doi:10.1155/2007/51406 Research Article Dynamics of a Discretization Physiological Control System

We study the dynamics of solutions of discrete
physiological control system obtained by Midpoint rule. It is shown
that a sequence of Hopf bifurcations occurs at the positive
equilibrium as the delay increases and we analyze the stability of
the solution of the discrete system and calculate the direction of
the Hopf bifurcations. The numerical results are presented.


Introduction
Recently, there has been an interest in the so-called dynamical diseases, which correspond to physiological disorders for which a generally stable control system becomes unstable [1][2][3][4][5].Here, the study of the dynamical features of the corresponding model such as equilibria, their local stability characteristics and bifurcation behaviors is extremely useful related to the dynamics of mathematical models of various biological systems and other applications.In this paper, we research an arterial CO 2 control system [1], which may be described by where p is the arterial CO 2 concentration, γ is the CO 2 production rate, τ is the time between oxygenation of blood in the lungs and stimulation of chemoreceptors in the brainstem, v m is the maximum ventilation, θ and n are parameters adjusted to fit experimental observations, and β is a constant.The equation reproduces certain qualitative features of normal and abnormal respiration.
Considering the need of scientific computation and real-time simulation, our interest is focused on the behaviors of discrete dynamics system corresponding to (1.1).Most of this time, it is desirable that a difference equation, which is derived from a differential equation, preserves the dynamical features of the corresponding continuous-time model.That is, the discrete-time model is "dynamically consistent" with the continuous-time model.
In [6], Wulf and Ford show that, if applying Euler forward method to solve the delay differential equation, then the discrete scheme is "dynamically consistent" with the continuous-time model.It means that for all sufficiently small step sizes, the discrete model undergoes a Hopf bifurcation of the same type with the corresponding continuous-time model, and the bifurcation point λ h of the discrete model is O(h) close to the bifurcation point λ * , which corresponds to the continuous-time model.
In this paper, we choose Midpoint rule [7][8][9] to make the discretization for system (1.1), this method can be considered as perturbation of the Euler forward method.With a similar analysis in [6], it is known that the discrete model is "dynamically consistent" with the continuous-time model for all sufficiently small step sizes, and we can expect the bifurcation point λ h of the discrete model is O(h 2 ) close to the bifurcation point λ * of the corresponding continuous-time model due to the Midpoint rule have the convergence of 2-order [7,9].On the other hand, the Midpoint rule is a symplectic method [8,9], it may preserve some important properties of the solutions of the original dynamical system.
The paper is organized as follows.In Section 2, we use the Hopf bifurcation theory of discrete system [10][11][12] to investigate the stability of equilibrium and the existence of the local Hopf bifurcations at the equilibrium.In Section 3, direction and stability of the Hopf bifurcation are established.In Section 4, some numerical simulations are provided to illustrate the results found.At last, roughly, we apply our results to explain the symptom of respiration.

