Node equilibria

In Chap. 12 we will study the global bifurcations of the disappearance of saddle-node equilibrium states and periodic orbits. First, we present a multidimensional analogue of a theorem by Andronov and Leontovich on the birth of a stable limit cycle from the separatrix loop of a saddle-node on the plane. Compared with the original proof in [130], our proof is drastically simplified due to the use of the invariant foliation technique. We also consider the case when a homoclinic loop to the saddle-node equilibrium enters the edge of the node region (non-transverse case). 

Fig. 11.2.4. Planar bifurcation of a saddle-node equilibrium state with 2 > 0.

GLOBAL BIFURCATIONS AT THE DISAPPEARANCE OF SADDLE-NODE EQUILIBRIUM STATES AND PERIODIC ORBITS... 

To study such bifurcations one should understand the structure of the limit set into which the periodic orbit transforms when the stability boundary is approached. In particular, such a limit set may be a homoclinic loop to a saddle or to a saddle-node equilibrium state. In another bifurcation scenario (called the blue sky catastrophe ) the periodic orbit approaches a set composed of homoclinic orbits to a saddle-node periodic orbit. In this chapter we consider homoclinic bifurcations associated with the disappearance of the saddle-node equilibrium states and periodic orbits. Note that we do not restrict our attention to the problem on the stability boundaries of periodic orbits but consider also the creation of invariant two-dimensional tori and Klein bottles and discuss briefly their routes to chaos. 

Bifurcations of a homoclinic loop to a saddle-node equilibrium state... 

Fig 12.1.1. Bifurcation sequence of a saddle-node equilibrium with a homoclinic trajectory (a) before, (b) at, and (c) after the bifurcation. 

Theorem 12.1. The disappearance of the saddle-node equilibrium with the homoclinic loop results in the appearance of a stable periodic orbit of period... 

Fig. 12.1.2. Two cross-sections 5o and Si to the homoclinic loop F are chosen near the saddle-node equilibrium O.

The smooth case corresponds, in particular, to a small time-periodic perturbations of an autonomous system possessing a homoclinic loop to a saddle-node equilibrium (see the previous section). Indeed, for a constant time shift map along the orbits of the autonomous system the equilibrium point becomes a saddle-node fixed point and the homoclinic loop becomes a smooth closed invariant curve, but the transversality of to F is, obviously, preserved under small smooth perturbations. 

Fig. 12.2.2. A nontransverse tangency of the unstable and strong-stable manifolds of a saddle-node fixed point may be obtained by a small time-periodic perturbation of the system with an on-edge homoclinic loop to a saddle-node equilibrium state, as shown in Fig. 12.1.4.

Fig. 12.4.3. A phemenological scenario of development of the blue sky catastrophe when the saddle-node equilibrium O disappears, the unstable manifold of the saddle-node periodic orbit L has the desired configuration, as the one shown in Fig. 12.4.1(a).

A periodic orbit merges with a homoclinic loop r e) to a saddle-node equilibrium state Og, where r(e) W (Oe). 

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... 

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. 

Subsequently, the condition of complete separation has to be coupled with the material balances derived for the nodes of the SMB unit and implemented in the Equilibrium Theory Model for Langmuir-type systems. That leads to the set of mathematical conditions given below, which the flow rate ratios have to fulfil in order to achieve complete separation, in particular ... 

Fig. 7. Cross-linker model for nucleosome arrangement in the chromatin fiber superstructure in the presence (a) or absence (b) of H1/H5, based on data in the literature (see text) and H5-containing mono-nucleosome stem structure in Fig. 3(c). In 3D, the plane of the nucleosomes is expected to rotate more or less regularly around the fiber axis, forming a solenoid-like superstructure. Nucleosomes 1, 2 and 5 are in the open conformation of Fig. 3(a), nucleosomes 4 and 7 in the open conformation of Fig. 2(b), and other nucleosomes in the closed negative (Fig. 2(c)) or positive conformations. Nucleosomes are expected to thermally fiuctuate between the different conformations, within an overall dynamic equilibrium of (ALkn) -l (see text). -I- and - refer to node polarities. (From Fig. 5 in Ref. [28].)...

For mechanisms involving three or more rapid-equilibrium segments, once the segments are properly represented as nodes in a scheme, the rate equation can be obtained by the usual systematic approach. For example, consider the case of one substrate—one product reaction in which a modifier M is in rapid equilibrium with E, EA, and EP. 

As is seen from the behaviour of the more sophisticated Heine-Abarenkov pseudopotential in Fig. 5.12, the first node q0 in aluminium lies just to the left of (2 / ) / and g = (2n/a)2, the magnitude of the reciprocal lattice vectors that determine the band gaps at L and X respectively. This explains both the positive value and the smallness of the Fourier component of the potential, which we deduced from the observed band gap in eqn (5.45). Taking the equilibrium lattice constant of aluminium to be a = 7.7 au and reading off from Fig. 5.12 that q0 at 0.8(4 / ), we find from eqn (5.57) that the Ashcroft empty core radius for aluminium is Re = 1.2 au. Thus, the ion core occupies only 6% of the bulk atomic volume. Nevertheless, we will find that its strong repulsive influence has a marked effect not only on the equilibrium bond length but also on the crystal structure adopted. 

These have only one true stationary-state solution, ass = pss = 0 (as t - go), corresponding to complete conversion of the initial reactant to the final product C. This stationary or chemical equilibrium state is a stable node as required by thermodynamics, but of course that tells us nothing about how the system evolves in time. If e is sufficiently small, we may hope that the concentrations of the intermediates will follow pseudo-stationary-state histories which we can identify with the results of the previous sections. In particular we may obtain a guide to the kinetics simply by replacing p by p0e et wherever it occurs in the stationary-state and Hopf formulae. Thus at any time t the dimensionless concentrations a and p would be related to the initial precursor concentration by... 

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