Extending the zero-derivative principle for slow–fast dynamical systems

Abstract

Document Version Peer reviewed version Link back to DTU Orbit Citation (APA):Benoît, E., Brøns, M., Desroches, M., & Krupa, M. (2015). Extending the zero-derivative principle for slow-fast dynamical systems. Zeitschrift fuer Angewandte Mathematik und Physik, 66(5), 2255-2270. DOI: 10.1007/s00033-015-0552-8 EXTENDING THE ZERO-DERIVATIVE PRINCIPLE FOR SLOW-FAST DYNAMICAL SYSTEMS ERIC BENOÎT, MORTEN BRØNS, MATHIEU DESROCHES, AND MARTIN KRUPAAbstract. Slow-fast systems often possess slow manifolds, that is invariant or locally invariant submanifolds on which the dynamics evolves on the slow time scale. For systems with explicit timescale separation the existence of slow manifolds is due to Fenichel theory and asymptotic expansions of such manifolds are easily obtained. In this paper we discuss methods of approximating slow manifolds using the so-called zero-derivative principle. We demonstrate several test functions that work for systems with explicit time scale separation including ones that can be generalized to systems without explicit timescale separation. We also discuss the possible spurious solutions, known as ghosts, as well as treat the Templator system as an example.

Local orthogonal rectification: Deriving natural coordinates to study flows relative to manifolds

“…We hope to take advantage of this fact in future work. It is natural to wonder whether the LOR flow is topologically conjugate to the flow induced by (1). It can be computationally difficult to find a codomain on which Ψ is a homeomorphism.…”

“…This example illustrates that when σ is a Frenet curve, K σ reduces to the matrix of curvatures. If we take σ(η) to be a trajectory of (1), then N f (η, ξ) = 0 andη| ξ=0 = 1.…”

“…In earlier work [9], we suggest that rivers in planar systems, which are centrally organizing orbits with no apparent source, are trajectories for which curvature and torsion vanish simultaneouly. Previous work has established the zero curvature set plays an organizational role in planar systems [1,4,5], we use the LOR frame to study where the zero curvature set is invariant. One advantage of this definition, is that it generalizes easily to arbirary dimension.…”

“…We intend to formalize the connection between rivers and canards in future work. Additionally, it would be interesting to investigate how the LOR frame, specialized to the context of fast-slow systems, relates to other geometrically-motivated manifold approximation schemes that apply in that setting, such as computational singular perturbation (CSP) methods [8,18] or the zero derivative principle [1], and whether LOR provides a way to extend ideas like CSP beyond the fast-slow context. For the time being, we hope that the computations and sketch of ideas in this section serve as an amuse-bouche to demonstrate how the LOR frame can highlight nontrivial features of phase space.…”

Inflection, Canards and Folded Singularities in Excitable Systems: Application to a 3D FitzHugh–Nagumo Model

“…In the 2D case, this leads to the possibility of unfolding canards associated to the excitability threshold. More generally, curvature-based methods [23] and similar approaches such as the zero-derivative method [9] can approximate Fenichel slow manifolds to any order in ε in general N -dimensional systems (even without explicit timescale separation). However, they require a substantial computational effort and they also produce many spurious solutions known as ghosts; see [9] for details.…”