Figure 1.
Functions from class $\Gamma$
-
Previous Article
Regular measurable dynamics for reaction-diffusion equations on narrow domains with rough noise
- DCDS Home
- This Issue
-
Next Article
Global dynamics in a two-species chemotaxis-competition system with two signals
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 |
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.
Keywords:
Differential delay equation,
homoclinic solution,
stable periodic solution,
global behavior,
Mackey-Glass type equation.
Mathematics Subject Classification:
Primary: 34K18, 34K13; Secondary: 37N25.
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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [6] |
A. Lasota and M. Wazewska-Czyzewska,
Matematyczne problemy dynamiki ukladu krwinek czerwonych, Matematyka Stosowana, 6 (1976), 23-40.
|
| [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.
|
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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [6] |
A. Lasota and M. Wazewska-Czyzewska,
Matematyczne problemy dynamiki ukladu krwinek czerwonych, Matematyka Stosowana, 6 (1976), 23-40.
|
| [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.
|
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, 2020, 13 (1) : 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] |
Jing Li, Boling Guo, Lan Zeng, Yitong Pei. Global weak solution and smooth solution of the periodic initial value problem for the generalized Landau-Lifshitz-Bloch equation in high dimensions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019230 |
| [8] |
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 |
| [9] |
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 |
| [10] |
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 |
| [11] |
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 |
| [12] |
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 |
| [13] |
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 |
| [14] |
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 |
| [15] |
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 |
| [16] |
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 |
| [17] |
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 |
| [18] |
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 |
| [19] |
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 |
| [20] |
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 |
2018 Impact Factor: 1.143
Tools
Metrics
Other articles
by authors
[Back to Top]