Bifurcation of a limit cycle

We have presented the first clear evidence of quasiperiodicity in a chemical dynamical system. The shape of the torus as well ais its evolution is consistent with the nature of some generic instabilities of this chemical system. Some preliminary experiments in our laboratory indicate that similair tori co ild be obtained after a Hopf bifurcation of a limit cycle in perfect agreement with the theoretical predictions. 

The bifurcation of a limit cycle from the homoclinic loop to the saddle-node was first discovered by Andronov and Vitt in their study of the Van der Pol equation with a small periodic force at a 1 1 resonance ... 

Bifurcation of a limit cycle from a scparatrix loop of a saddle with non-zero saddle value... 

FIGU RE 10 Illustration of the disappearance of a limit cycle via a turning point on a periodic branch near a subcritical Hopf bifurcation, (a) A stable limit cycle surrounding an unstable focus (b) the unstable focus undergoes a subcritical Hopf bifurcation and leaves an inner unstable limit cycle surrounding a stable focus (c) the two limit cycles combine into a metastable configuration and disappear altogether as the parameter is further increased. 

In conclusion of our short excursion into the qualitative theory of differential equations, we shall discussed the often-used term "bifurcation . It is ascribed to the systems depending on some parameter and is applied to point to a fundamental reconstruction of phase portrait when a given parameter attains its critical value. The simplest examples of bifurcation are the appearance of a new singular point in the phase plane, its loss of stability, the appearance (birth) of a limit cycle, etc. Typical cases on the plane have been discussed in detail in refs. [11, 12, and 14]. For higher dimensions, no such studies have been carried out (and we doubt the possibility of this). 

Figure 8.1. Numerical solutions showing bifurcation from a limit cycle in a plane into a limit cycle in the interior. The fixed parameters are a, = 0.3, Oi = 0.4, m = 10.0, / 2 = 4.5, and m- = 5.0. The parameter was varied between 0.46 and 0.48. The limit cycle moves from one plane (a face of R+) through the interior of (showing coexistence) and collapses into the opposite face of R +. ...

In this scenario, part of a limit cycle moves closer and closer to a saddle point. At the bifurcation the cycle touches the saddle point and becomes a homoclinic or-... 

In Section 5.6 we will discuss a catastrophe occurring in the reduced (van der Pol) system (5.41) when the parameter 5 changes sign. A catastrophe of this type the appearance of a limit cycle, associated with a loss of stability by the stationary point (0, 0), i.e. the Hopf bifurcation, also occurs in the original system (5.40). The resulting limit cycle is in this case localized on the surface z = x2, see Fig. 75. 

In Chapter 6 we will apply the method described above to the examination of stability of some chemical kinetics equations. Moreover, in the case of establishing the existence of a sensitive state, characteristic of the Hopf bifurcation, the presence of a limit cycle may sometimes be proved (without giving its more detailed characteristic) in a different way. For this purpose suffice it to demonstrate that the trajectories of a system cannot escape to infinity and remain in some limited region. In such a case, a limit cycle must exist inside this region. 

The functions f c, z,p) and g c) include the transport fluxes of cytoplasmic Ca2+ across the ER and plasma membrane, w c,p) the production and degradation of IP3. For realistic functions /, g and w the existence of a limit cycle must generally be shown numerically. However, the local stability properties of the steady state give an idea. One can show that, due to the Ca " transport across the plasma membrane, there is a unique steady state (c, z, p) [23]. Therefore, changes in stability of the unique steady state are likely to be connected with an Hopf bifurcation and the birth/death of a limit cycle. Generally, if the steady state is unstable it is to be expected that the trajectories move toward a stable limit cycle. Note that a stable limit cycle and a stable steady state can coexist. Our analysis can make no predictions in this regard. 

We choose the control parameters U and a such that the deterministic system exhibits no oscillations but is very close to a bifurcation thus yielding it very sensitive to noise. The transition from stationarity to oscillations in the system may occur either via a Hopf or via a saddle-node bifurcation on a limit cycle as depicted in the bifurcation diagram of Fig. 5.9. The different nature of these two bifurcations is reflected in the effect noise has in each case. The local character of the Hopf bifurcation is responsible for noise-induced high frequency oscillations of strongly varying amplitude around the stable fixed point. We try to characterize basic features of these oscillations such as coherence and time scales. The need to be able to adjust these features as one wishes will lead to the application of the time-delayed... 

Fig. 5.9. Bifurcation diagram in the (cr, t/) plane. Thick and hatched lines mark the transition from stationary to moving fronts via a Hopf or a saddle-node bifurcation on a limit cycle, respectively. The inset shows a blow-up of a small part of the hatched line revealing its saw-tooth-like structure. Dark and white correspond to stationary and moving fronts, respectively, where the numbers denote the positions of the stationary accumulation front in the superlattice. Upper inset shows the frequency / of the limit cycle which is born above the critical point (marked by a cross in the lower inset) as function of U. [57]...

