Scope of the coupled quadratic model: zero modes, degenerate eigenspaces, unstable quadratic directions and the limits of the approximation statements

Modal matrices, coefficients and phase norms use the fixed-unit numerical convention of The coupled quadratic oscillator model: mass metric, stiffness, generalized force, modal vocabulary and the small-oscillation model at a mechanical equilibrium. Configuration, velocity, modal amplitude, modal force and energy units are restored with its explicit scales. The finite-time error bound is in normalized phase coordinates, while the averaging radii and error constant have the rotating-amplitude unit and its horizon parameter has time units.

A zero mode of a coupled quadratic model is an index with modal stiffness ${\lambda }_i=0$ (The coupled quadratic oscillator model: mass metric, stiffness, generalized force, modal vocabulary and the small-oscillation model at a mechanical equilibrium clause (e)); the multiplicity of the value $0$ is $\dim \ker K$, and it counts the independent directions in which the quadratic model exerts no restoring force. By clause 2 of The free coupled quadratic model: global existence, the complete modal classification, mode energies, intrinsic degenerate mode subspaces, boundedness and the unstable direction the free motion in such a direction is affine, $z_i(t)=a_i+b_{i}t$, not oscillatory; for $b_i\mathrm{\ne }0$ it is unbounded on $[0,\mathrm{\infty })$. The classical instance is the translation mode of a chain or molecule: the linearisation of a translation-invariant model at a non-isolated minimum has a flat second-order direction, and the "mode" of zero frequency is the uniform motion of the whole system rather than a vibration. A constant generalized force component along a zero mode produces quadratic growth in time by clause 2(c) of The forced coupled quadratic model: modal responses, resonance, and boundedness for positive definite stiffness off resonance, the degenerate analogue of resonance; the model has no bounded steady response in that direction. Zero modes are therefore excluded whenever a statement about oscillations alone is intended: at a strict local minimum the stiffness is positive semidefinite and $\ker K$ can be nontrivial even at an isolated strict minimum, as $U(q)=q^4$ at $0$ shows; the Lyapunov stability of Small oscillations at a strict minimum: the linearized model, its normal modes and the Lyapunov stability of the nonlinear equilibrium remains valid in that case because it does not use positivity of the stiffness.

If a modal stiffness $\lambda$ has multiplicity $m>1$, the individual eigenvectors and hence the individual modal vectors $u_i$ of a chosen diagonalisation are not intrinsic: every orthonormal basis of the eigenspace gives an equally valid decoupling. What is intrinsic is the subspace $N_{\lambda }$ and its pure-$\lambda$ family of motions, exactly as clause 4 of The free coupled quadratic model: global existence, the complete modal classification, mode energies, intrinsic degenerate mode subspaces, boundedness and the unstable direction records: the pure-$\lambda$ motions are the solutions of $q^{\mathrm{\prime }\mathrm{\prime }}+\lambda q=0$ with values in $N_{\lambda }$, a family with $2m$ parameters, and for $\lambda >0$ all of them oscillate with the single frequency $\omega =\sqrt{\lambda }$. A statement about "the $i$-th mode" must therefore be read as a statement about a chosen basis vector, while the frequency, the subspace and the multi-parameter family are the basis-independent objects. Resonance of a degenerate family with a prescribed sinusoidal force is likewise basis-independent: it occurs when ${\mathrm{\Omega }}^2=\lambda$ and the modal force has a nonzero component in the corresponding modal eigenspace (equivalently, the mass-metric projection of $M^{-1}{J}_0$ onto $N_{\lambda }$ is nonzero) (clause 2 and the remarks of The forced coupled quadratic model: modal responses, resonance, and boundedness for positive definite stiffness off resonance).

