Saddle-focus

Fig. 8a, b. Possible types of SPs inside 4-simplex stable (1) and unstable (4) node, saddle of the first (2) and second (5) order, stable (5) and unstable (8) focus, saddle-focus of the first (6) and second (7) order... 

Figure 23. Singular trajectories of system of equations (104) with existence of two singulars at different values of fi (a), 10 (fe), 1 (c), 0.35 (d), 0.30 (e), 0.27. y = 1, i = 0.5, tj - 10, Singularities are site-saddle (a,b), stable focus-saddle (c), unstable focus with stable limiting cycle-saddle (d), and unstable focus-saddle (e).

As we have seen above, the dynamics near the homoclinic loop to a saddle with real leading eigenvalues is essentially two-dimensional. New phenomena appear when we consider the case of a saddle-focus. Namely, we take a C -smooth (r > 2) system with an equilibrium state O of the saddle-focus saddle-focus (2,1) type (in the notation we introduced in Sec. 2.7). In other words, we assume that the equilibrium state has only one positive characteristic exponent 7 > 0, whereas the other characteristic exponents Ai, A2,..., are with negative real parts. Besides, we also assume that the leading (nearest to the imaginary axis) stable exponents consist of a complex conjugate pair Ai and A2 ... 

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. 

Figure 29 Bifurcation diagram of the minimal model of glycolysis as a function of feedback strength and saturation 6 of the ATPase reaction. Shown are the transitions to instability via a saddle node (SN) and a Hopf (HO) bifurcation (solid lines). In the regions (i) and (iv), the largest real part with in the spectrum of eigenvalues is positive > 0. Within region (ii), the metabolic state is a stable node, within region (iii) a stable focus, corresponding to damped transient oscillations.

Fig. 9. Linear stability diagram illustrating (21). The bifurcation parameter B is plotted against the wave number n. (a) and (b) are regions of complex eigenvalues < > (b) shows region corresponding to an unstable focus (c) region corresponding to a saddle point. The vertical lines indicate the allowed discrete values of n. A = 2, D, = 8 103 D2 = 1.6 10-3.

We have discussed the case in which the unstable homogeneous steady state is a saddle point, that is, when a real eigenvalue con changes sign. Let us now consider what happens when the uniform steady state is an unstable focus and a time periodic regime sets in. 

Stationary-state solutions correspond to conditions for which both numerator and denominator of (3.54) vanish, giving doc/dp = 0/0, and so are singular points in the phase plane. There will be one singular point for each stationary state each of the different local stabilities and characters found in the previous section corresponds to a different type of singularity. In fact the terms node, focus, and saddle point, as well as limit cycle, come from the patterns on the phase plane made by the trajectories as they approach or diverge. Stable stationary states or limit cycles are often refered to as attractors , unstable ones as repellors or sources . The different phase plane patterns are shown in Fig. 3.4. 

FiC. 3.4. Representations of the different singular points in the concentration phase plane (a) stable node, sn (b) stable focus, sf (c) unstable focus uf (d) unstable node, un (e) saddle. point, sp. 

An example is shown in figure 3 for section AA near the bottom of the 2/1 resonance horn of figure 2. As the frequency is increased from left to right, the torus becomes phase locked as a pair of period 2 saddle nodes develop on it. The saddle nodes then separate with the saddles alternating with the node and the invariant circle is now composed of the unstable manifolds of the saddles whereas the stable manifolds of the saddles come from the unstable period 1 focus in the middle of the circle and from infinity. As the frequency is increased further, the saddles rotate around the circle and recombine with their neighbouring nodes in another saddle-node bifurcation. 

The three standard local codimensional-one bifurcations are the saddle-node, Hopf, and period doubling bifurcations and several have been continued numerically for this model and appear in figure 2. We have chosen not to show the curves of focus-node transitions because they do not represent any changes in stability, only changes in the approach to the steady behaviour. The saddle-node bifurcations that occur during phase locking of the torus at low amplitudes continue upward and either close upon themselves as in the case of the period 3 resonance horns or the terminate in some codimension-two bifurcation. 

FIGURE 8 (a) Detail of the tip of the 3/1 resonance horn illustrating typical way in which period 3 resonance horns close around a point with Floquet multipliers at the third root of unity (point F). (fa) and (c) The saddle-node pairings change from section AA to section BB and the unstable manifolds of the period 3 saddles no longer make up a phase locked torus. (d) A one-parameter vertical cut through the third root of unity point. The three saddles coalesce with the period one focus that is undergoing the Hopf bifurcation. 

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