Invariant surface

It is based on Denjoy s theorem, and ris the rotation number. This algorithm, implemented by Chan (1983) computes invariant circles with irrational rotation numbers. We may, of course, discretize and solve for the whole invariant surface and not just for a section of it. Instead of having to integrate the system equations, we will then be solving for a much larger number of unknowns resulting from the additional dimension we had suppressed in the shooting approach we used. 

We shall make more use of the notion of normally hyperbolic invariant manifold (NHIM). This invariant surface is the n-DOF generalization of the periodic orbit dividing surface, even if originally defined in a much more general framework (a bibliography may be found in Ref. 24). Its correct definition is put forward in Section IV.A and is used in all examples coming thereafter. 

The other important scenario in multidimensional tunneling is of the dynamical tunneling that is observed in 1.5D and 2D systems [21]. In this case, the classical phase space is separated not by the energy barrier but by the invariant surface (e.g., KAM tori). Such a situation is realized, for example, in periodically perturbed one-dimensional (1.5D) barrier potentials and also in 2D barrier systems when the total energy is taken over the potential saddle. In a series of recent articles [22-25], we have found a new class of tunneling phenomena... 

Goldman, B.B. and Wipke, W.T. (2000) Quadratic shape descriptors. 1. Rapid superposition of dissimilar molecules using geometrically invariant surface descriptors. J. Chem. Inf. Comput. Sci, 40, 644-658. 

Let us add a short comment on this result. Assume for a moment that are exact first integrals, (l = 1,..., n). Then the orbit lies on an invariant surface—actually a torus—which is e—close to the unperturbed torus pi = pi(0). Hence the actions pi(t) are not constants, being pi = 0(e). We say that this change is due to a deformation of the invariant surface, which shows up in a time 0(1/e), but remains bounded. Moreover, since the invariant surface is still a torus, the time evolution of the actions is actually quasi-periodic. 

The dynamics of such systems is described by the Kolmogorov-Arnold-Moser theory of nearly integrable conservative dynamical systems (see e.g. Ott (1993)). For e = 0 the fluid elements move along the streamlines and the trajectories in the phase space form tubes parallel to the time axis. Due to the periodicity in the temporal direction these tubes form tori that fill the whole phase space and are invariant surfaces for the motion of the fluid elements. Each torus... 

The main question is what happens with these invariant surfaces when the streamfunction has a small time-periodic component, 0 < e 1. Are there any invariant surfaces preserved when the perturbation is small or they disappear for arbitrarily small perturbations and the orbits may wander anywhere in the phase space The answers are given by important theorems from the field of Hamiltonian dynamical systems. 

According to the Poincare-Birkhoff fixed point theorem all resonant tori break up for arbitrarily small perturbations. If the rotation number is p/q the perturbation leaves q pairs of hyperbolic and elliptic periodic orbits. The unstable hyperbolic orbits are embedded in a layer filled by aperiodic, chaotic orbits that do not stay on an invariant surface, but cover a finite non-zero volume of the phase space in a chaotic layer around the original resonant torus. The elliptic points, however, are wrapped around by new concentric tori that form islands of regular orbits within the chaotic band (Fig. 2.5). 

Although in a weakly time-dependent flow all resonant tori disappear together with some of the nearly resonant tori around them, the Kolmogorov-Arnold-Moser theorem ensures that infinitely many invariant surfaces survive a small perturbation. For sufficiently small e the remaining invariant surfaces formed by quasiperiodic orbits, so called KAM tori, still occupy a non-zero volume of the phase space. The condition for a torus to survive a given perturbation is that its rotation number should be sufficiently far from any rational number so that the inequality... 

Thus some of the fluid elements move on aperiodic chaotic trajectories and others on quasiperiodic orbits. The quasiperiodic orbits are invariant surfaces in the phase space that form the boundaries of the chaotic layers and limit the motion of the chaotic trajectories. There is a similar structure around each elliptic periodic orbit resulting from broken resonant tori that are also surrounded by invariant tori forming isolated islands inside the chaotic region. 

In three-dimensional flows the velocity field cannot be defined through a streamfunction, therefore the advection of fluid elements does not have the simple Hamiltonian structure as in two dimensions. One significant result on mixing in three dimensions is related to the existence of invariant surfaces in steady inviscid flows (Arnold, 1965). The velocity field of such flows is a solution of the time-independent Euler equation... 

Assume that is repulsive. Since Tv-i is an invariant surface, motion precisely on Xy-i can never fall away. However, speaking somewhat loosely for the moment, a slight push along q will cause motion initially on tv, to roll away from the barrier top. The set of motions that will roll away most slowly are motions that are asymptotic to the repulsive manifold and the surface formed by these motions constitutes the multidimensional version of a sep-aratrix. As motion asymptotic to Tn,. i falls away, it will generate a surface embedded in the full phase space whose geometry is the direct product of the sphere and the real line, x that is, a hypercylinder. The dimension of this hypercylinder is thus IN - As we will see, this (2N - 2)-dimen-... 

A new approach to the perturbation theory of invariant surfaces. Communs Pure and Appl. Math., 18, (1965), 717-732. 

Draw schematically and explain the variation in groove angle formed at a symmetric tilt grain boundary with tilt angle 9. As the temperature increases, how does the relationship between and 9 change approximately Assume invariable surface energy. 

Surface reactions, agglomeration, and spinodal decomposition are known to yield fractal mass distributions but other processes, such as fracture, milling or etching, also form scale-invariant surfaces. ... 

Shape Descriptors. 1. Rapid Superposition of Dissimilar Molecules Using Geometrically Invariant Surface Descriptors. 

As a consequence of non-adiabaticity, the ions will not drift exactly along the longitudinal invariant surfaces, and this produces charge separation and a corresponding electric current density j. The latter arises from the variation in curvature of the longitudinal invariant surfaces in the perturbed state O. xhe resulting current pattern then produces a x... 

Liapunov characteristic exponents (LCE). Dissipative systems are characterized by the attraction of all trajectories passing through a certain domain toward an invariant surface or an attractor of lower dimensionality than the original space. 

Before analyzing Eq. (11), we shall briefly Illustrate the methods used and results for the corresponding deterministic system. In Eq. (11) (with 0=0), at the bifurcation point i.e. n = 0, one pair of eigenvalues is on the imaginary axis and their eigenvectors span a plane that is tangent at (u,v) = (0,0) to an invariant surface ... 

Then, in the extended phase space the direct product of the phase space and the parameter space) near the origin there exists a uniquely defined -smooth invariant surface of the form p = 0(a ), V (0) = 0, such that each its intersection with the plane p = constant consists of a set of closed orbits of the system (11.5.17), lying in a neighborhood of the origin at the given p. 

For instance, assuming the conditions of the theorem hold, it may happen that the invariant surface is given by the equation /x = 0. This means that all trajectories near the origin are closed, i.e. the equilibrium state at the origin is a center, whereas the system has no small closed orbits aroimd the origin at /X 0. Thus, the equilibrium state may only lose its stability without giving birth to a limit cycle at the instance how it occurs, for example, in the equation X + /XX + X 4- x = 0. 

Sacker, R. [1964] On invariant surfaces and bifurcations of periodic solutions of ordinary differential equations, IMM-NUY 333, New York State University. 

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