An unstable (exponential) direction is an index with ${\lambda }_i<0$ (The coupled quadratic oscillator model: mass metric, stiffness, generalized force, modal vocabulary and the small-oscillation model at a mechanical equilibrium clause (e)). Its free motion is the hyperbolic family $a\cosh (\kappa t)+b\sinh (\kappa t)$ with $\kappa =\sqrt{-{\lambda }_i}$, which has an exponentially growing component as $t\to +\mathrm{\infty }$ when $a+b\mathrm{\ne }0$; a nonzero family with $a=-b$ decays at $+\mathrm{\infty }$, although growing families exist in this direction, and clause 6 of The free coupled quadratic model: global existence, the complete modal classification, mode energies, intrinsic degenerate mode subspaces, boundedness and the unstable direction proves accordingly that the rest point of the free quadratic model is not Lyapunov stable (Equilibria, Lyapunov stability and asymptotic stability of an autonomous first-order system). For a mechanical system in a chart at a critical point $q_0$ of the potential that is not a minimum — for instance a symmetric saddle or a strict maximum — the stiffness $K=D^{2}U(q_0)$ may have negative eigenvalues (The Hessian matrix and critical points of a scalar field), and the corresponding exponential directions of the linearised model exhibit the local saddle behaviour. The linear calculation by itself certifies nothing about the nonlinear system: the nonlinear instability of $(q_0,0)$ follows from A hyperbolic equilibrium whose linearisation has an eigenvalue with positive real part is unstable only when the derivative $DG(q_0,0)$ of the mechanical field (The mechanical first-order field, Lagrangian and energy of a configuration-dependent kinetic metric in a chart) is hyperbolic — no eigenvalue on the imaginary axis, in particular no zero mode — and has an eigenvalue with positive real part. When a zero mode or an imaginary eigenvalue is present, the linearisation is not hyperbolic and the cited theorem does not cover that case. This is a limitation of that local supplier, not a claim that nonlinear terms can overturn the instability associated with a positive-real eigenvalue of a $C^1$ field; the more general linearisation-instability argument is not supplied here. Applicability of any such model instability to a physical system additionally requires its declared model and empirical conditions; no observation is asserted by this item.

Three separate qualifications belong to the results of this page.

• The quadratic model is exact only for quadratic data. The model $M{q}^{\mathrm{\prime }\mathrm{\prime }}+Kq=J$ is a declared model class; it coincides with a mechanical chart system $G_{\mathrm{k}\mathrm{i}\mathrm{n}}(q)q^{\mathrm{\prime }\mathrm{\prime }}+\cdots =-\mathrm{\nabla }U(q)$ only when the kinetic metric is constant and the potential is quadratic. The small-oscillation model at an equilibrium is a first-order Taylor model, and Small oscillations at a strict minimum: the linearized model, its normal modes and the Lyapunov stability of the nonlinear equilibrium identifies it with the derivative linearization without asserting that it approximates the nonlinear motion.

• The linearization error bound is finite-time and small-data. The estimate of Finite-time error control for the small-oscillation approximation: the quadratic remainder of the mechanical field and the variation-of-constants bound, $\parallel z(t)-y(t){\parallel }_2\le C{\delta }^{2}t{e}^{(\parallel A\parallel +2L)t}$ on $[0,T]$ under $\delta e^{LT}<r_0$, controls the discrepancy of the nonlinear and linear motions only for the stated horizon, datum size and regularity; the exponential factor grows with $T$, and no all-time closeness, stability or instability conclusion follows from it.

• The averaging estimate is a comparison, not an identity. The order-$\epsilon$ bound of Averaging for a weakly damped oscillator: the rotating-frame amplitude, the averaged exponential energy decay and the order-$\epsilon$ bound on the $O(1\mathrm{/}\epsilon )$ time scale holds on the $O(1\mathrm{/}\epsilon )$ horizon under the displayed smallness, trapping and constant hypotheses, and compares the exact weakly damped model with a different, averaged model; it is not an exact decay law of the damped oscillator, and its constants $M,L,K$ depend on the chosen ball and horizon.

Beyond the mathematical qualifications, the parameters of the models of this page — masses, stiffnesses, damping coefficients, drive amplitudes and frequencies — are model data. A value of such a parameter taken from a calibration experiment is an empirical premise carrying the apparatus conditions and the measurement uncertainty, and no claim of this page identifies a model frequency with an observed spectral line without a specified measurement model, which is not supplied here.