Hopf bifurcation cycle modeling

From a mathematical point of view, the onset of sustained oscillations generally corresponds to the passage through a Hopf bifurcation point [19] For a critical value of a control parameter, the steady state becomes unstable as a focus. Before the bifurcation point, the system displays damped oscillations and eventually reaches the steady state, which is a stable focus. Beyond the bifurcation point, a stable solution arises in the form of a small-amplitude limit cycle surrounding the unstable steady state [15, 17]. By reason of their stability or regularity, most biological rhythms correspond to oscillations of the limit cycle type rather than to Lotka-Volterra oscillations. Such is the case for the periodic phenomena in biochemical and cellular systems discussed in this chapter. The phase plane analysis of two-variable models indicates that the oscillatory dynamics of neurons also corresponds to the evolution toward a limit cycle [20]. A similar evolution is predicted [21] by models for predator-prey interactions in ecology. 

The size and period of the oscillations, or of the corresponding limit cycle, varies with the dimensionless reactant concentration pi. We may determine this dependence in a similar way to that used in 2.5. Close to the Hopf bifurcation points we can in fact determine the growth analytically, but in general we must employ numerical computation. For now we will merely present the basic result for the present model. The qualitative pattern of response is the same for all values of ku < g. 

There are no unstable limit cycles in this model, and the oscillatory solution born at one bifurcation point exists over the whole range of stationary-state instability, disappearing again at the other Hopf bifurcation. Both bifurcations have the same character (stable limit cycle emerging from zero amplitude), although they are mirror images, and are called supercritical Hopf bifurcations. 

The emerging limit cycle is born when the dimensionless reactant concentration has the value fx the cycle grows as n then varies away from n. There are two possibilities the limit cycle can grow as fx increases, i.e. for n > n, or as ix decreases, with n < n. Which of these two applies at any given bifurcation point is determined by the sign of a parameter ix2 (we retain the conventional notation for this quantity at the slight risk of confusion between this and the value of the dimensionless reactant concentration at the lower Hopf bifurcation point, fi ). The appropriate form for /x2 for the present model is... 

Some typical oscillatory records are shown in Fig. 4.6. For conditions close to the Hopf bifurcation points the excursions are almost sinusoidal, but this simple shape becomes distorted as the oscillations grow. For all cases shown in Fig. 4.6, the oscillations will last indefinitely as we have ignored the effects of reactant consumption by holding /i constant. We can use these computations to construct the full envelope of the limit cycle in /r-a-0 phase space, which will have a similar form to that shown in Fig. 2.7 for the previous autocatalytic model. As in that chapter, we can think of the time-dependent... 

Fig. 4.9. The development of oscillatory amplitude Ae and period T across the range of instability, 4.2 x 10 3 = n < n < jx = 0.0195, for the pool chemical model with k = 2x 10-3 and y = 0.21, typical of a system with a subcritical Hopf bifurcation at which an unstable limit cycle emerges at The broken curves give the limiting forms predicted by eqns (4.59)—(4.61).

We have already discussed the expressions resulting from a full Hopf bifurcation analysis of the thermokinetic model with the exponential approximation (y = 0). We may do the same for the exact. Arrhenius temperature dependence (y 0). Although the algebra is somewhat more onerous, we still arrive at analytical, expressions for the stability of the emerging or vanishing limit cycle and the rate of growth of the amplitude and period at... 

Fig. 5.3. Locus of Hopf bifurcation points in K-fi parameter plane for thermokinetic model with the full Arrhenius temperature dependence and y = 0.21. The nature of the Hopf bifurcation point and, hence, the stability of the emerging limit cycle changes along this locus at k = 2.77 x 10 3. Supercritical bifurcations are denoted by the solid curve, subcritical bifurcations occur along the broken segment, i.e. at the upper bifurcation point for the lowest k. The stationary-state solution is unstable and surrounded by a stable limit cycle for all parameter values within the enclosed region. Oscillatory behaviour also occurs in the small shaded region below the Hopf curve, where the stable stationary state is surrounded by both an unstable and...

