Exchange of Stabilities in Autonomous Systems

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Abstract
Equilibrium solutions y = ϕ(x) of an autonomous system of differential equations, depending on a parameter x, are considered. Bifurcation of a second family of solutions y = ψ(x) and exchange of stabilities is supposed to occur at (x,y) = (0,0). Considering x as slowly varying leads to a singularly perturbed initial‐value problem whose reduced path encounters a point of bifurcation. Rigorous asymptotic estimates are found for the difference between the (unique) solution of the full problem and that solution of the reduced problem which proceeds along stable segments of the reduced path.

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