Ambulatory-fed conditions and chaotic behavior

zˆ \ r 7 r 6 28 then, on considering Eqs. 18 and 20, we are assured that y 1 \ 0 and z 1 \ 0. Under these conditions, the shapes and relative positions of the equilibrium manifolds f = 0, g = 0, and h = 0 will be as depicted in Fig. 1, where three arrows indicate fast transitions, two arrows indicate intermediate ones, while a single arrow indicates slow ones. Thus, a system initially at a generic point, say point A of Fig. 1, will make a quick transition to the plane given by Eq. 14 on the fast manifold f = 0 point B in Fig. 1, since the signs of f assure us that this part of the manifold f = 0 is stable. As the plane where f = 0 is approached, y has slowly become active. A transition at an intermediate speed is made along f = 0 in the direction of decreasing y, since g B 0 here, to the point C on a stable part of the curve f = h = 0. From the point C, a slow transition is then made along this curve in the direction of decreasing y, since g is still negative here. Once the point D is reached, the stability of the manifold is lost, a catastrophic transition brings the system to point E on the manifold z = 0 where it intersects the plane given by Eq. 14. Conse- quently, the system will slowly develop along this line in the direction of increasing y, since g is now positive. At a point F on the plane z = 0, the stability will again be lost and a quick transition will bring the system back to the point G on the stable branch of the curve f = h = 0, before repeating the same previously described path, thereby forming a closed cycle GDEFG. Thus, the existence of a limit cycle in the system for o and d sufficiently small is assured. The exact solution trajectory of the system will be contained in a tube about this closed curve, the radius of which tends to zero with o and d. The above arguments can be summarized by the following theorem. Theorem 1. If inequalities Eqs. 26 – 28 hold, and o and d are sufficiently small, then the system of Eqs. 11 – 13 has a global attractor, in the positi6e octant of the phase space, which is a limit cycle composed of a concatenation of catastrophic transitions occurring at different speeds. A computer simulation of Eqs. 11 – 13 is presented in Fig. 2a, with parametric values cho- sen to satisfy inequalities Eqs. 26 – 28 iden- tified in the above theorem. The solution trajectory, projected onto the y, x-plane, is seen here to tend towards a limit cycle as theoretically predicted. The corresponding time series of insulin x and glucose y are shown in Fig. 2b.

5. Ambulatory-fed conditions and chaotic behavior

We now extend our model to incorporate tem- poral glucose absorption in order to model the responses of a subject under ambulatory-fed con- ditions. The term wt in Eq. 3 is then taken in this case to assume the sinusoidal form wt k sin vt 29 We then remove the explicit dependence on time of the above oscillatory term by letting u = cos vt and 6 = sin vt which transforms the model equations into the following five-dimensional system of autonomous differential equations. x;=z[r 1 y − r 2 x + c 1 ] 30 y;=o r 3 z − r 4 x + c 2 + c 3 u n 31 z;=od[r 5 y − yˆzˆ − z + r 6 zzˆ − z − r 7 z] 32 u;= −v6 33 6;=vu 34 and c 3 = k o We now carry out a numerical investigation to determine the ranges of parametric values where chaotic dynamics were likely. Since it has been Fig. 2. A computer simulation of the model system of Eqs. 11 – 13 with parametric values chosen to satisfy the conditions identified in the text for which periodic solutions exist. The solution trajectory, projected onto the y, x-plane, is seen in Fig. 2a to tend toward a stable limit cycle as theoretically predicted. The corresponding time series of plasma insulin x and glucose y are shown in Fig. 2b. Here, o = 0.1; d = 0.01; r 1 = 0.2; r 2 = 0.1; r 3 = 0.1; r 4 = 0.1; r 5 = 0.1; r 6 = 0.1; r 7 = 0.05; c l = 0.1; c 2 = 0.1; yˆ = 1.24; zˆ = 2.0; and N = 0.1. discovered by some researchers Hastings and Powell, 1991; Lenbury et al., 1999 that one may be able to generate chaos in a nonlinear system which already exhibits limit cycle behavior, we choose parametric values that would lead to cy- cling in the x, y and z components, guided by our work in the previous section. We then let the system run for 10 5 time steps, and examining only the last 8 × 10 4 time steps to eliminate transient behavior. Using the values of zˆ between 0.985 and 1.038, and changing zˆ in steps of 10 − 5 , the relative maximum values x max of x are collected during the last 8 × l0 4 time steps. They are then plotted as a function of zˆ as shown in Fig. 3. We discover in this bifurcation diagram the appearance of a period doubling route to chaos, which resembles the dynamics of one-di- mensional difference equations such as the logistic population model. Thus, the system of Eqs. 30 – 34 appears to exhibit chaotic dynamics for val- ues of zˆ between 1.000 and 1.038. A computer simulation of the model system Eqs. 30 – 34, with parametric values chosen under the above mentioned guidelines and zˆ = 1.01 in the chaotic range, is presented in Fig. 4a, showing the strange attractor projected onto the y, z-plane, and the corresponding chaotic time courses of plasma insulin x and glucose y are presented in Fig. 4b.

6. Discussion and conclusion