Discrete-time modeling of a hereditary impulsive feedback system

The paper deals with a broad class of hybrid systems where a linear hereditary plant with a cascade structure operates under impulsive feedback. The plant incorporates a distinct infinite-dimensional block that might be a pointwise or distributed time delay. Mathematical models that belong to this class of systems appear in mathematical biology and computational medicine. A discrete time (Poincaré) map is constructed to capture the system dynamics and investigate its periodic solutions. Simulation results indicate that the effects on the system dynamics incurred by distributed delays are quite similar to those previously observed for pointwise delays. Generally, it appears that the complexity of the nonlinear dynamics does not increase with an increasing delay value.

[1]  Wassim M. Haddad,et al.  Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control , 2006 .

[2]  Alexander Medvedev,et al.  Modeling of testosterone regulation by pulse-modulated feedback: An experimental data study , 2013 .

[3]  Alexander Medvedev,et al.  Periodic Modes and Bistability in an Impulsive Goodwin Oscillator with Large Delay , 2014 .

[4]  Alexander Medvedev,et al.  Periodical Solutions in a Pulse-Modulated Model of Endocrine Regulation With Time-Delay , 2014, IEEE Transactions on Automatic Control.

[5]  Alexander Medvedev,et al.  Bifurcation phenomena in an impulsive model of non-basal testosterone regulation. , 2012, Chaos.

[6]  Leon S. Farhy,et al.  Chapter 14 Biomathematical Modeling of Pulsatile Hormone Secretion , 2009 .

[7]  Alexander Medvedev,et al.  Mathematical model of non-basal testosterone regulation in the male by pulse modulated feedback , 2009, Autom..

[8]  D M Keenan,et al.  Hypothesis testing of the aging male gonadal axis via a biomathematical construct. , 2001, American journal of physiology. Regulatory, integrative and comparative physiology.

[9]  M. Husain,et al.  A model for the control of testosterone secretion. , 1986, Journal of theoretical biology.

[10]  Arkadiĭ Khaĭmovich Gelig,et al.  Stability and Oscillations of Nonlinear Pulse-Modulated Systems , 1998 .

[11]  S. Ruan,et al.  On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testosterone secretion. , 2001, IMA journal of mathematics applied in medicine and biology.

[12]  A. Samoilenko,et al.  Impulsive differential equations , 1995 .

[13]  Eduardo D Sontag,et al.  On the stability of a model of testosterone dynamics , 2004, Journal of mathematical biology.

[14]  Daniel M. Keenan,et al.  A Stochastic Biomathematical Model of the Male Reproductive Hormone System , 2000, SIAM J. Appl. Math..

[15]  Johannes D Veldhuis,et al.  Modelling the Nonlinear Time Dynamics of Multidimensional Hormonal Systems , 2012, Journal of time series analysis.

[16]  A. Medvedev.  Continuous least-squares observers with applications , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[17]  Tao Yang,et al.  In: Impulsive control theory , 2001 .

[18]  William R. Smith.  Hypothalamic regulation of pituitary secretion of luteinizing hormone—II feedback control of gonadotropin secretion , 1980 .

[19]  D. M. Keenan,et al.  A biomathematical model of time-delayed feedback in the human male hypothalamic-pituitary-Leydig cell axis. , 1998, American journal of physiology. Endocrinology and metabolism.

[20]  A. B. Roy,et al.  Stability and oscillations of a negative feedback delay model for the control of testosterone secretion. , 1994, Bio Systems.

[21]  Alexander Medvedev,et al.  An impulse-to-impulse discrete-time mapping for a time-delay impulsive system , 2014, Autom..

[22]  V. Lakshmikantham,et al.  Theory of Impulsive Differential Equations , 1989, Series in Modern Applied Mathematics.

[23]  B. Goodwin.  Oscillatory behavior in enzymatic control processes. , 1965, Advances in enzyme regulation.

[24]  W R Smith,et al.  Qualitative mathematical models of endocrine systems. , 1983, The American journal of physiology.

[25]  John R. Terry,et al.  Encoding and Decoding Mechanisms of Pulsatile Hormone Secretion , 2010, Journal of neuroendocrinology.

[26]  D. D. Perlmutter,et al.  Stability of time‐delay systems , 1972 .

[27]  N. Rashevsky,et al.  Mathematical biology , 1961, Connecticut medicine.

[28]  Leon S Farhy,et al.  Biomathematical modeling of pulsatile hormone secretion: a historical perspective. , 2009, Methods in enzymology.

[29]  B. Mukhopadhyay,et al.  A delayed mathematical model for testosterone secretion with feedback control mechanism , 2004, Int. J. Math. Math. Sci..

[30]  Alexander Medvedev,et al.  Finite-dimensional reducibility of time-delay systems under pulse-modulated feedback , 2013, 52nd IEEE Conference on Decision and Control.

[31]  W. Marsden.  I and J , 2012 .