July  2018, 38(7): 3637-3661. doi: 10.3934/dcds.2018157

Homoclinic and stable periodic solutions for differential delay equations from physiology

1. 

Justus Liebig University, 35392, Arndtstrasse 2, Giessen, Germany

2. 

National Research University Higher School of Economics, St. Petersburg, Russia

Received  November 2017 Revised  January 2018 Published  April 2018

A one-parameter family of Mackey-Glass type differential delay equations is considered. The existence of a homoclinic solution for suitable parameter value is proved. As a consequence, one obtains stable periodic solutions for nearby parameter values. An example of a nonlinear functions is given, for which all sufficient conditions of our theoretical results can be verified numerically. Numerically computed solutions are shown.

Citation: Vera Ignatenko. Homoclinic and stable periodic solutions for differential delay equations from physiology. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3637-3661. doi: 10.3934/dcds.2018157
References:
[1]

P. Bates and C. K. R. T. Jones, Invariant manifolds for semilinear partial differential equations, Dynamics Reported, 2 (1989), 1-38. Google Scholar

[2]

O. Diekmann, S. M. Verduyn Lunel, S. A. van Gils and H. -O. Walther, Delay Equations Functional-, Complex-, and Nonlinear Analysis, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2. Google Scholar

[3]

J. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. Google Scholar

[4]

T. Krisztin, H. -O. Walther and J. Wu, Shape, Smoothness and Invariant Stratification of an Attracting Set for Delayed Monotone Positive Feedback, American Mathematical Society, Providence, Rhode Island, 1999. Google Scholar

[5]

B. Lani-Wayda, Wandering solutions of delay equations with sine-like feedback, Mem. Amer. Math. Soc., 151 (2001), ⅹ+121 pp. doi: 10.1090/memo/0718. Google Scholar

[6]

A. Lasota and M. Wazewska-Czyzewska, Matematyczne problemy dynamiki ukladu krwinek czerwonych, Matematyka Stosowana, 6 (1976), 23-40. Google Scholar

[7]

M. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, New Series, (197) (1977), 286-289.Google Scholar

[8]

H.-O. Walther, Homoclinic and periodic solutions of scalar differential delay equations, Banach Center Publ., 23 (1989), 243-263. Google Scholar

show all references

References:
[1]

P. Bates and C. K. R. T. Jones, Invariant manifolds for semilinear partial differential equations, Dynamics Reported, 2 (1989), 1-38. Google Scholar

[2]

O. Diekmann, S. M. Verduyn Lunel, S. A. van Gils and H. -O. Walther, Delay Equations Functional-, Complex-, and Nonlinear Analysis, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2. Google Scholar

[3]

J. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. Google Scholar

[4]

T. Krisztin, H. -O. Walther and J. Wu, Shape, Smoothness and Invariant Stratification of an Attracting Set for Delayed Monotone Positive Feedback, American Mathematical Society, Providence, Rhode Island, 1999. Google Scholar

[5]

B. Lani-Wayda, Wandering solutions of delay equations with sine-like feedback, Mem. Amer. Math. Soc., 151 (2001), ⅹ+121 pp. doi: 10.1090/memo/0718. Google Scholar

[6]

A. Lasota and M. Wazewska-Czyzewska, Matematyczne problemy dynamiki ukladu krwinek czerwonych, Matematyka Stosowana, 6 (1976), 23-40. Google Scholar

[7]

M. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, New Series, (197) (1977), 286-289.Google Scholar

[8]

H.-O. Walther, Homoclinic and periodic solutions of scalar differential delay equations, Banach Center Publ., 23 (1989), 243-263. Google Scholar

Figure 1.  Functions from class $\Gamma$
Figure 2.  Approximate shape of the solution for $f_{\alpha_{0}}$
Figure 3.  Approximate shape of the solution for $f_{\alpha_1}$
Figure 4.  Invariant cone
Figure 5.  Solution with $\alpha = 0$
Figure 6.  Solution with $\alpha = 0.3649$
Figure 7.  Solution with $\alpha = 0.340435$
Figure 8.  Periodic solution for $\alpha = 0.34182$
[1]

