Saddle-node connection

Fig. 4.2 Phase portrait for logistic and bistable reaction terms. The front is a heterocUnic saddle-node connection for the logistic case. The front is a saddle-saddle connection for the bistable case...

Further rotation of the monkey saddle leads to the formation of a continuous hyperbolic surface that partitions space into two interpenetrating networks of tunnels. Each network consists of nodes connecting four tunnels meeting at tetrahedral angles (109.5°) (Fig. 1.17(c)). The nodes are arranged on... 

At p = p3, the system jumps to a new state of uniform precession of the director (UP2) with large reorientation (0 74 ) and slow precession rate. As displayed in Fig. 7, starting from the stable UP2 branch above pa and lowering the excitation intensity, one finds a large and rather complicated hysteretic cycle, which eventually flips back to the UPl solution at pg = 1.09. This part of the UP2 branch consists of alternatively stable and unstable regions exhibiting a series of saddle-node bifurcations. Eventually, this branch connects with the UPS one which makes a loop and connects with the UPl branch. 

For 0.53 < X < 0.72, one has the sequence U —> D —> O —> PR as before [see Fig. 13(b)], however there is an additional bifurcation between PR states. In fact, the limit cycle amplitude of the PR regime, now labeled PRi [curve 2 in Fig. 13(b)], abruptly increases. This results in another periodic rotating regime labeled PR2 with higher reorientation amplitude [curve 3 in Fig. 13(b)]. This is a hysteric transition connected to a double saddle-node structure with the (unstable) saddle separating the PRi and PR2 branches as already found... 

The two branches of the nontrivial steady states are not connected to the branch of the trivial steady state. They form a so-called isola and appear and disappear via a saddle-node bifurcation. 

The structure of trajectory bundle also has interesting peculiarities at the best nonsharp separation. In this case, trajectory bundle also has saddle-node and node points (Shafir et al., 1984). Figure 5.21a shows for the azeotropic mixture acetone(l)-benzene(2)-chloroform(3) the line of best bottom product (Poelhnann Blass, 1994), connecting the end of possible segment Reg at side 2-3 at sharp... 

Fig. 7.35 Homoclinic trajectraies connecting the (3, 1) saddle-nodes on the separatrix of the torus about the Li atom in the LiH molecule. The arrows indicate the direction of the eigenvectos of the transposed Jacobian matrix V/ at the stagnation points. An asymptotic wavy line flows across the stagnation loop, about its centre...

Fig. 7.41 Splitting of the central diamagnetic vortex into a central saddle line and two diamagnetic vortical lines in acetylene. The asymptotic blue trajectories passing through (3 1) saddle-node points mark the intersection of the separatrices containing the TVs with the yz plane. The truncated blue line is connected to the symmetrical pattern about the other C-H bond. The diamagnetic (paramagnetic) portions of the TV ate observed around green (red) SLs...

Fig. 8.1.2. A structurally unstable heteroclinic connection between a saddle-node Oi and a saddle O2.

Fig. 8.1.6. (a) A structurally unstable saddle connection after the disappearance of a saddle-node cycle in Pig. 8.1.4 (b) Phase plane after the splitting of the homoclinic loop in Fig. 8.1.5. 

As already mentioned, problems of this nature had appeared as early as in the twenties in connection with the phenomenon of transition from synchronization to an amplitude modulation regime. A rigorous study of this bifurcation was initiated in [3], under the assumption that the dynamical system with the saddle-node is either non-autonomous and periodically depending on time, or autonomous but possessing a global cross-section (at least in that part of the phase space which is under consideration). Thus, the problem was reduced to the study of a one-parameter family of C -diffeomorphisms (r > 2) on the cross-section, which has a saddle-node fixed point O at = 0 such that all orbits of the unstable set of the saddle-node come back to it as the number of iterations tends to -hoo (see Fig. 12.2.1(a) and (b)). 

We will study the case m = 0 in Sec. 12.4 in connection with the problem of the blue sky catastrophe . In the case m > 2, infinitely many saddle periodic orbits are born (see Theorem 12.5) when the saddle-node disappears moreover, even hyperbolic attractors may arise here (see [139]). We do not discuss such kind of bifurcations in this book. 

