Poincare map linearized

For this simple two-dimensional system, the linearized Poincare map degenerates to a 1 X1 matrix, i.e., a number. Exercise 8.7.1 asks you to show explicitly that... 

Let us suppose next that system (7.5.1) has a periodic trajectory L x = d t), of period r. The periodic orbit L is structurally stable if none of its (n — 1) multipliers lies on the unit circle. Recall that the multipliers of L are the eigenvalues of the (n — 1) x (n — 1) matrix A of the linearized Poincare map at the fixed point which is the point of intersection of L with the cross-section. The orbit L is stable (completely unstable) if all of its multipliers lie inside (outside of) the unit circle. Here, the stability of the periodic orbit may be understood in the sense of Lyapunov as well as in the sense of exponential orbital stability. In the case where some multipliers lie inside and the others lie outside of the unit circle, the periodic orbit is of saddle type. 

Since the estimated MLE is negative, Am < 0, we can say that this case displays as essential regular dynamical behavior. The Poincare map displays an orbit set contained in a short line. The above results can be imputed to the following fact The bubbles rise in an almost-linearly pathway and the liquid phase falls downward between bubbles streams (see scheme in the Figure 10a). This means that the bubbles interactions are feeble. In this way, the modes induced by one bubble stream can not affect another one. 

Since each of these mappings is from R" to R", where = 3 for the solution map and = 2 for the Poincare map, their linearizations are given by a matrix. The following is a basic result connecting the linearization of the two maps. 

Solution We linearize about the fixed point r = 1 of the Poincare map. Let... 

Operationally, to numerically construct a Poincare map, a number of initial conditions are chosen (they are often chosen uniformly on the Poincare map surface within the energetically allowed range of positions and momenta), and the trajectories are allowed to intersect the map many times. In praaice, the code must monitor whether a trajectory has crossed the surface, and then back-track to the surface with reasonable accuracy. The back-tracking can be carried out by a linear interpolation between the integration points immediately before and after the crossing, but this procedure lacks sufficient numerical accuracy for some applications. 

We have discussed the typical manifestation of periodic orbits on a Poincare map as fixed points that are either elliptic or hyperbolic. Let us now consider the properties of motion nearby these fixed points in terms of their stability properties. This is accomplished by a straightforward linear stability analysis about the fixed point. We can carry out such an analysis on any fixed point, whether or not the surrounding phase space is chaotic (as long as we can find the fixed point). 

The second approach, successfully followed in the analysis of complex oscillations observed in the model of the multiply regulated biochemical system, relies on a further reduction that permits the description of the dynamics of the three-variable system in terms of a single variable only, by means of a Poincare section of the original system. Based on the one-dimensional map thus obtained from the differential system, a piecewise linear map can be constructed for bursting. The fit between the predictions of this map and the numerical observations on the three-variable differential system is quite remarkable. This approach allows us to understand the mechanism by which a pattern of bursting with n peaks per period transforms into a pattern with (n + 1) peaks. 

The piecewise linear map does not account, however, for the appearance of chaotic behaviour. A slight modification of the unidimensional map, taking into account some previously neglected details of the Poincare section of the differential system, shows how chaos may appear besides complex periodic oscillations of the bursting type. 

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