Simple saddle-node

By decomposing the right-hand side of (11.4.11) into a Taylor series at the point we can verify that the Lyapunov value does not vanish, i.e. the point is a simple saddle-node. 

Consider a one-parameter family of (r > 2) smooth dynamical systems in (n > 1). Suppose that when the parameter vanishes the system possesses a non-rough equilibrium state at the origin with one characteristic exponent equal to zero and the other n exponents lying to the left of the imaginary axis. We suppose also that the equilibrium state is a simple saddle-node, namely the first Lyapimov value I2 is not zero (see Sec. 11.2). Without loss of generality we assume /2 > 0. 

We also assume that the family is transverse to the surface of the systems with a simple saddle-node. Therefore, near the origin such system is written... 

Since the return time from/to the cross-section S (i.e. the period of L j) grows proportionally to cj(/i), it must tend to infinity as /i —H-oo (see Sec. 12.2 if L is a simple saddle-node, then the period grows as tt/V/IZ ). Since the vector field vanishes nowhere in 17, it follows that the length of must tend to infinity also. Since L, does not bifurcate when /x > 0, we have an example of the blue sky catastrophe [152]. 

If the saddle-node L is simple, then all neighboring systems having a saddle-node periodic orbit close to L constitute a codimension-one bifurcational surface. By construction (Sec. 12.2), the function /o depends continuously on the system on this bifurcational surface. Thus, if the conditions of Theorem 12.9 are satisfied by a certain system with a simple saddle-node, they are also satisfied by all nearby systems on the bifurcational surface. This implies that Theorem 12.9 is valid for any one-parameter family which intersects the surface transversely. In other words, our blue sky catastrophe occurs generically... 

We assume also that L is a simple saddle-node. Thus, the function G can be represented in the form... 

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... 

In this case the limit of a periodic trajectory as e —0 is a homoclinic cycle r composed of a simple saddle-node equilibrium state and its... 

In this case, the topological limit of the bifurcating periodic motion as —> — 0 contains no equilibrium point, but a periodic trajectory of the saddle-node type which disappears when e < 0. The trajectory is a simple saddle-node in the sense that it has only one multiplier, equal to 1, and the first Lyapunov value is not equal to zero. 

Fig. 8.3. The approach to, or departure from, stationary-state solutions following small perturbations for simple cubic autocatalysis again showing the instability of the middle branch. The turning points (ignition and extinction) have one-sided stability as perturbations in one direction decay back to the saddle-node point, but those of the opposite sign depart for the other...

We assume that the equations (7.200) have a simple hysteresis type static bifurcation as depicted by the solid curves in Figures 10 to 12 (A-2). The intermediate static dashed branch is always unstable (saddle points), while the upper and lower branches can be stable or unstable depending on the position of eigenvalues in the complex plane for the right-hand-side matrix of the linearized form of equations (7.198) and (7.199). The static bifurcation diagrams in Figures 10 to 12 (A-2) have two static limit points which are usually called saddle-node bifurcation points. 

Fig. 2.7 (a) Temporal variation of the membrane potential V and the intracellular calcium concentration S in the considered simple model of a bursting pancreatic cell, (b) Bifurcation diagram forthe fast subsystem the black square denotes a Hopf bifurcation, the open circles are saddle-node bifurcations, and the filled circle represents a global bifurcation, (c) Trajectory plotted on top of the bifurcation diagram. The null-cline forthe slow subsystem is shown dashed. 

It is worthwhile noting that both Examples 5.1 and 5.2 are ternary problems, and profile behavior can as such be easily visualized in a two-dimensional space. It is also entirely possible, in the case of Example 5.1 where constant volatilities have been assumed, to plot the transformed triangles (TTs) for the problem to interpret where a particular profile will tend toward, or whether there may be instability issues because the profile is closely approaching a saddle node. Analogous analyses can be performed for nonideal systems too, as shown in Example 5.2. Generating profiles and pinch points with DODS-ProPlot is rather simple for such systems and is not shown explicitly here. 

Example 8.1 illustrates how one may go about synthesizing a simple reactive distillation process, or at the vary least allows one to assess whether the proposed process is at all feasible. It should be noted that if either k ox p turned out to be negative, the proposed process would be infeasible since these parameters only have realistic meaning when they are positive quantities. Moreover, if the k and p were found to be positive but the node at the desired composition is found to be an unstable node, the simple process would not be feasible either. If a saddle node were found, one has to be certain of the initial composition within the reactor because there is only a single profile which will end at the desired composition. Any other initial composition would result in an entirely different ending composition because the profile will veer away from the saddle point. Lasdy, the reader should also be aware that, in the case of the three component system, there are two independent equations that may be written, implying that one may solve for two unknown process... 

The proofs of Theorems 10.2, 10.3, and 10.4 are found in [348]. Equation (10.17) is of particular interest. Near the Takens-Bogdanov point, the frequency of the limit-cycle oscillations along the line of Hopf bifurcations, a = 0, is given by >h = see above. On the line of saddle-node bifurcations we have Aj = 0. An equation like (10.17) is expected from simple dimensional arguments. The only intrinsic length scales in reaction-diffusion systems come from the diffusion coefficients. The inverse time is determined by the rate coefficients of the reaction kinetics. Thus (10.17) provides an estimate of the intrinsic length of the Turing pattern near a double-zero point ... 

Note that in the n-dimensional case, where n > 4, other topological configurations of may be realized. Such saddle-node bifurcations will definitely lead the system out of the class of systems with simple dynamics. For example, it is shown in [139, 152] that a hyperbolic attractor of the Smale-Williams type may appear just after the disappearance of a saddle-node periodic orbit. ... 

If the saddle-node is not simple, then there may be more saddle and stable periodic orbits when < 0, in this case 7p is the union of all of them and all their unstable manifolds. 

In its full generality, this lemma is proven in [140] and it implies almost immediately the basic Theorem 12.4 below. The proof is based on a lengthy calculations and we omit them here. A simple proof of an analogous statement is given in Sec. 12.5 under some additional assumptions. Namely, it is assumed there that the system is sufficiently smooth with respect to all variables and and that the saddle-node L is simple. Moreover, instead of proving that all of the derivatives tend to zero, the vanishing of only a sufficiently large number of derivatives is established. Of course, all this does not represent a severe restriction. 

When the saddle-node is simple, (12.2.22) is just a condition of transversality of the one-parameter family under consideration to the bifurcation surface of systems with the saddle-node, which allows the Poincare map on to be written in the form (12.2.2). 

So, the results of Theorems 12.3, 12.5 and 12.7 are summarized as follows IfW is a smooth toruSy then a smooth attracting invariant torus persists after the disappearance of the saddle-node L. If is homeomorphic to a torus but it is non-smoothy then chaotic dynamics appears after the disappearance of L, Herey either the torus is destroyed and chaos exists for all small /i > 0 the big lobe condition is sufficient for that)y or chaotic zones on the parameter axis alternate with regions of simple dynamics. 

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