Global bifurcations of cycles

In two-dimensional systems, there are four common ways in which limit cycles are created or destroyed. The Hopf bifurcation is the most famous, but the other three deserve their day in the sun. They are harder to detect because they involve large 

A bifurcation in which two limit cycles coalesce and annihilate is called a fold or saddle-node bifurcation of cycles, by analogy with the related bifurcation of fixed points. An example occurs in the system 

At p a half-stable cycle is born out of the clear blue sky. As p. increases it splits into a pair of limit cycles, one stable, one unstable. Viewed in the other direction, a stable and unstable cycle collide and disappear as p decreases through p, Notice that the origin remains stable throughout it does not participate in this bifurcation. 

As p decreases, the limit cycle r = 1 develops a bottleneck at 8 = njl that becomes increasingly severe as 1. The oscillation period lengthens and finally becomes infinite at p = 1, when a fixed point appears on the circle hence the term infinite-period bifurcation. For p 1, the fixed point splits into a saddle and a node. 

As the bifurcation is approached, the amplitude of the oscillation stays 9(1) but the period increases like p -, for the reasons discussed in Section 4.3. 

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In Chap. 12 we will study the global bifurcations of the disappearance of saddle-node equilibrium states and periodic orbits. First, we present a multidimensional analogue of a theorem by Andronov and Leontovich on the birth of a stable limit cycle from the separatrix loop of a saddle-node on the plane. Compared with the original proof in [130], our proof is drastically simplified due to the use of the invariant foliation technique. We also consider the case when a homoclinic loop to the saddle-node equilibrium enters the edge of the node region (non-transverse case). 

In the bifurcation diagram shown in Fig. 85, the plane of control parameters was divided into regions of a qualitatively different character of phase trajectories (the shapes of these trajectories are given in the respective regions) and the lines on which occur sensitive states of the Hopf bifurcation and the saddle bifurcation were marked. The diagram also depicts the line of sensitive states of the global bifurcation the appearance of a cycle from the branches of saddle separatrices. 

Figure 38, Chapter 3. A bifurcation diagram for the model of the Calvin cycle with product and substrate saturation as global parameters. Left panel Upon variation of substrate and product saturation (as global parameter, set equalfor all irreversible reactions), the stable steady state is confined to a limited region in parameter space. All other parameters fixed to specific values (chosen randomly). Right panel Same as left panel, but with all other parameters sampled from their respective intervals. Shown is the percentage r of unstable models, with darker colors corresponding to a higher percentage of unstable models (see colorbar for numeric values).

In the case of steady state bifurcations, certain eigenvalues of the linear-approximation matrix reduce to zero. If we consider relaxations towards a steady state, then near the bifurcation point their rates are slower. This holds for the linear approximation in the near neighbourhood of the steady state. Similar considerations are also valid for limit cycles. But is it correct to consider the relaxation of non-linear systems in terms of the linear approximations To be more precise, it is necessary to ask a question as to whether this consideration is sufficient to get to the point. Unfortunately, it is not since local problems (and it is these problems that can be solved in terms of the linear approximations) are more simple than global problems and, in real systems, the trajectories of interest are not always localized in the close neighbourhood of their attractors. 

The bursting dynamics ends in a different type of process, referred to as a global (or homoclinic) bifurcation. In the interval of coexisting stable solutions, the stable manifold of (or the inset to) the saddle point defines the boundary of the basins of attraction for the stable node and limit cycle solutions. (The basin of attraction for a stable solution represents the set of initial conditions from which trajectories asymptotically approach the solution. The stable manifold to the saddle point is the set of points from which the trajectories go to the saddle point). When the limit cycle for increasing values of S hits its basin of attraction, it ceases to exist, and... 

The limit cycle found in the previous section holds only for 103 — 031 small. Obviously, once the limit cycle exists, it can be continued, either globally or until certain bad things happen such as the period tending to infinity or the orbit collapsing to a point. It is very difficult to show analytically that these events do not occur. Moreover, the computations necessary to actually prove the asymptotic stability of the bifurcating orbit are very difficult. We discuss briefly some numerical computations, shown in Figure 8.1, which suggest answers to both these problems. 

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. 

Fig. 7. (a) The amplitudes and (b) the period T of the oscillations found from integrating Eq. (27), N= 3, for varying values of n. Except for the value n = 4, global limit cycle attractors were found. The arrows on the right-hand side of the diagrams represent the theoretical values for the piecewise linear equation in the limit The arrow on the left-hand ride of (b) is the period predicted by the Hopf bifurcation theorem. 

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