Eigenvalues, critical case

It is now shown how the abrupt changes in the eigenvalue distribution around the central critical point relate to changes in the classical mechanics, bearing in mind that the analog of quantization in classical mechanics is a transformation of the Hamiltonian from a representation in the variables pR, p, R, 0) to one in angle-action variables (/, /e, Qr, 0) such that the transformed Hamiltonian depends only on the actions 1r, /e) [37]. Hamilton s equations diR/dt = (0///00 j), etc.) then show that the actions are constants of the motion, which are related to the quantum numbers by the Bohr correspondence principle [23]. In the present case,... 

As in previous simpler cases with fewer DOFs, everything begins with equilibrium points (critical points in mathematical language). We straightforwardly generalize earlier results. Let H = H p, ..., pn,q, , qn) = H x) be the Hamiltonian and suppose that F is an equilibrium point, V//(x) p = 0. We linearize motion around P, and we analyze the eigenvalues of the 2n x 2n matrix M, Eq. (4). If we have the following situation ... 

A [3, —3 critical point. In this case all three eigenvalues are negative and hence there is a local maximum in... 

The local canonical curvatures can be compared to a reference curvature parameter b [156,199]. For each point r of the molecular surface G(a) a number X = x(r,b) is defined as the number of local canonical curvatures [the number of eigenvalues of the local Hessian matrix H(r) that are less than this reference value b. The special case of b=0 allows one to relate this cla.ssification of points to the concept of ordinary convexity. If b=0, then p is the number of negative eigenvalues, also called the index of critical point r. As mentioned previously, in this special case the values 0, 1, or 2 for p(r,0) indicate that at the point r the molecular surface G(a) is locally concave, saddle-type, or convex, respectively [199]. 

An eigenvalue and its associated eigenvector of the Hessian of p (a principal curvature and its associated axis) at a critical point define a onedimensional system. If the eigenvalue or curvature is negative, then p is a maximum at the critical point on this axis and a gradient vector will approach and terminate at this point from both its left- and right-hand side as illustrated in Fig. 2.6 for the case (1, — 1), a system of rank 1 and signature... 

The eigenvectors u, span the space in the neighbourhood of a critical point and, by fixing the values of the constants c,-, one chooses some initial point on a particular trajectory. One may then follow the succession of points r(s) and determine the trajectory by varying the path parameter s. If Cj = C3 = 0, for example, then r(s) lies on the axis defined by Uj. If the corresponding eigenvalue < 0, then the coefficient e decreases as s increases. Hence the trajectory of the points r(s) approaches r,. along u, and the trajectory terminates at when s = + 00 as illustrated in Fig. 2.6 for the case (1, — 1). 

The two eigenvectors associated with the two positive eigenvalues of a (3, +1) critical point also define a unique surface but in this case all of the trajectories in this surface originate at the critical point and p(r) is a minimum at Tj. The third eigenvector defines a unique axis and the two trajectories on this axis terminate at r. Thus the phase portraits of a (3, + 1) critical point are the reverse of those found for a (3, — 1) critical point. 

Now, the problem (12-225)-( 12-228) is an eigenvalue problem. In this case, given a2, a nonzero solution will exist only a for certain value of Ra. Because we have already set a = 0, the critical value Racrit is then the minimum of Ref for all possible a. 

The eigenvalue problem that must be solved in this case to determine the conditions for instability of the fluid layer is (12-289)-(12-292) and (12-295). Because we have already put a = 0, we pick a and Bi and find the corresponding value of Ma such that nontrivial solutions exist for f(z) and h(z). The critical condition for instability is thus to determine the minmum Ma as a function of a (with Bi being fixed for any particular configuration). 

In a general case of equation (5.62), the necessary condition on the Hopf bifurcation is that the stability matrix has, for the critical value of the control parameter c = c0, one pair of purely imaginary eigenvalues, whereas the remaining eigenvalues must have a non-zero real part. 

Discrete-time Markov processes are a third type of problem we shall discuss. One of the challenges in this case is to compute the correlation time of such a process in the vicinity of a critical point, where the correlation time goes to infinity, a phenomenon called critical slowing down. Computationally, the problem amounts to the evaluation of the second largest eigenvalue of the Markov matrix, or more precisely its difference from unity. The latter goes to zero as the correlation time approaches infinity. 

There is thus a critical value of the diffusion coefficient ratio, = 8 > 1, above which H k ) > 0 over a range of wavenumbers < k < /fema. Perturbations with wavenumbers within this range will grow because the associated temporal eigenvalues are positive. Perturbations with wavenumbers outside this range will decay exponentially to the homogeneous steady state. Figure 11 shows the variation oiH(k ) with k for the cases of 8 > 8, and 8 <... 

For = 1 the eigenvalues lie on the unit circle. Since the parameter k determines whether the fixed point is stable or not, it is called the critical parameter. Analog to that, the value A = 1 is called the critical (or bifurcation) value of the parameter k. Figures 1 and 2 show the orbit of the dynamic system for a stable and an unstable case, respectively. Nevertheless, in both cases the Xf = (0 0) is a fixed point. 

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