# American Institute of Mathematical Sciences

January  2015, 20(1): 93-105. doi: 10.3934/dcdsb.2015.20.93

## Complex dynamics of a forced discretized version of the Mackey-Glass delay differential equation

 1 Math. Dept., Faculty of Science, Damanhour University, Damanhour, Egypt

Received  June 2013 Revised  April 2014 Published  November 2014

In this paper, the chaotic behaviour of a forced discretized version of the Mackey-Glass delay differential equation is considered for different levels of noise intensity. The existence and stability of the equilibria of the skeleton are studied. The modified straight-line stabilization method is used to control chaos. The autocorrelation structure is discussed. Numerical simulations are employed to show the model's complex dynamics by means of the largest Lyapunov exponents, bifurcations, time series diagrams and phase portraits. The effects of noise intensity on its dynamics and the intermittency phenomenon are also discussed via simulation.
Citation: Ahmed Elhassanein. Complex dynamics of a forced discretized version of the Mackey-Glass delay differential equation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 93-105. doi: 10.3934/dcdsb.2015.20.93
##### References:
 [1] I. Bashkirtseva and L. Ryashko, Stochastic sensitivity analysis of noise-induced intermittency and transition to chaos in one-dimensional discrete-time systems, Physica A, 392 (2013), 295-306. doi: 10.1016/j.physa.2012.09.001.  Google Scholar [2] J. Brockwell and A. Davis, Time Series: Theory and Methods, $2^{nd}$ edition, Springer-Verlag, New York, 2006. doi: 10.1007/978-1-4419-0320-4.  Google Scholar [3] J. H. E. Cartwright, Nonlinear stiffness, Lyapunov exponents, and attractor dimension, Phys. Lett. A, 264 (1999), 298-302. doi: 10.1016/S0375-9601(99)00793-8.  Google Scholar [4] S. Chatterjee and M. Yilmaz, Chaos, fractals and statistics, Statist. Sci., 7 (1992), 49-68., Available from: , ().  doi: 10.1214/ss/1177011443.  Google Scholar [5] R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesely Reading, 1989.  Google Scholar [6] J. Du, T. Huang and Z. Sheng, Analysis of decision-making in economic chaos control, Nonlinear Anal. Real World Appl., 10 (2009), 2493-2501. doi: 10.1016/j.nonrwa.2008.05.007.  Google Scholar [7] S. N. Elaydi, An Introduction to Difference Equations, $3^{rd}$ edition, Springer-Verlag, New York, 2005.  Google Scholar [8] A. Elhassanein, Complex dynamics of logistic self-exciting threshold autoregressive model,, J. Comput. Theor. Nanosci., ().   Google Scholar [9] A. Elhassanein, On the control of forced process feedback nonlinear autoregressive model,, J. Comput. Theor. Nanosci., ().   Google Scholar [10] A. Elhassanein, Complex dynamics of a stochastic discrete modified Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting, Computational Ecology and Software, 4 (2014), 116-128., Available from: , (): 2014.   Google Scholar [11] A. Elhassanein, On the theory of continuous time series, Indian J. Pure Appl. Math., 45 (2014), 297-310. doi: 10.1007/s13226-014-0064-9.  Google Scholar [12] A. Elhassanein, Nonparametric spectral analysis on disjoint segments of observations, JAMSI, 7 (2011), 81-96., Available from: , ().   Google Scholar [13] W. A. Fuller, Introduction to Statistical Time Series, John Wiley & Sones, 1996.  Google Scholar [14] J. Gao, J. Hu, W. Tung and Y. Zheng, Multiscale analysis of economic time series by scale-dependent Lyapunov exponent, Quantitative Finance, 13 (2013), 265-274. doi: 10.1080/14697688.2011.580774.  Google Scholar [15] M. A. Ghazal and A. Elhassanein, Dynamics of EXPAR models for high frequency data, Int. J. Appl. Math. Stat., 14 (2009), 88-96., Available from: , ().   