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For a function of n variables, the number of negative eigenvalues of the Hessian matrix at a critical point is called the index of the critical point. A non-degenerate critical point is a local maximum if and only if the index is n, or, equivalently, if the Hessian matrix is negative definite; it is a local minimum if the index is zero, or ...
A less trivial example of a degenerate critical point is the origin of the monkey saddle. The index of a non-degenerate critical point of is the dimension of the largest subspace of the tangent space to at on which the Hessian is negative definite.
In systems in equilibrium, the critical point is reached only by precisely tuning a control parameter. However, in some non-equilibrium systems, the critical point is an attractor of the dynamics in a manner that is robust with respect to system parameters, a phenomenon referred to as self-organized criticality. [6]
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The second statement is that when f is a Morse function, so that the singular points of f are non-degenerate and isolated, then the question can be reduced to the case n = 1. In fact, then, a choice of g can be made to split the integral into cases with just one critical point P in each.
The degeneracy of these critical points can be unfolded by expanding the potential function as a Taylor series in small perturbations of the parameters. When the degenerate points are not merely accidental, but are structurally stable , the degenerate points exist as organising centres for particular geometric structures of lower degeneracy ...
Critical point may refer to: Critical phenomena in physics; Critical point (mathematics), in calculus, a point where a function's derivative is either zero or nonexistent; Critical point (set theory), an elementary embedding of a transitive class into another transitive class which is the smallest ordinal which is not mapped to itself
Figure 6. Imperfect transcritical bifurcation phase portraits. Five values of r are shown, given relative to the two critical points. Note that the y-intercept value is the same as h, or the magnitude of the imperfection. The green and blue points are stable, while the green red and magenta are unstable. The black dots indicate semistable fixed ...