Unique invariant manifold

The solutions of Eqs.(2.1.34), (2.1.35) with appropriate boundary conditions such as Eqs. (2.4.3), (2.4.4) will be called the steady states of the system. Various properties of the steady states, such as the invariant manifolds and a priori bounds, the existence and uniqueness of solutions, the asymptotic behavior, and the stability will be treated in the sections that follow. There is a strong similarity in the properties of the uniform open systems investigated in Sections 1.8,1.9 and the distributed systems to be studied now. In both types of systems the interplay between reaction and transport rates (or flow rates) creates the possibility of multiple steady states for certain types of reaction kinetics. Furthermore, the conditions for uniqueness and stability of the steady state have a common mathematical and physical basis. 

The concept of the index can be used for deriving uniqueness criteria. If, for example, each steady state has index -h 1, there is exactly one steady state. This observation can be applied in spite of the unavailability of the steady states by using their a priori properties, for instance the fact that they lie in the invariant manifold r(uo)- first result is the theorem... 

Thus, we have established the existence and uniqueness of the invariant manifolds — the surfaces in the space (v , r) of the form... 

Since the right-hand side of system (10.5.10) is periodic with respect to t, the set of the trajectories which start at t = 0 from the points on the image of the curve n (when mapping along the trajectories of the system over the period) also comprise an invariant manifold. By virtue of uniqueness, this must be the same invariant manifold, and hence Cn = Cn- But the map over the period of the system (10.5.10) is the map (10.5.3). So, we have established the existence of the six invariant manifolds Co, - - -, Cs (three stable and three unstable) of the fixed point w = 0 of the given map, from which the statement of the theorem follows. 

Trajectories initiated close to an elliptic fixed point behave in a qualitatively different manner from those near a hyperbolic fixed point. For one thing, elliptic fixed points are invariably surrounded by invariant tori, with frequency ratios not far from that of the fixed point all motion on each torus stays on the same torus. Hyperbolic fixed points may or may not be surrounded by tori. However, they are always associated with a single unique set of manifolds composed of motion asymptotic to them in positive and negative time. These manifolds are called stability manifolds or separatrix manifolds, and their continued existence in the presence of a coupling term is guaranteed by the Stable Manifold theorem. The nature of the asymptotic manifolds will be seen to be of special interest and importance, and we discuss them at length in the following section. 

The parametrisation of the whole set of solutions is not unique (i.e. the selection of the parameters is not unique), only the number of degrees of freedom (D,ot) is invariant. In mathematical terms, the set is a differentiable manifold of dimension it is called the solution manifold of the set of balance equations. The main result of the structural analysis is that the set of equations (constraints) (8.2.2) and (8.3.1) is minimal by (8.3.30) the number of degrees of freedom equals the number of variables minus the number of (scalar) constraints. 

The manifold M- a has locally stable and unstable manifolds that are close to the unperturbed locally stable and unstable manifolds and if these manifolds intersect transversely, then the Smale-Birkhoff theorem predicts the existence of horseshoes and their chaotic dynamics in the perturbed dissipative system. A 2D hyperbolic invariant torus Tja(Pi, P2) may be located on by averaging the perturbed dissipative vector field 7 > 0 and a > 0 restricted to M q, over the angular variables Qi and Q2- The averaged equations have a unique stable hyperbolic fixed point (Pi,P2) = (0,0) with two negative eigenvalues provided that the... 

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