Scope of the stability, perturbation and chaos statements: conditional model deductions, the limits of finite-time and numerical evidence, and the empirical qualifications

This remark records what the page's statements do and do not establish. It adds no new claim of its own.

1. Model deductions only. The force-model results of the page are conditional deductions about the framework class of The model vocabulary of stability and chaos: the associated flow of a declared force model, equilibria, periodic and homoclinic orbits, and invariant sets: a force model with stated masses, configuration domain, regularity and — where relevant — a time-shift-invariant periodic restriction. The two-rotor KAM result instead adopts a reduced angle-action Hamiltonian on a torus phase domain. The pulsed horseshoe result is a full one-label force model on ${\mathbb{R}}^3$; its axial pulse dynamics is embedded by explicitly adopted transverse inverted-oscillator forces. The conclusions about equilibria, tori, horseshoes and recurrence are properties of these declared models. Continuous-time periodic-orbit, limit-cycle and homoclinic vocabulary is used for autonomous models; periodic motions of a driven model are treated by its stroboscopic map. The energy and variational theorems, the pendulum check and the stable-without-attraction harmonic counterexample use declared mechanical chart data: an open $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$, a $C^2$ positive-definite kinetic metric $M$, and a $C^2$ potential $U$. The growth and isotropic-oscillator checks are declared force models of the framework. None asserts that nature realizes the model, and none uses a measured value: couplings, energy scales, pulse parameters and the thresholds $c$ and ${\delta }_0$ are model data; the page has no measurement, calibration, sampling or uncertainty budget.

2. Stability is not attraction, and one notion is not another. Energy stability and instability of the mechanical model at an equilibrium: the strict minimum is Lyapunov stable, a negative Hessian eigenvalue gives linear instability, a negative definite potential Hessian gives nonlinear instability, and exact energy conservation rules out attraction establishes Lyapunov stability of a strict minimum, linear instability from a negative Hessian eigenvalue, nonlinear instability from a negative definite potential Hessian, and the absence of asymptotic stability under exact energy conservation. A zero Hessian eigenvalue alone gives no stability conclusion; strict minima remain covered even when their Hessians are degenerate. For a mixed saddle (a negative eigenvalue together with oscillatory directions), only the linear-instability statement is made. The harmonic-oscillator counterexample shows that the page's stability vocabulary cannot be collapsed.

3. Linearisation has separate local and finite-time conclusions. Linearisation along a motion of the mechanical model: the flow derivative solves the variational equation, with Lipschitz dependence, the constant-metric second-order form, and the conditional finite-time error bound of the linear approximation gives finite-time flow derivative, Lipschitz and linear-approximation estimates under its declared ball and remainder hypotheses. By contrast, Hyperbolic linearisation of an equilibrium of the model: exponential dichotomy of the linearised flow, local exponential asymptotic stability in the stable case, and instability in the saddle case gives all-time exponential estimates for the linearized flow and, under the hyperbolic spectral hypotheses, local exponential asymptotic stability or nonlinear instability; it gives no centre case or global conclusion. A finite-time growth rate is attached to a state, tangent vector and interval (Finite-time tangent growth rates and Lyapunov exponents of the model's flow: the rate per unit time of a transported deviation, with its existence convention and its limits); positive growth is not a definition of chaos. At an equilibrium where the field is $C^1$ nearby, smooth chart changes conjugate derivative matrices and preserve their spectrum, while norms and finite-time growth bounds can depend on the chart.

4. Chaos is a proved property of a stated map and set. A smooth periodically forced one-label Cartesian model with a transverse homoclinic horseshoe: the physical stroboscopic map has a compact invariant set conjugate to the shift, chaotic in the stated sense proves the four properties of Chaos in the stated sense for maps of the model: topological mixing, dense periodic points and sensitive dependence, as one precise list among several for the full model's stroboscopic map on an explicitly constructed compact invariant set contained in its invariant axis, by transferring the supplied shift conjugacy. Its full six-dimensional fixed points are hyperbolic because the two transverse inverted-oscillator modes have multipliers off the unit circle; the coded homoclinic intersections are transverse in the full phase space by the product construction. This is existence for the declared pulsed model under its small-pulse condition, not a claim about all driven oscillators, every separatrix, Hénon-Heiles energies or any numerical portrait. Quantified trajectories and plotted sections are computations under chosen parameters, not experiments or proofs. The page has no experiment item and reports no observed data.

5. KAM persistence is a positive-measure conditional statement. The two-rotor model: a positive-measure family of invariant Lagrangian tori with Diophantine frequencies for sufficiently small coupling, with the explicit units and the stated scope gives, for the adopted two-rotor potential and sufficiently small coupling, a positive-measure set of Diophantine frequencies whose tori persist, with quasi-periodic motions on them and explicit unit restoration. It says nothing about the complement of that set, about resonant gaps, about all initial data, about chaotic diffusion, or about a laboratory rotor. The threshold $\epsilon \le c{\alpha }^2$ is a model-data condition; the constant $c$ is not computed numerically and no empirical smallness is asserted.

6. Recurrence needs finite volume and invariance. Poincare recurrence for the model's Hamiltonian flow on a bounded invariant set of finite phase volume: almost every state of any Borel subset returns to it infinitely often assumes a bounded flow-invariant Borel set with compact closure in the potential phase domain and a flow complete on that set. Its proof extends the suppliers' local phase-volume preservation by a finite cover and time subdivision. The conclusion is almost-everywhere with respect to phase volume and gives no return time, rate or individual-state statement. Free and unconfined models fail the finite-volume hypothesis, and the theorem says nothing about them or thermodynamic irreversibility.

7. Choice conventions. The horseshoe theorem assumes the Axioms of Countable and Dependent Choice, inherited from its supplier, and the recurrence theorem assumes Countable Choice, inherited from Liouville's theorem for a potential presentation: the phase-volume form ${\omega }^{3N}\mathrm{/}(3N)!$ and the Lebesgue measure in each canonical chart are preserved by the Hamiltonian evolution on its actual domains. The KAM theorem also assumes Countable Choice, inherited from its analytic and measure-theoretic suppliers. The other statements of the page are choice-free: their suppliers use finite-dimensional linear algebra, explicit constructions and finite estimates. No item reaches a recorded or unproved set-theoretic item.

8. Relation to the sources. The four complete works consulted for this page (Tong, Likharev, SICM and Stewart) supply the model formulations, worked examples and caveats; their informal stability and chaos discussions are replaced by the exact supplier statements, and their numerical portraits are treated as illustrations, never as evidence. The empirical status of any model on this page would require a separately specified apparatus, sampling model and statistical analysis, which this page does not provide.