Equilibrium manifold invariant

Comparative simplicity of MEIS-based computing experiments is due primarily to the simplicity of the main initial assumption of its construction on the equilibrium of all states belonging to the set of thermodynamic attainability Dt(y) and the identity of their physico-mathematical description. These states belong to the invariant manifold that contains trajectories tending to the extremum of characteristic thermodynamic function of the system and satisfying the monotonic variation of this function. The use of the mentioned assumption consistent with the second thermodynamics law allows one, as was noted, not to include in the formulation of the problem solved different more particular principles, such as the Gibbs... 

The Invariant manifold Meth cMi composed of equilibrium states (xi)eth(y ) e... 

Normally, hyperbolic invariant manifolds persist under perturbation [22]. If we are in the setting where the form of Eq. (1) must hrst be obtained by applying Normal Form theory, then we are restricted to a sufficiently small neighborhood of the equilibrium point. In this case the nonlinear terms are much smaller than the linear terms. Therefore, the sphere present in the linear problem becomes a deformed sphere for the nonlinear problem and still has (2n — 2)-dimensional stable and unstable manifolds in the (2n — l)-dimensional energy surface since normal hyperbolicity is preserved under perturbations. 

The conventional theory of reaction processes relies on equilibrium statistical physics where the equi-energy surface is uniformly covered by orbits as shown in Fig. 34. To the contrary, the phase space in multidimensional chaos has various invariant structures, and orbits wander around these structures as shown in Fig. 35. In these processes, those degrees of freedom that constitute the movement along stable or unstable manifolds vary from NHIM to NHIM. Their variance reveals how reaction coordinates change during successive processes in reaction dynamics. 

Let p = x(0) be an arbitrary initial point with Jr,(0) > 0. Then the initial data do not belong to either stable manifold. Hence w p) is not equal to either Eq or E2, but it does lie on E = 0. Since it is invariant and since every solution of (5.2) on E = 0 converges to an equilibrium, u(p) contains an equilibrium. By the Butler-McGehee theorem, Eq uIp) since M (Eq) is unbounded. If o p) contains E2, then o p) also contains either Eq or an unbounded orbit, again by the Butler-McGehee theorem (see Figure 5.2). Since this is impossible, E must be in 0 p). However, is a local attractor, so u p) = Ey This completes the proof. ... 

The main problem in the solution of non-linear ordinary and partial differential equations in combustion is the calculation of their trajectories at long times. By long times we mean reaction times greater than the time-scales of intermediate species. This problem is especially difficult for partial differential equations (pdes) since they involve solving many dimensional sets of equations. However, for dissipative systems, which include most applications in combustion, the long-time behaviour can be described by a finite dimensional attractor of lower dimension than the full composition space. All trajectories eventually tend to such an attractor which could be a simple equilibrium point, a limit cycle for oscillatory systems or even a chaotic attractor. The attractor need not be smooth (e.g., a fractal attractor in a chaotic system) and is in some cases difficult to compute. However, the attractor is contained in a low-dimensional, invariant, smooth manifold called the inertial manifold M which locally attracts all trajectories exponentially and is easier to find [134,135]. It is this manifold that we wish to investigate since the dynamics of the original system, when restricted to the manifold, reduce to a lower dimensional set of equations (the inertial form). The inertial manifold is, therefore, a useful notion in the field of mechanism reduction. 

Let be a four-dimensional smooth symplectic manifold on which a Hamiltonian system v = sgrad H is given, with H being a smooth Hamiltonian. Equilibrium positions xq of the system v are critical points of the function H. Since fT is an integral of the system u, it follows that the field v may be restricted to an invariant three-dimensional constant-energy surface Q, that is, Q = x M H x) = const. Being a symplectic manifold, is orientable, and therefore the manifold Q is abo orientable. 

Theorem 1.4.1. If the rate functions/i,. ..,/u and the internal energy function U satisfy the conditions of Postulate 1.2.1 then each invariant manifold Amq) contains at least one equilibrium point. 

An invariant manifold r(ii ) containing the equilibrium point u, can be described by the following extent variables... 

Open systems do not in general possess thermodynamic potential functions, a fact which often manifests itself with the existence of more than one equilibrium points in the same invariant manifold. Open systems still satisfy the condition of positive entropy production, Eq.(1.5.5). Some interesting implications of the positive entropy production for the steady states of open distributed systems are discussed in Section 2.8. 

In Section 1.4 the topological concept of rotation was used to prove the existence of equilibrium states. When the reaction kinetics are not restricted by Postulate 1.5.1, each invariant manifold may include more than one equilibrium states and it is interesting to obtain information about the number and stability of these states. In the present section we shall use one more topological concept, the index of a fixed point, to show that the equilibrium states are odd in number, 2m+1, among which m at least are unstable. As in the preceding sections, the discussion concerns isolated systems, but extension to other closed systems should not present difficulties. 

