﻿ 造血系统中的周期解分岔及反馈控制 Bifurcation of Periodic Solution of a Hematopological System and Feedback Control

Vol.05 No.03(2016), Article ID:18458,7 pages
10.12677/AAM.2016.53059

Bifurcation of Periodic Solution of a Hematopological System and Feedback Control

Suqi Ma

Mathematical Department, Chinese Agricultural University, Beijing

Received: Aug. 11th, 2016; accepted: Aug. 25th, 2016; published: Aug. 31st, 2016

ABSTRACT

The oscillating phenomenon of a hematological cell model is investigated. Based on effective G-SCF administration, the mathematical blood cell model is described as DDEs with multi-delays. Commonly, in blood cell models, the intrinsic mechanism is composed of triggering differentiating mechanism and maturation mechanism, and hematoplogical cell model differentiating into three types of necessary blood cells in the body. By applying numerical software DDE-Biftool, the bifurcation of periodic solutions with long period are derived, and the period doubling bifurcation of periodic solutions is found at critical values. Chaotic solutions also appear afterwards period- doubling bifurcation, and the stabilization of dynamical chaos to the expected periodic solution is finished successfully by applying Pyragas feedback control method.

Keywords:Hematological Cell Model, Periodic Solution, Periodic Doubling Bifurcation, Delay

1. 引言

2. 造血系统的两房数学模型

(1)

3. 数值模拟

3.1. 造血系统的吸引子及混沌

3.2. 反馈控制系统

(2)

(a) Periodic solutions of type I(b) periodic solutions of tyoe II (c) periodic solutions of type III(a) I型周期解 (b) II型周期解 (c) III型周期解

Figure 1. Three types of different periodic solutions of system (1),

(a) (b)

Figure 2. Continuation extending of periodic solutions with varying parameter. (a) Two branches of continuation periodic solutions respectively obtained by extending periodic solution of type I and type II; (b) One branch of periodic solutions obtained by continuation extending of type III solution

(a) (b)

Figure 3. The whole solutions of continuation of two types of periodic solution, respectively, manifested by (a) periodic solution branch 1; (b) periodic solution 2

(a) (b)(c) (d)

Figure 4. Phase portraits of stable periodic solutions of continuation branch of system (1) by varying parameter. (a); (b); (c); (d)

(3)

，选取，从同一初始函数开始，得到解支２上的周期吸引子，其控制效果见图6，但在这一小范围内产生了如图6(a)所示的混沌解。图6(b)为时的吸引子解。进一步采用相空间压缩法，让

(4)

(a) (b)

Figure 5. Chaotic control to stable periodic orbits due to negative feedback control. (a) Chaos with; (b) Poincare section of chaotic control

(a) (b)(c) (d)

Figure 6. Phase space contraction to expected stable periodic orbits. (a) Poincare section obtained by controlling method which formulated by (3); (b) Phase portrait onto Poincare section (a) by choosing; (c) Poincare section obtained by controlling method which formulated by (3) and (4); (b) Phase portrait onto Poincare section (b) by choosing

4. 结论

Bifurcation of Periodic Solution of a Hematopological System and Feedback Control[J]. 应用数学进展, 2016, 05(03): 487-493. http://dx.doi.org/10.12677/AAM.2016.53059

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