Entry–Exit Functions in Fast–Slow Systems with Intersecting Eigenvalues

• Published in: Journal of dynamics and differential equations Vol. 37; no. 1; pp. 559 - 576
• Main Authors: Kaklamanos, Panagiotis, Kuehn, Christian, Popović, Nikola, Sensi, Mattia
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.03.2025; Springer Nature B.V; Springer Verlag
• Subjects: Applications of Mathematics; Differential equations; Eigenvalues; Fields (mathematics); Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Partial Differential Equations; Real variables; Stability; Transcritical bifurcation; Fast–slow systems; Ordinary differential equations; Entry–exit function; Geometric singular perturbation theory; Secondary 34A26; 34C60; Delayed loss of stability; Primary 34C23; 34E15
• ISSN: 1040-7294; 1572-9222
• DOI: 10.1007/s10884-023-10266-2

We study delayed loss of stability in a class of fast–slow systems with two fast variables and one slow one, where the linearisation of the fast vector field along a one-dimensional critical manifold has two real eigenvalues which intersect before the accumulated contraction and expansion are balanced along any individual eigendirection. That interplay between eigenvalues and eigendirections renders the use of known entry–exit relations unsuitable for calculating the point at which trajectories exit neighbourhoods of the given manifold. We illustrate the various qualitative scenarios that are possible in the class of systems considered here, and we propose novel formulae for the entry–exit functions that underlie the phenomenon of delayed loss of stability therein.