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Structural stability of a dynamical system near a non-hyperbolic fixed point. (English) Zbl 1347.37041
Summary: We prove structural stability under perturbations for a class of discrete-time dynamical systems near a non-hyperbolic fixed point. We reformulate the stability problem in terms of the well-posedness of an infinite-dimensional nonlinear ordinary differential equation in a Banach space of carefully weighted sequences. Using this, we prove existence and regularity of flows of the dynamical system which obey mixed initial and final boundary conditions. The class of dynamical systems we study, and the boundary conditions we impose, arise in a renormalization group analysis of the 4-dimensional weakly self-avoiding walk and the 4-dimensional \(n\)-component \(|\varphi|^4\) spin model.

MSC:
37C20 Generic properties, structural stability of dynamical systems
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
82B28 Renormalization group methods in equilibrium statistical mechanics
81T17 Renormalization group methods applied to problems in quantum field theory
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