FlO. 5.4. The birth and growth of oscillatory solutions for the thermokinetic model with the full Arrhenius temperature dependence, (a) The Hopf bifurcations /x and ft are both supercritical, with [12 < 0, and the stable limit cycle born at one dies at the other, (b) The upper Hopf bifurcation is subcritical, with fl2 > 0. An unstable limit cycle emerges and grows as the dimensionless reactant concentration ft increases—at /rsu this merges with the stable limit cycle born at the lower supercritical Hopf bifurcation point ft. ... 

Fig. 12.4. Stationary-state solutions and limit cycles for surface reaction model in presence of catalyst poison K3 = 9, k2 = 1, k3 = 0.018. There is a Hopf bifurcation on the lowest branch p = 0.0237. The resulting stable limit cycle grows as the dimensionless partial pressure increases and forms a homoclinic orbit when p = 0.0247 (see inset). The saddle-node bifurcation point is at...

In chapter 12 we discussed a model for a surface-catalysed reaction which displayed multiple stationary states. By adding an extra variable, in the form of a catalyst poison which simply takes place in a reversible but competitive adsorption process, oscillatory behaviour is induced. Hudson and Rossler have used similar principles to suggest a route to designer chaos which might be applicable to families of chemical systems. They took a two-variable scheme which displays a Hopf bifurcation and, thus, a periodic (limit cycle) response. To this is added a third variable whose role is to switch the system between oscillatory and non-oscillatory phases. 

The effects of forced oscillations in the partial pressure of a reactant is studied in a simple isothermal, bimolecular surface reaction model in which two vacant sites are required for reaction. The forced oscillations are conducted in a region of parameter space where an autonomous limit cycle is observed, and the response of the system is characterized with the aid of the stroboscopic map where a two-parameter bifurcation diagram for the map is constructed by using the amplitude and frequency of the forcing as bifurcation parameters. The various responses include subharmonic, quasi-peri-odic, and chaotic solutions. In addition, bistability between one or more of these responses has been observed. Bifurcation features of the stroboscopic map for this system include folds in the sides of some resonance horns, period doubling, Hopf bifurcations including hard resonances, homoclinic tangles, and several different codimension-two bifurcations. 

Figure 12.6a shows the temporal variation of the proximal tubular pressure Pt as obtained from the single-nephron model for a = 12 and T = 16 s. All other parameters attain their standard values as listed in Table 12.1. Under these conditions the system operates slightly beyond the Hopf bifurcation point, and the depicted pressure variations represent the steady-state limit cycle oscillations reached after the initial transient has died out For physiologically realistic parameter values the model reproduces the observed self-sustained oscillations with characteristic periods of 30-40 s. The amplitudes in the pressure variation also correspond to experimentally observed values. Figure 12.6b shows the phase plot Here, we have displayed the normalized arteriolar radius r against the proximal intratubular pressure. Again, the amplitude in the variations of r appears reasonable. The motion... 

Recently, Bernard et al. [499] studied oscillations in cyclical neutropenia, a rare disorder characterized by oscillatory production of blood cells. As above, they developed a physiologically realistic model including a second homeostatic control on the production of the committed stem cells that undergo apoptosis at their proliferative phase. By using the same approach, they found a local supercritical Hopf bifurcation and a saddle-node bifurcation of limit cycles as critical parameters (i.e., the amplification parameter) are varied. Numerical simulations are consistent with experimental data and they indicate that regulated apoptosis may be a powerful control mechanism for the production of blood cells. The loss of control over apoptosis can have significant negative effects on the dynamic properties of hemopoiesis. 

The empty-site requirement in Eq. (28) can be physically interpreted in one of two different ways either the adsorbed A and B have to rearrange prior to reaction, or they are bound to more than one adsorption site. For the latter case, the intermediate concentration is low, thus allowing a pseudo-steady-state assumption. Through the application of bifurcation analysis and catastrophe theory this model was found to predict a very rich bifurcation and dynamic behavior. For certain parameter values, sub- and supercritical Hopf bifurcations as well as homoclinic bifurcations were discovered with this simple model. The oscillation cycle predicted by such a model is sketched in Fig. 6c. This model was also used to analyze how white noise would affect the behavior of an oscillatory reaction system... 