To conclude, noise-induced front motion and oscillations have been observed in a spatially extended system. The former are induced in the vicinity of a global saddle-node bifurcation on a limit cycle where noise uncovers a mechanism of excitability responsible also for coherence resonance. In another dynamical regime, namely below a Hopf bifurcation, noise induces oscillations of decreasing regularity but with almost constant basic time scales. Applying time-delayed feedback enhances the regularity of those oscillations and allows to manipulate the time scales of the system by varying the time delay t. 

Another route to chaos that is important in chemical systems involves a torus attractor which arises via bifurcation from a limit cycle attractor. Again chaos is found to be associated with periodic behavior and to arise from it through a sequence of transformations and associated bifurcations of a periodic state of the system. The specific sequence is different in this case, however, and somewhat more complex. 

The Hopf bifurcation theorem is only valid in the neighborhood of the bifurcation value n = 4. However, it is possible to prove existence of a limit cycle for finite values n >4, and uniqueness and stability for infinite n. 

A second weak point of the geometrical phase space analysis is the fact that it will be very difficult if not impossible to detect the onset of a limit cycle at a critical or bifurcation point as defined in Section 6.4 for two variables by... 

In terms of the original variable (p — — ut, the stationary value of (the equilibrium state of system (12.1.9)) corresponds to an oscillatory regime with the same frequency as that of the external force. The periodic oscillations of (the limit cycle in (12.1.9)) correspond to a two-frequency regime. Hence, the above bifurcation scenario of a limit cycle from a homoclinic loop to a saddle-node characterizes the corresponding route from synchronization to beat modulations in Eq. (12.1.7). 

The quasiperiodic route to chaos is historically important. It arises from a succession of Hopf birfurcations. As already noted, a single Hopf bifurcation results in a limit cycle. The next Hopf bifurcation produces a phase flow tliat can be represented on tire surface of a toms (douglmut). This flow is associated witli two frequencies if tire ratio of tliese frequencies is irrational tlien tire toms surface is densely covered by tire phase trajectory, whereas if... 

Leaving the details, the equation describing the motion of one particle in two electrostatic waves allows perturbation methods to be applied in its study. There are three main types of behavior in the phase space - a limit cycle, formation of a non-trivial bounded attracting set and escape to infinity of the solutions. One of the goals is to determine the basins of attraction and to present a relevant bifurcation diagram for the transitions between different types of motion. 

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. 

If the cell cycle in amphibian embryonic cells appears to be driven by a limit cycle oscillator, the question arises as to the precise dynamical nature of more complex cell cycles in yeast and somatic cells. Novak et al. [144] constructed a detailed bifurcation diagram for the yeast cell cycle, piecing together the diagrams obtained as a function of increasing cell mass for the transitions between the successive phases of the cell cycle. In these studies, cell mass plays the role of control parameter a critical mass has to be reached for cell division to occur, provided that it coincides with a surge in cdkl activity which triggers the G2/M transition. 

As the dimensionless concentration of the reactant decreases so that pi just passes through the upper Hopf bifurcation point pi in Fig. 3.8, so a stable limit cycle appears in the phase plane to surround what is now an unstable stationary state. Exactly at the bifurcation point, the limit cycle has zero size. The corresponding oscillations have zero amplitude but are born with a finite period. The limit cycle and the amplitude grow smoothly as pi is decreased. Just below the bifurcation, the oscillations are essentially sinusoidal. The amplitude continues to increase, as does the period, as pi decreases further, but eventually attains a maximum somewhere within the range pi% < pi < pi. As pi approaches the lower bifurcation point /zf from above, the oscillations decrease in size and period. The amplitude falls to zero at this lower bifurcation point, but the period remains non-zero. 

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... 

When we come to look at the stability of the limit cycle which is born at the Hopf bifurcation point, we shall meet a quantity known as the Floquet multiplier , conventionally denoted p2, which plays a role similar to that played for the stationary state by the eigenvalues and k2. If / 2 is negative, the limit cycle will be stable and should correspond to observable oscillations if P2 is positive the limit cycle will be unstable. 

At the point of Hopf bifurcation, the emerging limit cycle has zero amplitude and an oscillatory period given by 2n/a>0. As we begin to move away from the bifurcation point the amplitude A and period T grow in a form we can calculate according to the formulae... 

Equations (5.44) and (5.45) give only the leading-order growth of A and T. The higher-order terms become more important as the magnitude of the departure from the bifurcation point (fi — /i ) increases and then, in general, we need to compute the size of the limit cycles. 

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