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

[2]

Tarik Mohammed Touaoula. Global stability for a class of functional differential equations (Application to Nicholson's blowflies and Mackey-Glass models). Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4391-4419. doi: 10.3934/dcds.2018191

[3]

Changrong Zhu, Bin Long. The periodic solutions bifurcated from a homoclinic solution for parabolic differential equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3793-3808. doi: 10.3934/dcdsb.2016121

[4]

Benjamin B. Kennedy. A periodic solution with non-simple oscillation for an equation with state-dependent delay and strictly monotonic negative feedback. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 47-66. doi: 10.3934/dcdss.2020003

[5]

Guofu Lu. Nonexistence and short time asymptotic behavior of source-type solution for porous medium equation with convection in one-dimension. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1567-1586. doi: 10.3934/dcdsb.2016011

[6]

Zhijun Zhang. Optimal global asymptotic behavior of the solution to a singular monge-ampère equation. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1129-1145. doi: 10.3934/cpaa.2020053

[7]

Xuecheng Wang. Global solution for the $3D$ quadratic Schrödinger equation of $Q(u, \bar{u}$) type. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 5037-5048. doi: 10.3934/dcds.2017217

[8]

P. Dormayer, A. F. Ivanov. Symmetric periodic solutions of a delay differential equation. Conference Publications, 1998, 1998 (Special) : 220-230. doi: 10.3934/proc.1998.1998.220

[9]

Cong He, Hongjun Yu. Large time behavior of the solution to the Landau Equation with specular reflective boundary condition. Kinetic & Related Models, 2013, 6 (3) : 601-623. doi: 10.3934/krm.2013.6.601

[10]

Bhargav Kumar Kakumani, Suman Kumar Tumuluri. Asymptotic behavior of the solution of a diffusion equation with nonlocal boundary conditions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 407-419. doi: 10.3934/dcdsb.2017019

[11]

Federico Bassetti, Lucia Ladelli. Large deviations for the solution of a Kac-type kinetic equation. Kinetic & Related Models, 2013, 6 (2) : 245-268. doi: 10.3934/krm.2013.6.245

[12]

T. Diogo, P. Lima, M. Rebelo. Numerical solution of a nonlinear Abel type Volterra integral equation. Communications on Pure & Applied Analysis, 2006, 5 (2) : 277-288. doi: 10.3934/cpaa.2006.5.277

[13]

Defei Zhang, Ping He. Functional solution about stochastic differential equation driven by $G$-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 281-293. doi: 10.3934/dcdsb.2015.20.281

[14]

Shaoyong Lai, Yong Hong Wu, Xu Yang. The global solution of an initial boundary value problem for the damped Boussinesq equation. Communications on Pure & Applied Analysis, 2004, 3 (2) : 319-328. doi: 10.3934/cpaa.2004.3.319

[15]

Tomás Caraballo, Marta Herrera-Cobos, Pedro Marín-Rubio. Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1801-1816. doi: 10.3934/dcdsb.2017107

[16]

Maicon Sônego. Stable solution induced by domain geometry in the heat equation with nonlinear boundary conditions on surfaces of revolution. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5981-5988. doi: 10.3934/dcdsb.2019116

[17]

Ábel Garab. Unique periodic orbits of a delay differential equation with piecewise linear feedback function. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2369-2387. doi: 10.3934/dcds.2013.33.2369

[18]

Út V. Lê. Regularity of the solution of a nonlinear wave equation. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1099-1115. doi: 10.3934/cpaa.2010.9.1099

[19]

Giuseppe Maria Coclite, Lorenzo di Ruvo. A note on the convergence of the solution of the Novikov equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2865-2899. doi: 10.3934/dcdsb.2018290

[20]

Bao-Zhu Guo, Li-Ming Cai. A note for the global stability of a delay differential equation of hepatitis B virus infection. Mathematical Biosciences & Engineering, 2011, 8 (3) : 689-694. doi: 10.3934/mbe.2011.8.689

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (89)
  • HTML views (180)
  • Cited by (0)

Other articles
by authors

[Back to Top]