Limit cycles also appear in Chapter 4, but no bifurcation theorems were used (although the Hopf bifurcation theorem could have been used). Uniqueness of these cycles is a question of major interest and importance. To more accurately model the chemostat as it is used in commercial production, the plasmid model discussed in Chapter 10 should be combined with the inhibitor model of Chapter 4. More specifically, consider two organisms - differing only by the presence or absence of a plasmid that confers immunity to the inhibitor - competing in a chemostat (equation (4.2) of Chapter 9 modified for the presence of the inhibitor). The techniques of analysis used in Chapter 4 do not apply, since the system is not competitive in the mathematical sense. Yet an understanding of this system would be very important. 

The behaviour at the upper Hopf point is also that of a supercritical Hopf bifurcation although the loss of stability of the steady-state and the smooth growth of the stable limit cycle now occurs as the parameter is reduced. This is sketched in Fig. 5.10(b). We can join up the two ends of the limit cycle amplitude curve in the case of this simple Salnikov model to show that the amplitude of the limit cycle varies smoothly across the range of steady-state instability, as indicated in Fig. 5.11(a). The limit cycle born at one Hopf point survives across the whole range and dies at the other. Although this is the simplest possibility, it is not the only one. Under some conditions, even for only very minor elaboration on the Salnikov model [16b], we encounter a subcritical Hopf bifurcation. At such an event, the limit cycle that is born is not stable but is unstable. It still has the form of a closed loop in the phase plane but the trajectories wind away from it, perhaps back in towards the steady-state as indicated in Fig. 

Fig. 5.11. Variation of the oscillatory (limit cycle) solution with ju for the simple Salnikov model showing that the stable limit cycle born at one supercritical Hopf bifurcation exists over the whole range of the unstable steady-state, shrinking to zero amplitude at the other Hopf point (b) in this case, each Hopf point gives rise to a different limit cycle, with a stable limit cycle born at fi growing as increases and an unstable limit cycle born at /x also increasing in size as fx increases. At some fx> fx the two limit cycles collide and are...

The main purpose of this paper is to present a simple theoretical model to describe this interesting phenomenon which provides a further example of the synergy between an equilibrium phase transition and a pattern forming instability. It is based on a Landau type equation for the polymer volume fraction coupled to a simple two variables reactive system that, on its own, can undergo a Hopf bifurcation giving rise to chemical oscillations of the limit cycle type. These oscillating systems have now been extensively studied both from the theoretical and experimental points of view (7). 

Thus, y is the bifurcation parameter of the non-linear model. The Hopfs bifurcation occurs at y = 1, stability of the steady state being lost and a limit cycle being... 

The two-dimensional (c,T) phase space for each of these systems can be partitioned into basins of attraction of the distinct attractors. Naturally, the basin structure is more complicated than that of the one-dimensional model studied in Sec. 3, but it is still quite simple. In the one-dimensional model the two stable fixed points were separated by an unstable fixed point, which formed a boundary separating the basins of attraction of the fixed points. In the GK model for the specified parameter setting the situation differs in that one of the stable fixed points has undergone a Hopf bifurcation to yield a stable limit cycle. The basin boundary in the two-dimensional space now consists of the unstable fixed point along with its stable manifolds these are shown in Fig. 1 and labelled Bi and B2. 

Fig. 20. Existence diagram of various wave forms in the excitable region of model (5). N no waves, F flat solitary pulses, S rigidly rotating spirals, M meandering spirals, T turbulence. For even larger e a limit cycle forms in a saddle-loop bifurcation and subsequently vanishes in a Hopf bifurcation. Turbulence also exists between these two lines after [114]. ...

Figure 39, Chapter 3. Bifurcation diagrams for the model of the Calvin cycle for selected parameters. All saturation parameters are fixed to specific values, and two parameters are varied. Shown is the number of real parts of eigenvalues larger than zero (color coded), with blank corresponding to the stable region. The stability of the steady state is either lost via a Hopf (HO), or via saddle node (SN) bifurcations, with either two or one eigenvalue crossing the imaginary axis, respectively. Intersections point to complex (quasiperiodic or chaotic) dynamics. See text for details.

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