Google Scholar [16] M. A. Ghazal and A. Elhassanein, Spectral analysis of time series in joint segments of observations, J. Appl. Math. & Informatics, 26 (2008), 933-943. Available from: http://www.kcam.biz/contents/table_contents_view.php?Len=&idx=818. Google Scholar [17] M. A. Ghazal and A. Elhassanein, Nonparametric spectral analysis of continuous time Series, Bull. Stat. Econ., 1 (2007), 41-52., Available from: , ().   Google Scholar [18] M. A. Ghazal and A. Elhassanein, Periodogram analysis with missing observations, J. Appl. Math. Comput., 22 (2006), 209-222. doi: 10.1007/BF02896472.  Google Scholar [19] D. Gulick, Encounters with Chaos, McGraw Hill, New York, 1992. Google Scholar [20] L. Junges and J. A. C. Gallas, Intricate routes to chaos in the Mackey-Glass delayed feed back system, Physics letters A, 376 (2012), 2109-2116. doi: 10.1016/j.physleta.2012.05.022.  Google Scholar [21] J. L. Kaplan and Y. A. Yorke, A regime observed in a fluid flow model of Lorenz, Comm. Math. Phys., 67 (1979), 93-108., Available from: , ().  doi: 10.1007/BF01221359.  Google Scholar [22] M. Karanasos and C. Kyrtsou, Analyzing the link between stock volatility and volume by a Mackey-Glass GARCH-type model: The case of Korea, Quantitative and Qualitative Analysis in Social Sciences, 5 (2011), 49-69., Available from: , ().   Google Scholar [23] C. Kyrtsou and M. Terraza, Seasonal Mackey-Glass-GARCH process and short-term dynamics, Emp. Econ., 38 (2010), 325-345. doi: 10.1007/s00181-009-0268-8.  Google Scholar [24] C. Kyrtsou, Re-examining the sources of heteroskedasticity: The paradigm of noisy chaotic models, Physica A, 387 (2008), 6785-6789. doi: 10.1016/j.physa.2008.09.008.  Google Scholar [25] C. Kyrtsou, Evidence for neglected linearity in noisy chaotic models, Int. J. Bifurcation Chaos, 15 (2005), 3391-3394. doi: 10.1142/S0218127405013964.  Google Scholar [26] C. Kyrtsou, W. Labys and M. Terraza, Terraza, Noisy chaotic dynamics in commodity markets, Emp. Econ., 29 (2004), 489-502. doi: 10.1007/s00181-003-0180-6.  Google Scholar [27] C. Kyrtsou and M. Terraza, Is it possible to study chaotic and ARCH behaviour jointly? Application of a noisy Mackey-Glass equation with heteroskedastic errors to the Paris stock exchange returns series, Computational Economics, 21 (2003), 257-276. doi: 10.1023/A:1023939610962.  Google Scholar [28] P. S. Landa and M. G. Rosenblum, Modefied Mackey-Glass model of respiration control, Physical Review E, 52 (1995), 36-39. doi: 10.1103/PhysRevE.52.R36.  Google Scholar [29] J. Losson, M. C. Mackey and A. Longtin, Solution multistability in first order nonlinear differential delay equations, Chaos, 3 (1993), 167-176. doi: 10.1063/1.165982.  Google Scholar [30] M. C. Mackey, M. Santill'an and N. Yildirim, Modeling operon dynamics: The trytophan and lactose operation as paradigms, C. R. Biologies, 327 (2004), 211-224. doi: 10.1016/j.crvi.2003.11.009.  Google Scholar [31] M. C. Mackey and L. Glass, Oscillation and chaos in physiological control system, Science, 197 (1977), 287-289. Google Scholar [32] A. E. Matouk and H. N. Agiza, Bifurcations, chaos and synchronization in ADVP circuit with parallel resistor, J. Math. Anal. Appl., 341 (2008), 259-269. doi: 10.1016/j.jmaa.2007.09.067.  Google Scholar [33] A. Matsumoto, Controlling the Cournot-Nash chaos, J. Optim. Theory Appl., 128 (2006), 379-392. doi: 10.1007/s10957-006-9021-z.  Google Scholar [34] R. K. Miller and A. N. Michael, Ordinary Differential Equations, Academic Press, New York, 1982.  Google Scholar [35] I. Ncube, Stability switching and Hopf bifurcation in a multiple-delayed neural network with distributed delay, J. Math. Anal. Appl., 407 (2013), 141-146. doi: 10.1016/j.jmaa.2013.05.021.  Google Scholar [36] D. T. Nguyen, Mackey-Glass equation driven by fractional Brownian motion, Physica A, 391 (2012), 5465-5472. doi: 10.1016/j.physa.2012.06.013.  Google Scholar [37] B. Niu and W. Jiang, Nonresonant Hopf-Hopf bifurcation and a chaotic attractor in neutral functional differential equations, J. Math. Anal. Appl., 398 (2013), 362-371. doi: 10.1016/j.jmaa.2012.08.051.  Google Scholar [38] G. Qi, Z. Chen and Z. Yuan, Adaptive high order differential feedback control for affine nonlinear system, Chaos, Solitons & Fractals, 37 (2008), 308-315. doi: 10.1016/j.chaos.2006.09.027.  Google Scholar [39] H. Xu, G. Wang and S. Chen, Controlling chaos by a modified straight-line stabilization method, The European Physical Journal B, 22 (2001), 65-69. doi: 10.1007/PL00011136.  Google Scholar

