, Nonlinear analysis: real world applications, 5, 277-308, (2004) · Zbl 1087.91042
[28] Szydlowski, M.; Krawiec, A.; Tobola, J., Nonlinear oscillations in business cycle model with time lags, Chaos, solitons and fractals, 12, 505-517, (2001) · Zbl 1036.91038
[29] Diethelm, K.; Ford, N.J.; Freed, A.D., A predictor – corrector approach for the numerical solution of fractional differential equations, Nonlinear dynamics, 29, 3-22, (2002) · Zbl 1009.65049
[30] Tavazoei, M.S.; Haeri, M., A necessary condition for double scroll attractor existence in fractional order systems, Physics letters A, 367, 102-113, (2007) · Zbl 1209.37037
[31] D. Matignon, Stability results for fractional differential equations with applications to control processing, In: Computational Engineering in Systems and Application Multiconference, vol. 2, in: IMACS, IEEE-SMC Proceedings, Lille, France, July 1996, pp. 963-968.
[32] Deng, W.; Li, C.; Lü, J., Stability analysis of linear fractional differentional differential system with multiple time delays, Nonlinear dynamics, 48, 409-416, (2007) · Zbl 1185.34115
[33] Tavazoei, M.S.; Haeri, M., Chaotic attractors in incommensurate fractional order systems, Physica D, 237, 2628-2637, (2008) · Zbl 1157.26310
[34] Ma, J.H.; Chen, Y.S., Study for the bifurcation topological structure and the global complicated character of a kind of nonlinear finance system, I, Applied mathematics and mechanics, 22, 11, 1240-1251, (2001) · Zbl 1001.91501
[35] Ma, J.H.; Chen, Y.S., Study for the bifurcation topological structure and the global complicated character of a kind of nonlinear finance system, II, Applied mathematics and mechanics, 22, 12, 1375-1382, (2001) · Zbl 1143.91341
[36] Wolf, A.; Swift, J.B.; Swinney, H.L.; Vastano, J.A., Determining Lyapunov exponents from a time series, Physica D, 16, 285-317, (1985) · Zbl 0585.58037
[37] Kim, H.S.; Eykholt, R.; Salas, J.D., Nonlinear dynamics, delay times, and embedding windows, Physica D, 127, 48-60, (1999) · Zbl 0941.37054
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.