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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 ...
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]
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.
Otherwise it is non-degenerate, and called a Morse critical point of . The Hessian matrix plays an important role in Morse theory and catastrophe theory, because its kernel and eigenvalues allow classification of the critical points. [2] [3] [4]
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 ...
Let (, , ) be a real Hilbert space, and let be an open neighbourhood of the origin in . Let : be a (+)-times continuously differentiable function with ; that is, + (;). Assume that () = and that is a non-degenerate critical point of ; that is, the second derivative () defines an isomorphism of with its continuous dual space by (,).
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