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##### References:
 [1] I. Bashkirtseva and L. Ryashko, Stochastic sensitivity analysis of noise-induced intermittency and transition to chaos in one-dimensional discrete-time systems, Physica A, 392 (2013), 295-306. doi: 10.1016/j.physa.2012.09.001.  Google Scholar [2] J. Brockwell and A. Davis, Time Series: Theory and Methods, $2^{nd}$ edition, Springer-Verlag, New York, 2006. doi: 10.1007/978-1-4419-0320-4.  Google Scholar [3] J. H. E. Cartwright, Nonlinear stiffness, Lyapunov exponents, and attractor dimension, Phys. Lett. A, 264 (1999), 298-302. doi: 10.1016/S0375-9601(99)00793-8.  Google Scholar [4] S. Chatterjee and M. Yilmaz, Chaos, fractals and statistics, Statist. Sci., 7 (1992), 49-68., Available from: , ().  doi: 10.1214/ss/1177011443.  Google Scholar [5] R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesely Reading, 1989.  Google Scholar [6] J. Du, T. Huang and Z. Sheng, Analysis of decision-making in economic chaos control, Nonlinear Anal. Real World Appl., 10 (2009), 2493-2501. doi: 10.1016/j.nonrwa.2008.05.007.  Google Scholar [7] S. N. Elaydi, An Introduction to Difference Equations, $3^{rd}$ edition, Springer-Verlag, New York, 2005.  Google Scholar [8] A. Elhassanein, Complex dynamics of logistic self-exciting threshold autoregressive model,, J. Comput. Theor. Nanosci., ().   Google Scholar [9] A. Elhassanein, On the control of forced process feedback nonlinear autoregressive model,, J. Comput. Theor. Nanosci., ().   Google Scholar [10] A. Elhassanein, Complex dynamics of a stochastic discrete modified Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting, Computational Ecology and Software, 4 (2014), 116-128., Available from: , (): 2014.   Google Scholar [11] A. Elhassanein, On the theory of continuous time series, Indian J. Pure Appl. Math., 45 (2014), 297-310. doi: 10.1007/s13226-014-0064-9.  Google Scholar [12] A. Elhassanein, Nonparametric spectral analysis on disjoint segments of observations, JAMSI, 7 (2011), 81-96., Available from: , ().   Google Scholar [13] W. A. Fuller, Introduction to Statistical Time Series, John Wiley & Sones, 1996.  Google Scholar [14] J. Gao, J. Hu, W. Tung and Y. Zheng, Multiscale analysis of economic time series by scale-dependent Lyapunov exponent, Quantitative Finance, 13 (2013), 265-274. doi: 10.1080/14697688.2011.580774.  Google Scholar [15] M. A. Ghazal and A. Elhassanein, Dynamics of EXPAR models for high frequency data, Int. J. Appl. Math. Stat., 14 (2009), 88-96., Available from: , ().   Google Scholar [16] M. A. Ghazal and A. Elhassanein, Spectral analysis of time series in joint segments of observations, J. Appl. Math. & Informatics, 26 (2008), 933-943. Available from: http://www.kcam.biz/contents/table_contents_view.php?Len=&idx=818. Google Scholar [17] M. A. Ghazal and A. Elhassanein, Nonparametric spectral analysis of continuous time Series, Bull. Stat. Econ., 1 (2007), 41-52., Available from: , ().   Google Scholar [18] M. A. Ghazal and A. Elhassanein, Periodogram analysis with missing observations, J. Appl. Math. Comput., 22 (2006), 209-222. doi: 10.1007/BF02896472.  