Structurally unstable equilibrium point

This second-level modeling of the feedback mechanisms leads to nonlinear models for processes, which, under some experimental conditions, may exhibit chaotic behavior. The previous equation is termed bilinear because of the presence of the b [y (/,)] r (I,) term and it is the general formalism for models in biology, ecology, industrial applications, and socioeconomic processes [601]. Bilinear mathematical models are useful to real-world dynamic behavior because of their variable structure. It has been shown that processes described by bilinear models are generally more controllable and offer better performance in control than linear systems. We emphasize that the unstable inherent character of chaotic systems fits exactly within the complete controllability principle discussed for bilinear mathematical models [601] additive control may be used to steer the system to new equilibrium points, and multiplicative control, either to stabilize a chaotic behavior or to enlarge the attainable space. Then, bilinear systems are of extreme importance in the design and use of optimal control for chaotic behaviors. We can now understand the butterfly effect, i.e., the extreme sensitivity of chaotic systems to tiny perturbations described in Chapter 3. 

It is well known (see, e.g.. Ref. 13) that the normal form transformations do not converge in the sense that normalization to all orders generally does not yield a meaningful result. However, this is of no consequence for our purposes. We view the technique more as the input to a numerical method for realizing the NHIM, its stable and unstable manifolds, and the TS. In this sense the limitations of machine precision make normalization beyond a certain finite order meaningless. This is a local result valid in the neighborhood of the equilibrium point of center center saddle type. However, once the phase-space structure is established locally, it can be numerically continued outside of the local region. 

Phase Space. The PODS structure is easily lifted into phase space and described in a way very analogous to the linear case. We begin by finding the equilibrium points of the original Hamiltonian. Let P,- be such points. In a general case of chemical relevance, there will be a point P whose linear stability will be of stable/unstable (center/saddle) character. That is. 

Figure 13. The whole structure of the phase space in a nutshell. P is the equilibrium point, E is the energy, (q) are the collective bath coordinates, and ( ) is the collective transition coordinates. The cental manifold of P is Cp, and the stable and unstable manifolds are indicated by S/U.

Transport. We need now to construct the NHIM, its stable/unstable manifolds, and the center manifold. Let P be the main relative equilibrium point. The first task is to find the short periodic orbits lying above P in energy. These p.o. are unstable. We did so by exploring phase space at energies 4, 10, and 14 cm above E (1 atomic unit = 2.194746 x 10 cm ). It is not possible to go much higher in E, since the center manifold disappears shortly above E + 14cm , because of the structure of the potential energy surface. 

Certain mathematical-physical considerations and the subsequent fitting of f (p T) allow us to conclude that the coexistence envelope diameter point (pd(T), pa(T)) is an (orbitally unstable) improper node, i.e. that all solution paths leaving (pD(T), pequilibrium points (pG(T), p (T)), (pD(T), P (T)), and (pL(T), Pa(T)) converge to the critical point (1, 1). This multiple equilibrium point is an orbitally stable, but structurally (topologically) unstable, multiple node. The parameter T thus can be considered as a bifurcation parameter, and T = 1 as a bifurcation value of dynamic System 3. 

The heteroclinic cycles including the saddles whose unstable manifolds have different dimensions were first studied in [34, 35]. This study mostly focused on systems with complex dynamics. Let us, however, discuss here a case where the dynamics is simple. Let a three-dimensional infinitely smooth system have two equilibrium states 0 and O2 with real characteristic exponents, respectively, 7 > 0 > Ai > A2 and 772 > 1 > 0 > (i.e. the unstable manifold of 0 is onedimensional and the unstable manifold of O2 is two-dimensional). Suppose that the two-dimensional manifolds (Oi) and W 02) have a transverse intersection along a heteroclinic trajectory To (which lies neither in the corresponding strongly stable manifold, nor in the strongly unstable manifold). Suppose also that the one-dimensional unstable separatrix of Oi coincides with the one-dimensional stable separatrix of ( 2j so that a structurally unstable heteroclinic orbit F exists (Fig. 13.7.24). The additional non-degeneracy assumptions here are that the saddle values are non-zero and that the extended unstable manifold of Oi is transverse to the extended stable manifold of O2 at the points of the structurally unstable heteroclinic orbit F. 

Remark. A great deal of attention has been paid in recent years to non-equilibrium stationary processes that are unstable and also extended in space. They can give rise to different phases that exist side by side, so that translation symmetry is broken. The name dissipative structures has been coined for them, and the prime examples are the Benard cells and the Zhabotinski reactions, but they also occur in biology and meteorology. However, these are features of the macroscopic equations. They are only relevant for fluctuation theory inasmuch as the fluctuation becomes very large at the point where the instability sets in. The critical fluctuations in XIII.5 are an example. There are many more varieties, in particular in the case of more variables. 

The steady-state solution that is an extension of the equilibrium state, called the thermodynamic branch, is stable until the parameter A reaches the critical value A,. For values larger than A<, there appear two new branches (61) and (62). Each of the new branches is stable, but the extrapolation of the thermodynamic branch (a ) is unstable. Using the mathematical methods of bifurcation theory, one can determine the point A, and also obtain the new solution, (i.e., the dissipative structures) in the vicinity of A, as a function of (A - A,.). One must emphasize that... 

Structure is interpreted based on the concept of free energy, that is, the concept of equihbrium statistical mechanics. However, if the folding processes take place far from equilibrium, the concept of equilibrium statistical mechanics cannot be applied. On the other hand, from our point of view, the funnel-like structure would give a typical example where the strong dynamical correlation exists. If we can analyze how stable and unstable manifolds intersect in the folding processes, we can have an alternative explanation concerning the funnel, which is a future project from our approach. 

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