Stability of the positive equilibrium and local Hopf bifurcations
In this section, we will see that, when Midpoint rule is applied to system (1.1), it gives rise to a discrete dynamics system (2.8) and we study the stability of the positive equilibrium and the existence of local Hopf bifurcations of system (2.8), which inherits certain dynamics of system (1.1) [13].
Under transformation p(t) = θx(t), (1.1) becomes where Then there is ẋ(t) = u(t/τ)(1/τ).Inserting them into (2.1),we have X. Ding and H. Su 3 So (2.1) can be rewritten as Thanks to the role of system (1.1) in practice, we only take an interest in the positive equilibrium point of (1.1).Without loss of generality, assume that u is the positive equilibrium point of (2.3), that is, which has a unique positive equilibrium point.
In fact, by defining function F(x) := bx n+1 − ax n − a, from and F(0) = −a < 0, it follows that (2.3) has a unique positive equilibrium point.
If we apply the Midpoint rule to autonomous delay differential equations we could get where h = (1/m)(m ∈ N + ) stands for stepsize, and u k denotes the approximate value to u(kh).Hence, using Midpoint rule (2.7) to (2.3) yields difference equation where F = (F 0 ,F 1 ,...,F m ) T and (2.12) Clearly, the origin is a fixed point of map (2.11), and the linear part of map (2.11) is where whose characteristic equation is (2.15) Lemma 2.1.All roots of (2.15) have modulus less than one for sufficiently small positive τ > 0.
Proof.For τ = 0, (2.15) is equated with The equation has an m-fold root and a simple root λ = 1.
A Hopf bifurcation occurs when two roots of the characteristic equation (2.15) cross the unit circle.We have to find values of τ such that there are roots on the unit circle.The roots on the unit circle are given by e iω , ω ∈ (−π,π].Since we are dealing with complex roots of a real polynomial which occur in complex conjugate pairs, we only need to look for ω ∈ (0,π].For ω ∈ (0,π], e iω is a root of (2.15) if and only if So the values of τ are Because τ is assumed to be real, we get the following relations for ω(∈ (0,π]) and τ: (2.23) Let λ i (τ)=r i (τ)e iωi(τ) be a root of (2.15) near τ = τ i satisfying r i (τ i ) = 1 and ω i (τ i ) = ω i .Then there are ) where [•] denotes the greatest integer function, and we have the following result.

.27)
Substituting this equation into (2.17),we have Since tan(ω/2) is monotonically increasing for ω ∈ (0,π] and (2.24), we know that the smallest positive τ i with roots on unit circle is τ 0 .By Lemmas 2.1 and 2.3, we know that if τ ∈ [0,τ 0 ), then all the roots of (2.15) have modulus less than one; if τ = τ 0 , then all roots of (2.15) except e ±iω0 have modulus less than one; furthermore, by Rouché's theorem (Dieudonné [14, Theorem 9.17.4]), the statement on the number of eigenvalues with modulus more than one as follows.
Spectral properties in Lemma 2.4 immediately lead to stability properties of the zero solution of (2.10), and equivalently, of the positive fixed point u = u * of (2.8).

Direction and stability of the Hopf bifurcation in discrete model
In the previous section, we obtained conditions of Hopf bifurcation occurring when τ = τ i , i = 0,1,2,...,[(m − 1)/2].In this section, we study the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions when τ = τ 0 , using techniques from normal form and center manifold theory (see, e.g., Kuznetsov [12]).To prove the main result, we need some preliminary lemmas.
At last, in order to illustrate the merits of Midpoint Rule we try to use another methods to get discrete systems of (1.1), such as explicit Euler method.Compare Figures 4.3 and 4.5, we can find that for τ = 50, using the same step size h = 1/10, the Euler discrete model cannot get the bifurcation solution, if we will show the realities, we must reduce the step size (see Figure 4.6).But the Midpoint discrete can get it.These show that, τ 0 in discrete system (2.8) converges the Hopf bifurcation of the original system more fast than ones in another discrete system, with the same step size.

Conclusions
From the above analysis, we can find that the delay does not influence the system's stability when the coefficients of system satisfy condition 1 + u n * > n.But when the coefficients of system satisfy condition 1 + u n * < n, for small delay the positive fixed point of system is stable.With the increasing of delay, the positive fixed point loses its stability and a family of periodic solutions occurs.We could compute the bifurcation for a given system (1.1).Therefore, we infer that when the time between oxygenation of blood in the lungs and stimulation of chemoreceptors in the brainstem is less than τ 0 , the arterial CO 2 concentration in the patient will attain stability; but when the time is more than τ 0 , the arterial CO 2 concentration in the person will be unstable, that is, the patient has an abnormal respiration (Cheyne-Stokes respiration).
then an invariant closed curve, topologically equivalent to a circle, for map(2.11)existsfor τ in a one side neighborhood of τ * .The radius of the invariant curve grows like O( |τ − τ * |).