Google Scholar [19] D. Gulick, Encounters with Chaos, McGraw Hill, New York, 1992. Google Scholar [20] L. Junges and J. A. C. Gallas, Intricate routes to chaos in the Mackey-Glass delayed feed back system, Physics letters A, 376 (2012), 2109-2116. doi: 10.1016/j.physleta.2012.05.022.  Google Scholar [21] J. L. Kaplan and Y. A. Yorke, A regime observed in a fluid flow model of Lorenz, Comm. Math. Phys., 67 (1979), 93-108., Available from: , ().  doi: 10.1007/BF01221359.  Google Scholar [22] M. Karanasos and C. Kyrtsou, Analyzing the link between stock volatility and volume by a Mackey-Glass GARCH-type model: The case of Korea, Quantitative and Qualitative Analysis in Social Sciences, 5 (2011), 49-69., Available from: , ().   Google Scholar [23] C. Kyrtsou and M. Terraza, Seasonal Mackey-Glass-GARCH process and short-term dynamics, Emp. Econ., 38 (2010), 325-345. doi: 10.1007/s00181-009-0268-8.  Google Scholar [24] C. Kyrtsou, Re-examining the sources of heteroskedasticity: The paradigm of noisy chaotic models, Physica A, 387 (2008), 6785-6789. doi: 10.1016/j.physa.2008.09.008.  Google Scholar [25] C. Kyrtsou, Evidence for neglected linearity in noisy chaotic models, Int. J. Bifurcation Chaos, 15 (2005), 3391-3394. doi: 10.1142/S0218127405013964.  Google Scholar [26] C. Kyrtsou, W. Labys and M. Terraza, Terraza, Noisy chaotic dynamics in commodity markets, Emp. Econ., 29 (2004), 489-502. doi: 10.1007/s00181-003-0180-6.  Google Scholar [27] C. Kyrtsou and M. Terraza, Is it possible to study chaotic and ARCH behaviour jointly? Application of a noisy Mackey-Glass equation with heteroskedastic errors to the Paris stock exchange returns series, Computational Economics, 21 (2003), 257-276. doi: 10.1023/A:1023939610962.  Google Scholar [28] P. S. Landa and M. G. Rosenblum, Modefied Mackey-Glass model of respiration control, Physical Review E, 52 (1995), 36-39. doi: 10.1103/PhysRevE.52.R36.  Google Scholar [29] J. Losson, M. C. Mackey and A. Longtin, Solution multistability in first order nonlinear differential delay equations, Chaos, 3 (1993), 167-176. doi: 10.1063/1.165982.  Google Scholar [30] M. C. Mackey, M. Santill'an and N. Yildirim, Modeling operon dynamics: The trytophan and lactose operation as paradigms, C. R. Biologies, 327 (2004), 211-224. doi: 10.1016/j.crvi.2003.11.009.  Google Scholar [31] M. C. Mackey and L. Glass, Oscillation and chaos in physiological control system, Science, 197 (1977), 287-289. Google Scholar [32] A. E. Matouk and H. N. Agiza, Bifurcations, chaos and synchronization in ADVP circuit with parallel resistor, J. Math. Anal. Appl., 341 (2008), 259-269. doi: 10.1016/j.jmaa.2007.09.067.  Google Scholar [33] A. Matsumoto, Controlling the Cournot-Nash chaos, J. Optim. Theory Appl., 128 (2006), 379-392. doi: 10.1007/s10957-006-9021-z.  Google Scholar [34] R. K. Miller and A. N. Michael, Ordinary Differential Equations, Academic Press, New York, 1982.  Google Scholar [35] I. Ncube, Stability switching and Hopf bifurcation in a multiple-delayed neural network with distributed delay, J. Math. Anal. Appl., 407 (2013), 141-146. doi: 10.1016/j.jmaa.2013.05.021.  Google Scholar [36] D. T. Nguyen, Mackey-Glass equation driven by fractional Brownian motion, Physica A, 391 (2012), 5465-5472. doi: 10.1016/j.physa.2012.06.013.  Google Scholar [37] B. Niu and W. Jiang, Nonresonant Hopf-Hopf bifurcation and a chaotic attractor in neutral functional differential equations, J. Math. Anal. Appl., 398 (2013), 362-371. doi: 10.1016/j.jmaa.2012.08.051.  Google Scholar [38] G. Qi, Z. Chen and Z. Yuan, Adaptive high order differential feedback control for affine nonlinear system, Chaos, Solitons & Fractals, 37 (2008), 308-315. doi: 10.1016/j.chaos.2006.09.027.  Google Scholar [39] H. Xu, G. Wang and S. Chen, Controlling chaos by a modified straight-line stabilization method, The European Physical Journal B, 22 (2001), 65-69. doi: 10.1007/PL00011136.  Google Scholar
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