Structural stability of a dynamical system near a non-hyperbolic fixed point

Abstract

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 renormalisation group analysis of the 4-dimensional weakly self-avoiding walk and the 4-dimensional n-component $|\phi {|}^4$ spin model.

31 pages, to appear in Ann. Henri Poincare

References in corpus (3)

• Logarithmic correction for the susceptibility of the 4-dimensional weakly self-avoiding walk: a renormalisation group analysis
• Scaling limits and critical behaviour of the 4-dimensional n-component $|\phi {|}^4$ spin model
• Critical two-point function of the 4-dimensional weakly self-avoiding walk

Cited by in corpus (15)

• Logarithmic correction for the susceptibility of the 4-dimensional weakly self-avoiding walk: a renormalisation group analysis
• Scaling limits and critical behaviour of the 4-dimensional n-component $|\phi {|}^4$ spin model
• Introduction to a renormalisation group method
• Critical two-point function of the 4-dimensional weakly self-avoiding walk
• Critical exponents for long-range O(n) models below the upper critical dimension
• A renormalisation group method. III. Perturbative analysis
• A renormalisation group method. IV. Stability analysis
• A renormalisation group method. V. A single renormalisation group step
• Critical correlation functions for the 4-dimensional weakly self-avoiding walk and n-component $|\phi {|}^4$ model
• Finite-order correlation length for 4-dimensional weakly self-avoiding walk and $|\phi {|}^4$ spins
• Four-dimensional weakly self-avoiding walk with contact self-attraction
• Three-dimensional tricritical spins and polymers
• Renormalisation group analysis of 4D spin models and self-avoiding walk
• Percolation transition for random forests in $d\ge 3$
• A renormalization group method by harmonic extensions and the classical dipole gas