In order to show the asymptotic approach to equilibrium, we have to modify the general method based on Eq. (2.8.20) because the rates vanish at more than one point of the invariant manifold. Actually, we shall adopt a direct approach based on the behavior of the functions... 

A related approach termed the method of invariant grids (MIG) is also discussed in Gorban and Karlin (2003), Gorban et al. (2004a, c), Chiavazzo et al. (2007,2009) based on the method of invariant manifold (MIM). In MIG, a quasi-equilibrium approach is used to define a first approximation to the SIM on a grid in concentration space, and then improved estimations of the SIM are obtained using either Newton iteration or relaxation methods. The MIG was compared to CSP-based methods, the ILDM method and the entropy-based methods in Chiavazzo et al. (2007). 

The multi-dimensional extension of two-dimensional rough systems is the Morse-Smale systems discussed in Sec. 7.4. The list of limit sets of such a system includes equilibrium states and periodic orbits only furthermore, such systems may only have a finite number of them. Morse-Smale systems do not admit homoclinic trajectories. Homoclinic loops to equilibrium states may not exist here because they are non-rough — the intersection of the stable and unstable invariant manifolds of an equilibrium state along a homoclinic loop cannot be transverse. Rough Poincare homoclinic orbits (homoclinics to periodic orbits) may not exist either because they imply the existence of infinitely many periodic orbits. The Morse-Smale systems have properties similar to two-dimensional ones, and it was presumed (before and in the early sixties) that they are dense in the space of all smooth dynamical systems. The discovery of dynamical chaos destroyed this idealistic picture. 

Axiom 2. All periodic orbits and equilibrium states in G are structurally stable and any intersection of their stable and unstable invariant manifolds is transverse. 

The stable invariant manifolds of equilibrimn states and periodic orbits may have common points with the unstable manifolds. Clearly, if a point xq is such a common point of two invariant manifolds, then the trajectory x = xq) belongs to both manifolds entirely. In the simplest case, O = Wq n W, i.e. the stable and imstable manifolds of an equilibrium state intersect at only trajectory, which is the equilibrium state itself. 

By Axiom 2, the transversality condition (7.5.3) holds at all intersections of the stable and vmstable invariant manifolds of equilibrium states and periodic orbits in system (7.5.1). 

As shown in Chap. 5, the above critical equilibrium state lies in an invariant C -smooth center manifold defined by an equation of the form y = (a ), where (x) vanishes at the origin along with its first derivative. 

It follows from formula (9.2.9) that if the right-hand side of the system (9.2.6) is analytic, and if all Lyapunov values vanish, then g[x,ip x)) = 0. Hence, since y = (p x) is the solution of the system (9.2.7), it follows that the curve y = ip x) is filled out by the equilibrium states of the system (9.2.6). Thus, it is an invariant manifold of this system. Since it is tangent to y = 0 at O, it is the center manifold by definition. It follows that for the case imder consideration, the system has an analytic center manifold W y = (p(x) which consists of equilibrium states as illustrated in Fig. 9.2.5. 

The saddle equilibrium states are the saddle fixed points of the shift map, and respectively, their separatrices are the invariant manifolds. Returning to the original (non-rescaled) variables we find that the fixed points must lie apart from the origin at some distance of order e. If the third iteration (10.6.2) of the map (10.6.1) were the shift map of the reduced system (10.6.5), then the above theorem would follow from our arguments because the fixed points Oi, O2,03 of the third iterations correspond to the cycle of period three of the original map. 

Starting with any (x, y), a trajectory of system (12.4.8) converges typically to an attractor of the fast system corresponding to the chosen value of x. This attractor may be a stable equilibrium, or a stable periodic orbit, or of a less trivial structure — we do not explore this last possibility here. When an equilibrium state or a periodic orbit of the fast system is structurally stable, it depends smoothly on x. Thus, we obtain smooth attractive invariant manifolds of system (12.4.8) equilibrium states of the fast system form curves Meq and the periodic orbits form two-dimensional cylinders Mpo, as shown in Fig. 12.4.6. Locally, near each structurally stable fast equilibrium point, or periodic orbit, such a manifold is a center manifold with respect to system (12.4.8). Since the center manifold exists in any nearby system (see Chap. 5), it follows that the smooth attractive invariant manifolds Meqfe) and Mpo( ) exist for all small e in the system (12.4.7) [48]. 

In the original proof the system under consideration was assumed to be analytic. Later on, other simplified proofs have been proposed which are based on a reduction to a non-local center manifold near the separatrix loop (such a center manifold is, generically, 3-dimensional if the stable characteristic exponent Ai is real, and 4-dimensional if Ai = AJ is complex) and on a smooth linearization of the reduced system near the equilibrium state (see [120, 147]). The existence of the smooth invariant manifold of low dimension is important here because it effectively reduces the dimension of the problem. ... 

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