A functional analytic theory for differential equations on Banach spaces with slowly evolving parameters

TL;DR

The paper develops a rigorous, functional-analytic framework for fast–slow differential equations on Banach spaces with slowly evolving parameters, proving the existence and $C^{k}$-smoothness of attracting slow manifolds and a reduction map that semi-conjugates full dynamics to the slow flow. A Lyapunov–Perron contraction approach is used to construct slow manifolds and their reductions, extending Fenichel theory to infinite-dimensional fast subsystems. It provides a detailed multi-tiered regularity theory (from $C^{0,1}$ up to $C^{k}$) and a structured reduction (via $P_{\varepsilon}$) with invariant foliations, along with local applications to spatially extended models and a pathway to center-unstable generalizations. The results yield a rigorous toolkit for analyzing spatially extended systems with slowly varying parameters, with potential extensions to non-autonomous and PDE contexts.

Abstract

This paper provides a functional analytic approach to differential equations on Banach space with slowly evolving parameters. We develop a Fenichel-like theory for attracting subsets of critical manifolds via a Lyapunov-Perron method. This functional analytic approach to invariant manifold theory for fast-slow systems of differential equations has not been fully developed before, especially for the case that the fast subsystem lives on an infinite-dimensional Banach space. We provide rigorous functional analytic proofs for both the persistence of attracting critical manifolds as smooth slow manifolds, as well as the validity of slow manifold reduction near slow manifolds. Several aspects of our proofs are new in the literature even for the finite-dimensional case. The theory as developed here provides a rigorous framework that allows one (for example) to derive formal statements on the dynamics of biologically meaningful spatially extended models with slowly varying parameters.

Figures (4)

• Figure 1: The relationship between the (semi)flow $\varphi$ of the system and the reduction map $P$, for arbitrary $t \geq 0$.
• Figure 1: Suppose that $h \in BC^0(\overline{V}, \mathbb{X})$ parameterizes a slow manifold, and the mappings $\psi : \overline{V} \times \{ h \}\rightarrow C^0(\mathbb{R}, \overline{V})$ as well as $\phi: C^0(\mathbb{R}, \overline{V}) \rightarrow BC^0(\mathbb{R}, \mathbb{X})$ are as outlined in steps 1-3. Then we should have for every $\eta \in \overline{V}$ and $t \in \mathbb{R}$ that $x(t;(h(\eta),\eta)) = \phi(t;\psi(\cdot;\eta,h))$. The Figure displays this relationship as a commutative diagram. In particular, setting $t=0$ gives $h(\eta) = \phi(0;\psi(\cdot;\eta,h)) = \Lambda(h)(\eta)$, with $\Lambda$ defined as in step 3.
• Figure 1: Suppose that $h \in BC^1(\overline{V}, \mathbb{X})$ parameterizes a slow manifold, and the mappings $z : \overline{V} \times \{ Dh \} \rightarrow BC_{\gamma}^0 (\mathbb{R}, L(\mathbb{R}^n, \mathbb{R}^n))$ with $\gamma = -N_1(\| Dh \|_{\infty} + 1)$, as well as $\tilde{\Gamma}_{\eta}: BC_{\gamma}^0 (\mathbb{R}, L(\mathbb{R}^n, \mathbb{R}^n)) \rightarrow BC_{\gamma}^0(\mathbb{R},L(\mathbb{R}^n,\mathbb{X}))$ (for arbitrary $\eta \in \overline{V}$) are as outlined in step 1 and \ref{['rem:w-unique-bdd-sol']}. Then we should have for every $\eta \in \overline{V}$ and $t \in \mathbb{R}$ that $\partial_{\eta} x(t;\eta) = \tilde{\Gamma}_{\eta}(z({\mkern 2mu\cdot\mkern 2mu}; \eta, Dh))(t)$. The Figure displays this relationship as a commutative diagram. In particular, if $w=\partial_{\eta}x({\mkern 2mu\cdot\mkern 2mu};\eta)$, then $w = \Gamma_{\eta}(w) = \tilde{\Gamma}_{\eta}(z({\mkern 2mu\cdot\mkern 2mu};\eta,w))$ and setting $t=0$ gives $Dh(\eta) = w(0) = h^1(\eta)$, with $\Gamma_{\eta}$ and $h^1$ as defined in step 2.
• Figure 1: The relationship between the (semi)flow of the system and the reduction map $P$, for arbitrary $t \geq 0$. The reduction map $P$ is a topological semi-conjugation between the (semi)flow of the system and the slow flow on $S_h$ defined by $\mathbb{R} \times \overline{V} \ni (t; \eta) \mapsto (h(\psi(t; \eta, h)),\psi(t; \eta, h))$.

Theorems & Definitions (106)

• Theorem 1.1: Existence of smooth slow manifolds
• Theorem 1.2: Reduction map
• Lemma 2.1
• Proof 1
• Lemma 3.1
• Proof 2
• Remark 3.2
• Remark 3.3
• Remark 3.4
• Definition 3.5