Saddle-focus point

Assume a point disclination located in a nematic droplet of radius R. The point disclination can be classified according to their Poincare characteristic angle a as a knot point (a = 0), focus point (0 < a < 7t/2), center (a = 7t/2), saddle-focus point (tt/2 < a < tt) or saddle point (a = 7t/2). For a knot point, one has a spherically symmetrical radial configuration and then... 

As S continues to increase we reach a point marked by an open circle. Here, the equilibrium point undergoes a saddle-node bifurcation. Somewhere before this bifurcation, the unstable focus point has turned into an unstable node with two positive real eigenvalues. In the saddle-node bifurcation, one of these eigenvalues... 

When P > 0, using Table 5.5 one can immediately point out the type of the inner azeotrope, and when p < 0, only the node (N) and saddle (S) in this table are replaced with the focus (F) and saddle-focus (SF), respectively. The sign of Ind (X ) of this azeotrope is opposite to the sign of a3 and is determined by indexes of the boundary SPs (see Table 5.6) in accordance with the rule of azeotropy (5.18) which in the case of tetrapolymerization yields ... 

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). 

Let us examine next the bifurcations of the system (11.5.1) in the multidimensional case. If Li < 0 (Fig. 11.5.4), then when // < 0, the equilibrium state O is stable (rough focus when p < 0, and a weak focus aX p = 0) and it attracts all trajectories in a small neighborhood of the origin. When > 0 the point O becomes a saddle-focus with a two-dimensional unstable manifold and an m-dimensional stable manifold. The edge of the unstable manifold is the stable periodic orbit which now attracts all trajectories, except those in the stable manifold of O. One multiplier of the periodic orbit was calculated in Theorem 11.1, this is po p) = 1 — 47r /a (0) -h o p). To find the others we... 

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... 

The limit points of this process correspond to the existence of a homoclinic loop which is the o -limit set for a separatrix of the other saddle-focus. 

This is similar to Case 1, but with Li(0) >0. As e —> —0, a saddle periodic trajectory shrinks into a stable point O. Upon moving through e = 0, the equilibrium state becomes a saddle-focus it spawns a two-dimensional unstable invariant manifold (i.e. the boundary Ss is dangerous). 

A homoclinic bifurcation is a composite construction. Its first stage is based on the local stability analysis for determining whether the equilibrirun state is a saddle or a saddle-focus, as well as what the first and second saddle values are, and so on. On top of that, one deals with the evolution of a -limit sets of separatrices as parameters of the system change. A special consideration should also be given to the dimension of the invariant manifolds of saddle periodic trajectories bifurcating from a homoclinic loop. It directly correlates with the ratio of the local expansion versus contraction near the saddle point, i.e. it depends on the signs of the saddle values. 

Introduce the second saddle value <72 as the sum of the three leading characteristic exponents at the saddle-focus. In the three-dimensional case, it is the divergence of the vector field at the origin. Here, the curve 0 2 = 0, given by the equation a = 6, intersects HS at (a = 6,6 = 7.19137). Above this point, <72 > 0. 

The point NSF a = 0 corresponds to a neutral saddle-focus. At this codimension-two point the dynamics of the trajectories near the homoclinic loops to the saddle-focus becomes chaotic. This bifurcation indeed proceeds the origin of the chaotic double scroll attractor in Chua s circuit. In the general case, this bifurcation was first considered in [29]. The complete unfolding of... 

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... 

Another codimension-two homoclinic bifurcation in the Shimizu-Morioka model occurs at (a 0.605,6 0.549) on the curve H2 corresponding to the double homoclinic loops. At this point, the separatrix value A vanishes and the loops become twisted, i.e. we run into inclination-flip bifurcation [see Figs. 13.4.8 and C.7.11]. The geometry of the local two-dimensional Poincare map is shown in Fig. 13.4.5 and 13.4.6. To find out what our case corresponds to in terms of the classification in Sec. 13.6, we need also to determine the saddle index 1/ at this point. Again, as in the case of a homoclinic loop to the saddle-focus, it is very crucial to determine whether 1/ < 1/2 or 1/ > 1/2. Simple calculation shows that z/ > 1/2 for the given parameter values. Therefore,... 

C.7. 86.1 Assume there is a homoclinic loop to a saddle-focus in the Shimizu-Morioka model (like a T-point). Without computing the characteristic exponent of the saddle-focus, what can we say about the local structure is it trivial (one periodic orbit), or complex (infinitely many periodic orbits) ... 

Since singular points are identified with the positions of equilibria, the significance of the three principal singular points is very simple, namely the node characterizes an aperiodically damped motion, the focus, an oscillatory damped motion, and the saddle point, an essentially unstable motion occurring, for instance, in the neighborhood of the upper (unstable) equilibrium position of the pendulum. 

Saddle point method, 68 Self-focusing, 83, 84, 86 Self-injection, 150... 

Instead of nodal lines in closed systems we are interested in the statistics of NPs for open chaotic billiards since they form vortex centers and thereby shape the entire flow pattern (K.-F. Berggren et.al., 1999). Thus we will focus on nodal points and their spatial distributions and try to characterize chaos in terms of such distributions. The question we wish to ask is simply if one can find a distinct difference between the distributions for nominally regular and irregular billiards. The answer to this question is clearly positive as it is seen from fig. 3. As shown qualitatively NPs and saddles are both spaced less regularly in chaotic billiard in comparison to the integrable billiard. The mean density of NPs for a complex RGF (9) equals to k2/A-k (M.V. Berry et.al., 1986). This formula is satisfied with good accuracy in both chaotic and integrable billiards. 

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