Invariant manifold computation

Two alternatives present themselves in formulating algorithms for the tracking of segments of stable and unstable manifolds. The first involves observing the initial value problem for an appropriately chosen familv of initial conditions, henceforth referred to as simulation of invariant manifolds. A second generation of algorithms for the computation of invariant manifolds involves numerical fixed point techniques. 

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... 

Hobson, D. (1993) An efficient method for computing invariant manifolds of planar... 

The GMM determines the existence of transverse homoclinic/heteroclinic points that are transverse intersections between the stable and unstable manifolds to any invariant sets of the perturbed system when a homoclinic/heteroclinic orbit exists to a hyperbolic invariant manifold in the unperturbed (undamped a = 0 and unperturbed 7 = 0) system. The unperturbed vector field may be computed by setting the perturbation (wavemaker forcing) parameter 7 = 0 and the dissipation parameter a = 0 in (3.45), and are... 

On M-y there are locally stable and unstable manifolds that are of equal dimensions and are close to the impertm-bed locally stable and unstable manifolds. The perturbed normally hyperbolic locally invariant manifold intersects each of the 5D level energy sm-faces in a 3D set of which most of the two-parameter family of 2D nonresonant invariant tori persist by the KAM theorem. The Melnikov integral may be computed to determine if the stable and unstable manifolds of the KAM tori intersect transversely. 

Bykov, V., Gol dshtein, V. Fast and slow invariant manifolds in chranical kinetics. Comput. Math. Appl. 65, 1502-1515 (2013)... 

Chiavazzo, E., Gorban, A.N., Karlin, I.V. Comparison of invariant manifolds for model reduction in chemical kinetics. Commun. Comput. Phys. 2, 964-992 (2007)... 

Roussel, M.R., Tang, T. The functional equation truncation method for approximating slow invariant manifolds a rapid method for computing intrinsic low-dimensional manifolds. J. Chem. Phys. 125, 214103 (2006)... 

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. 

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