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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.
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 ...
The Picard–Lefschetz formula describes the monodromy at a critical point. Suppose that f is a holomorphic map from an (k+1)-dimensional projective complex manifold to the projective line P 1. Also suppose that all critical points are non-degenerate and lie in different fibers, and have images x 1,...,x n in P 1. Pick any other point x in P 1.
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 (,).
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.
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]
Self-organized criticality (SOC) is a property of dynamical systems that have a critical point as an attractor.Their macroscopic behavior thus displays the spatial or temporal scale-invariance characteristic of the critical point of a phase transition, but without the need to tune control parameters to a precise value, because the system, effectively, tunes itself as it evolves towards ...
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