Limit cycles global bifurcations

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. 

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). 

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).

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