An explicit graph transform on the state-dependent-delay solution manifold

Definitions and mathematical statement

The source requests alternative proofs of infinite-dimensional local invariant manifolds at equilibria of differential equations with state-dependent delay, specifically by graph transform or the Lyapunov--Perron method for local stable manifolds. Its statement that no such result exists is a source-era literature assertion, not a verified present-day claim.

A standard framework making the target precise is the following. Fix $r>0$, let $\mathcal C=C([-r,0],{\mathbb{R}}^m)$, $\mathcal C^1=C^1([-r,0],{\mathbb{R}}^m)$, and let $f:U\subseteq\mathcal C^1\to{\mathbb{R}}^m$. For a solution define $x_t(\theta)=x(t+\theta)$ and consider \[ x'(t)=f(x_t),\qquad X_f=\{\phi\in U:\phi'(0)=f(\phi)\}. \] The compatibility manifold $X_f$ is the relevant phase space. Assume the differentiability, extended-derivative and well-posedness hypotheses ensuring that it is a $C^1$ manifold and that the equation generates a local $C^1$ semiflow there. The aid specifically uses extension of $Df(\phi)$ to continuous histories; a usual sufficient version requires a continuous linear extension $D_ef(\phi):\mathcal C\to{\mathbb{R}}^m$ and joint continuity of $(\phi,\psi)\mapsto D_ef(\phi)\psi$. The complete precise hypotheses are not supplied in the record.

Let $\widehat a$ be the constant history of an equilibrium, $f(\widehat a)=0$. The tangent space is \[ T_{\widehat a}X_f =\{\chi\in\mathcal C^1:\chi'(0)=Df(\widehat a)\chi\}. \] Assume the linearized semiflow is hyperbolic and has a splitting $E^s\oplus E^u$, with $E^s$ infinite dimensional and $E^u$ finite dimensional, with exponential contraction and expansion respectively. The desired alternative construction is a local stable manifold in $X_f$, locally positively invariant while trajectories remain in the chosen neighborhood, consisting of nearby compatible histories with exponentially convergent forward trajectories toward $\widehat a$.

The aid reports a graph-transform construction after a compatibility chart flattens $X_f$ at $\widehat a$. For a suitable local time map $\Phi_{t_0}$, the charted map has the form \[ (s,u)\longmapsto(As+g(s,u),Bu+h(s,u)),\qquad (s,u)\in E^s\oplus E^u, \] where $A$ contracts, $B$ is invertible with contracting inverse, and the nonlinear terms are sufficiently small locally. It constructs a Lipschitz graph \[ W^s_{\mathrm{loc}} =\{\Psi(s,\eta(s)):s\in V^s\subseteq E^s\},\qquad \eta(0)=0, \] for a chart $\Psi$, with stable contraction, preservation of a Lipschitz cone and uniqueness controlled by backward contraction in the unstable block. The summary claims explicit graph-transform inequalities and a uniform construction, but does not supply those inequalities or constants; none are invented here.

The reported ingredients are a narrow-bump verification of the compatibility chart using the extended derivative, and a radial Lipschitz cutoff on the local map. The aid emphasizes avoiding the modified global semiflows used in a known center-stable proof. It reports a Lipschitz local stable graph at a hyperbolic equilibrium; it does not supply a $C^1$ conclusion, a full center-manifold theorem, or a Lyapunov--Perron construction for all state-dependent delay equations. Passage from discrete-time graph invariance to the full local semiflow and the characterization by exponential convergence remain part of the proof obligations for review.

Short English statement

Construct local stable manifolds for state-dependent delay equations directly through graph transform or Lyapunov--Perron methods. The aid reports a graph transform on a flattened compatibility manifold, giving an infinite-dimensional Lipschitz stable graph under hyperbolicity and small-nonlinearity assumptions. The original historical literature claim and the stronger smoothness or center-manifold questions remain separate.

Sources

• Local UnsolvedMath registry, permanent ID $20001441$, code catalogue entry; canonical aim-dynamical-systems-notes.json, notes[98].
• AIM, "Low dimensional structures in dynamical systems with variable time lags," item (9), https://aimath.org/WWN/variabletimelag/variabletimelag.pdf. The supplied verification reports exact wording on page 2 of the two-page PDF, with no substantive OCR loss or spillover. This draft does not claim PDF inspection.
• The supplied aid refers to a known center-stable proof with modified global semiflows but provides no author, title or full authoritative citation. No separate literature citation is supplied for the graph-transform construction.
• Research attempt 1: "partial_result," "partially_solved," rigor label "proved," novelty confidence "low." The research aid is machine generated, not peer reviewed; independently verify before citation.