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An attractor network is a type of recurrent dynamical network, that evolves toward a stable pattern over time.Nodes in the attractor network converge toward a pattern that may either be fixed-point (a single state), cyclic (with regularly recurring states), chaotic (locally but not globally unstable) or random (). [1]
Then one can use the Atiyah–Bott fixed-point theorem, of Michael Atiyah and Raoul Bott, to reduce, or localize, the computation of a GW invariant to an integration over the fixed-point locus of the action. Another approach is to employ symplectic surgeries to relate X to one or more other spaces whose GW invariants are more easily computed ...
These slopes arise from the linearizations of the stable manifold and unstable manifold of the fixed point. The unstable manifold of the fixed point in the attractor is contained in the strange attractor of the Hénon map. The Hénon map does not have a strange attractor for all values of the parameters a and b. For example, by keeping b fixed ...
Complex eigenvalues of an arbitrary map (dots). In case of the Hopf bifurcation, two complex conjugate eigenvalues cross the imaginary axis. In the mathematical theory of bifurcations, a Hopf bifurcation is a critical point where, as a parameter changes, a system's stability switches and a periodic solution arises. [1]
If the Jacobian matrix of a dynamical system at a fixed point is Hurwitz, then the fixed point is asymptotically stable. Markus-Yamabe conjecture asks if a similar result holds globally . Precisely, the conjecture states that if a continuously differentiable map on an n {\displaystyle n} -dimensional real vector space has a fixed point , and ...
Stability means that the trajectories do not change too much under small perturbations. The opposite situation, where a nearby orbit is getting repelled from the given orbit, is also of interest. In general, perturbing the initial state in some directions results in the trajectory asymptotically approaching the given one and in other directions ...
The birational point of view can afford to be careless about subsets of codimension 1. To have a moduli space as a scheme is on one side a question about characterising schemes as representable functors (as the Grothendieck school would see it); but geometrically it is more like a compactification question, as the stability criteria revealed.
A function need not have a least fixed point, but if it does then the least fixed point is unique. One way to express the Knaster–Tarski theorem is to say that a monotone function on a complete lattice has a least fixed point that coincides with its least prefixpoint (and similarly its greatest fixed point coincides with its greatest ...