Saddle-focus fixed point

Figure 26. Skeleton bifurcation diagram in the t/-p parameter plane for the model equation (16). Shown are Hopf and saddle-node bifurcations (SUN = saddle-unstable-node bifurcation) as well as the border of the focus-node transition (dashed line) mixed-mode wave forms exist close to the dark region (which marks the region where a fixed point is a ShQ nikov saddle focus). The phase portraits sketch the Unear stability of the fixed point(s). (Reprinted with permission from M. T. M. Koper and P. Gaspard, J. Chem. Phys. 96, 7797, 1992. Copyright 1992, American Institute of Physics.)...

In the case where the Lyapunov value Lk is positive, the fixed point of the original map is a weak saddle-focus. Its stable and unstable manifolds are and respectively, as shown in Fig. 10.4.2. 

Formula (10.4.20) is similar to the formula (10.4.14) for the non-resonant case and the only difference is that in. the case of a weak resonance only a finite number of the Lyapunov values Li,..., Lp is defined (for example, only L is defined when N = b). If at least one of these Lyapunov values is non-zero, then Theorem 10.3 holds i.e. depending on the sign of the first non-zero Lyapunov value the fixed point is either a stable complex focus or an unstable complex focus (a complex saddle-focus in the multi-dimensional case). 

We have seen in the previous sections that the qualitative behavior of a strongly resonant critical fixed point differs essentially from that of a non-resonant or a weakly resonant one. It is therefore natural to ask the question what happens at a strongly resonant point as the frequency varies In particular, in the case of the resonance a = 27t/3 the fixed point is a saddle with six separatrices in general, but when an arbitrarily small detuning is introduced the point becomes a weak focus (stable or unstable, depending on the sign of the first Lyapunov value). The question we seek to answer is how does the dynamics evolve before and after the critical moment ... 

The above theorem is related to the map on the center manifold. Reconstructing the behavior of trajectories of the original map (11.6.2) is relatively simple. Here, if L < 0, then the fixed point is stable when /i < 0. When /i > 0 it becomes a saddle-focus with an m-dimensional stable manifold (defined by T = 0) and with a two-dimensional unstable manifold which consists of a part of the plane y = 0 bounded by the stable invariant curve C,... 

If Li > 0, then when /i > 0, the fixed point is a saddle-focus of the above type, but its unstable manifold is the whole plane y = 0. Upon entering the region M < 0, the fixed point becomes stable. Meanwhile a saddle invariant curve C bifurcates from the fixed point its unstable manifold is (m -h 1)-dimensional and consists of the layers x — constant, restored at the points of the invariant curve. The stable manifold separates the attraction basin of the point O all trajectories from the inner region tend to O, and all those from outside of Wq leave a neighborhood of the origin. 

Remark. Note that the problem on the fixed points of the Poincare map near the homoclinic loop to a saddle-focus is reduced to the study of the fixed point... 

To answer it, one must examine the two-dimensional Poincare map instead of the one-dimensional one, and evaluate the Jacobian of the former map. If its absolute value is larger than one, the map has no stable periodic points, and hence there are no stable orbits in a neighborhood of the homoclinic trar jectory because the product of the multipliers of the fixed point is equal to the determinant of the Jacobian matrix of the map. One can see from formula (13.4.2) that the value of the Jacobian is directly related to whether — 1 >0 or 2i/ — 1 < 0, or, equivalently, i/ > 1/2 or z/ < 1/2. Rephrasing in terms of the characteristic exponents of the saddle-focus, the above condition translates into whether the second saddle value o-q = Ai + 2ReA is positive or negative. It can be shown [100] that if <7 > 0 but ct2 < 0 (a < 6 in Fig. C.7.4), there may be stable periodic orbits near the loop, along with saddle ones. However, when (72 > 0 > O5 automatically), totally imstable periodic orbits emerge... 

The third assumption is that the energy states of the reactants as well as the (shortlived) activated complex are populated according to the Boltzmann distribution. Now, we focus on activated complexes where the reaction coordinate q is fixed at the saddle-point value and where the associated momentum ispj, i.e., with a position and momentum in the range (q, q - -dq ) and (p, p +dp ). The probability of finding such... 

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