Fig. 12.4.2. The option of chaotic behavior resulted from the disappearance of a saddle-node fixed point of the corresponding Poincare map, assuming the contraction condition is not satisfied but the big lobe condition holds each leaf of the foliation must intersect at least two of the connected components of n S.

As we mentioned. Lemma 12.3 may be reformulated as a possibility to embed a sufficiently smooth one-dimensional map near a simple saddle-node fixed point into a smooth one-dimensional flow for ji > 0. An analogous result was proved in [74] in connection with the problem on the appearance of... 

Afraimovich, V, S. and Shilnikov, L. P. [1974] On some global bifurcations connected with the disappearance of fixed point of a saddle-node type, Soviet Math. Dokl. 15, 1761-1765. 

The electron density is a continuous function that is experimentally observable, hence uniquely defined, at all points in space. Its topology can be described in terms of the distribution of its critical points, i.e. the points at which the electron density has a zero gradient in all directions. There are four kinds of critical point which include maxima (A) usually found near the centres of atoms, and minima (D) found in the cavities or cages that lie between the atoms. In addition there are two types of saddle point. The first (B) represents a saddle point that is a maximum in two directions and a minimum in the third, the second (C) represents a saddle point that is a minimum in two direction and a maximum in the third. One can draw lines of steepest descent connecting the maxima (A) to the minima (D), lines whose direction indicates the direction in which the electron density falls off most rapidly. Of the infinite number of lines of steepest descent that can be drawn there exists a unique set that has the property that, in passing from the maximum to the minimum, each line passes successively through a B and a C critical point. This set forms a network whose nodes are the critical points and whose links are the lines of steepest descent connecting them. 

A distillation boundary connects two fixed points node, stable or unstable, to a saddle. The distillation boundaries divide the separation space into separation regions. The shape of the distillation boundary plays an important role in the assessment of separations. 

The geometric properties of a RCM allow its simple sketch. Figure 9.5 shows the construction for the mixture methyl-isopropyl-ketone (MIPK), methyl-ethyl-ketone (MEK) and water. Firstly, the position of the binary azeotropes and of the ternary azeotrope is located. Then the boiling points for pure components and azeotropes are noted (Fig. 9.5a). The behaviour of characteristic points (node or saddle) is determined by taking into account the direction of temperatures. Finally, straight distillation boundaries are drawn by connecting saddles with the corresponding nodes (Fig. 9.5b). 

V < 2V. The state (1,0) is always a saddle point. To be physically acceptable, a front must always be nonnegative. Consequently, only nonnegative heteroclinic orbits are acceptable. Such orbits can only exist if (0,0) is a stable node. In other words, fronts only exist for v > 2 /z5r. Since there exists a heteroclinic connection or front for each value of v with v > 2 /Dt, this analysis does not yield a unique propagating velocity. In fact, the front velocity v depends on the initial condition, specifically on the tail of the initial condition. 

The presence of (non-)reactive singular points i.e. pure components and (non-)reactive azeotropes) in ROMs allows the division of the (transformed) composition diagram into separate (reactive) distillation regions by introducing (reactive) distillation separatrices, which connect two singular points in the composition space. The explanatory note 5.1 gives a brief overview of the concepts of nodes and saddle points, as features of singular points. 

Distillation boundaries always exist if there is more than one origin or terminus of residue curves in the system, which means more than one feasible bottom or top product. Typical residue curves for the system benzene-cyclohexane-acetone are shown in Figure 11.11. The most important lines are the distillation boundaries, which cannot be crossed by distillation. While the residue curves are connections between the high-boiling compound (stable node) and the low-boiling compound (unstable node), the distillation boundaries are connecting lines between the saddle point and the stable or the unstable node, respectively. 

The first topological equation that connects a possible number of stationary points of various types for three-component mixtures (N, node S, saddle upper index is the number of components in a stationary point) was deduced (Gurikov, 1958) ... 

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