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Complex dynamics of a forced discretized version of the Mackey-Glass delay differential equation
1. | Math. Dept., Faculty of Science, Damanhour University, Damanhour, Egypt |
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. |
[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. |
[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. |
[4] |
S. Chatterjee and M. Yilmaz, Chaos, fractals and statistics, Statist. Sci., 7 (1992), 49-68. Available from: http://www.jstor.org/discover/10.2307/2245990?uid=2&uid=4&sid=21104445119883.
doi: 10.1214/ss/1177011443. |
[5] |
R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesely Reading, 1989. |
[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. |
[7] |
S. N. Elaydi, An Introduction to Difference Equations, $3^{rd}$ edition, Springer-Verlag, New York, 2005. |
[8] |
A. Elhassanein, Complex dynamics of logistic self-exciting threshold autoregressive model, J. Comput. Theor. Nanosci., accepted. |
[9] |
A. Elhassanein, On the control of forced process feedback nonlinear autoregressive model, J. Comput. Theor. Nanosci., accepted. |
[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: http://www.iaees.org/publications/journals/ces/articles/2014-4(2)/dynamics-of-a-stochastic-discrete-modified-Leslie-Gower-model.pdf. |
[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. |
[12] |
A. Elhassanein, Nonparametric spectral analysis on disjoint segments of observations, JAMSI, 7 (2011), 81-96. Available from: http://jamsi.fpv.ucm.sk/docs/v07n01_05_2011/v07_n01_07_ELHASSANEIN.pdf. |
[13] |
W. A. Fuller, Introduction to Statistical Time Series, John Wiley & Sones, 1996. |
[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. |
[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: http://www.ceser.in/ceserp/index.php/ijamas/article/view/165. |
[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. |
[17] |
M. A. Ghazal and A. Elhassanein, Nonparametric spectral analysis of continuous time Series, Bull. Stat. Econ., 1 (2007), 41-52. Available from: http://www.ceserp.com/cp-jour/index.php?journal=bse&page=article&op=view&path[]=483. |
[18] |
M. A. Ghazal and A. Elhassanein, Periodogram analysis with missing observations, J. Appl. Math. Comput., 22 (2006), 209-222.
doi: 10.1007/BF02896472. |
[19] |
D. Gulick, Encounters with Chaos, McGraw Hill, New York, 1992. |
[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. |
[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: http://projecteuclid.org/euclid.cmp/1103905158.
doi: 10.1007/BF01221359. |
[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: http://www.qass.org.uk/2011/paper4.pdf. |
[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. |
[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. |
[25] |
C. Kyrtsou, Evidence for neglected linearity in noisy chaotic models, Int. J. Bifurcation Chaos, 15 (2005), 3391-3394.
doi: 10.1142/S0218127405013964. |
[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. |
[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. |
[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. |
[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. |
[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. |
[31] |
M. C. Mackey and L. Glass, Oscillation and chaos in physiological control system, Science, 197 (1977), 287-289. |
[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. |
[33] |
A. Matsumoto, Controlling the Cournot-Nash chaos, J. Optim. Theory Appl., 128 (2006), 379-392.
doi: 10.1007/s10957-006-9021-z. |
[34] |
R. K. Miller and A. N. Michael, Ordinary Differential Equations, Academic Press, New York, 1982. |
[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. |
[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. |
[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. |
[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. |
[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. |
show all references
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. |
[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. |
[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. |
[4] |
S. Chatterjee and M. Yilmaz, Chaos, fractals and statistics, Statist. Sci., 7 (1992), 49-68. Available from: http://www.jstor.org/discover/10.2307/2245990?uid=2&uid=4&sid=21104445119883.
doi: 10.1214/ss/1177011443. |
[5] |
R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesely Reading, 1989. |
[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. |
[7] |
S. N. Elaydi, An Introduction to Difference Equations, $3^{rd}$ edition, Springer-Verlag, New York, 2005. |
[8] |
A. Elhassanein, Complex dynamics of logistic self-exciting threshold autoregressive model, J. Comput. Theor. Nanosci., accepted. |
[9] |
A. Elhassanein, On the control of forced process feedback nonlinear autoregressive model, J. Comput. Theor. Nanosci., accepted. |
[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: http://www.iaees.org/publications/journals/ces/articles/2014-4(2)/dynamics-of-a-stochastic-discrete-modified-Leslie-Gower-model.pdf. |
[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. |
[12] |
A. Elhassanein, Nonparametric spectral analysis on disjoint segments of observations, JAMSI, 7 (2011), 81-96. Available from: http://jamsi.fpv.ucm.sk/docs/v07n01_05_2011/v07_n01_07_ELHASSANEIN.pdf. |
[13] |
W. A. Fuller, Introduction to Statistical Time Series, John Wiley & Sones, 1996. |
[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. |
[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: http://www.ceser.in/ceserp/index.php/ijamas/article/view/165. |
[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. |
[17] |
M. A. Ghazal and A. Elhassanein, Nonparametric spectral analysis of continuous time Series, Bull. Stat. Econ., 1 (2007), 41-52. Available from: http://www.ceserp.com/cp-jour/index.php?journal=bse&page=article&op=view&path[]=483. |
[18] |
M. A. Ghazal and A. Elhassanein, Periodogram analysis with missing observations, J. Appl. Math. Comput., 22 (2006), 209-222.
doi: 10.1007/BF02896472. |
[19] |
D. Gulick, Encounters with Chaos, McGraw Hill, New York, 1992. |
[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. |
[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: http://projecteuclid.org/euclid.cmp/1103905158.
doi: 10.1007/BF01221359. |
[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: http://www.qass.org.uk/2011/paper4.pdf. |
[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. |
[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. |
[25] |
C. Kyrtsou, Evidence for neglected linearity in noisy chaotic models, Int. J. Bifurcation Chaos, 15 (2005), 3391-3394.
doi: 10.1142/S0218127405013964. |
[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. |
[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. |
[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. |
[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. |
[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. |
[31] |
M. C. Mackey and L. Glass, Oscillation and chaos in physiological control system, Science, 197 (1977), 287-289. |
[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. |
[33] |
A. Matsumoto, Controlling the Cournot-Nash chaos, J. Optim. Theory Appl., 128 (2006), 379-392.
doi: 10.1007/s10957-006-9021-z. |
[34] |
R. K. Miller and A. N. Michael, Ordinary Differential Equations, Academic Press, New York, 1982. |
[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. |
[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. |
[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. |
[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. |
[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. |
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