Appendix 1: Proof of the main result
1.1 First-order autonomous form
By the nondegeneracy of M, the matrices \(M_{11}\) and \(M_{22}\) are necessarily invertible, which enables us to split (1) in the form
$$\begin{aligned} \left[ M_{11}-M_{12}M_{22}^{-1}M_{21}\right] \ddot{x}= & {} -F_{1}-M_{12}M_{22}^{-1}F_{2},\\ \left[ M_{22}-M_{21}M_{11}^{-1}M_{12}\right] \ddot{y}= & {} -F_{2}-M_{21}M_{11}^{-1}F_{1}. \end{aligned}$$
The nondegeneracy of M also implies that the two matrices on the left-hand side of this system must be invertible, leading to the explicit second-order dynamical system
$$\begin{aligned} \ddot{x}= & {} M_{1}^{-1}\left( x,\frac{y}{\epsilon },t;\epsilon \right) Q_{1}\left( x,\dot{x},\frac{y}{\epsilon },\dot{y},t;\epsilon \right) ,\nonumber \\ \ddot{y}= & {} M_{2}^{-1}\left( x,\frac{y}{\epsilon },t;\epsilon \right) Q_{2}\left( x,\dot{x},\frac{y}{\epsilon },\dot{y},t;\epsilon \right) , \end{aligned}$$
(64)
with \(M_{i}\) and \(Q_{i}\) defined in (5).
In order to convert this system into a first-order autonomous system, we first introduce a phase variable \(\varphi \in \mathcal {C}\) such that
$$\begin{aligned} \mathcal {C}=\left\{ \begin{array}{ll} S^{1}, &{}\quad M_{i},Q_{i}\text { are periodic in }t,\\ \mathbb {T}^{k}, &{}\quad M_{i},Q_{i}\text { are quasiperiodic with }\\ &{}\quad k \hbox { independent frequencies in }t,\\ {[}a,b], &{}\quad M_{i},Q_{i}\text { are aperiodic in }t.\\ \end{array}\right. \end{aligned}$$
We then let
$$\begin{aligned} v=\dot{x},\quad w=\dot{y}, \end{aligned}$$
and rewrite Eq. (64) as a first-order autonomous system on the extended phase space \(\mathcal {P}=\mathbb {R}^{s}\times \mathbb {R}^{s}\times \mathbb {R}^{f}\times \mathbb {R}^{f}\times \mathcal {C}\) in the form
$$\begin{aligned}&\dot{x} = v,\\&\dot{v} = M_{1}^{-1}\left( x,\frac{y}{\epsilon },\varphi ;\epsilon \right) Q_{1}\left( x,v,\frac{y}{\epsilon },w,\varphi ;\epsilon \right) ,\\&\dot{y} = w,\\&\dot{w} = M_{2}^{-1}\left( x,\frac{y}{\epsilon },\varphi ;\epsilon \right) Q_{2}\left( x,v,\frac{y}{\epsilon },w,\varphi ;\epsilon \right) ,\\&\dot{\varphi } = \omega , \end{aligned}$$
where
$$\begin{aligned} \mathcal {\omega }={\left\{ \begin{array}{ll} \omega _{1}, &{}\quad \mathcal {C}=S^{1},\\ \left( \omega _{1},\ldots ,\omega _{k}\right) , &{}\quad \mathcal {C}=\mathbb {T}^{k},\\ 1, &{}\quad \mathcal {C}=[a,b]. \end{array}\right. } \end{aligned}$$
1.2 Timescale separation
Up to this point, the splitting \(q=(x,y)\) has been arbitrary. We now seek conditions under which the x-degrees of freedom serve as coordinates for a reduced-order model. For such a reduced-order model to capture effectively the long-term system dynamics, we require the y variables to become enslaved to the x variables and to the phase variable \(\varphi \) over a timescale that is an order of magnitude faster than the characteristic timescale of the reduced-order model (cf. the requirement (R2) in the Introduction). To this end, we introduce a characteristic fast timescale \(\tau \) by letting \(t=\epsilon \tau ,\) with small, non-dimensional parameter \(0<\epsilon \ll 1.\) Denoting differentiation with respect to \(\tau \) by prime, we obtain the rescaled equations
$$\begin{aligned}&x^{\prime } = \epsilon v,\nonumber \\&v^{\prime } = \epsilon M_{1}^{-1}\left( x,\frac{y}{\epsilon },\varphi ;\epsilon \right) Q_{1}\left( x,v,\frac{y}{\epsilon },w,\varphi ;\epsilon \right) ,\nonumber \\&\varphi ^{\prime } = \epsilon \omega ,\nonumber \\&y^{\prime } = \epsilon w,\nonumber \\&W^{\prime } = \epsilon M_{2}^{-1}\left( x,\frac{y}{\epsilon },\varphi ;\epsilon \right) Q_{2}\left( x,v,\frac{y}{\epsilon },w,\varphi ;\epsilon \right) .\nonumber \\ \end{aligned}$$
(65)
In this new scale, the evolution in the (y, w) variables should be taking place at an \(\mathcal {O}(1)\) speed with respect to \(\epsilon ,\) whereas the (x, v) variables should experience an \(\mathcal {O}(\epsilon )\) rate of change. By the structure of system (65), this timescale separation will only arise if we localize y by letting \(y=\epsilon \eta \). With this scaling, we obtain the equations
$$\begin{aligned}&x^{\prime } = \epsilon v,\nonumber \\&v^{\prime } = \epsilon M_{1}^{-1}\left( x,\eta ,\varphi ;\epsilon \right) Q_{1}\left( x,v,\eta ,w,\varphi ;\epsilon \right) ,\nonumber \\&\varphi ^{\prime } = \epsilon \omega ,\nonumber \\&\eta ^{\prime } = w,\nonumber \\&w^{\prime } = \epsilon M_{2}^{-1}\left( x,\eta ,\varphi ;\epsilon \right) Q_{2}\left( x,v,\eta ,w,\varphi ;\epsilon \right) . \end{aligned}$$
(66)
To ensure that w also varies at \(\mathcal {O}(1)\) speeds for small enough \(\epsilon \), the function \(\epsilon M_{2}^{-1}\left( x,\eta ,\varphi ;\epsilon \right) Q_{2}(x,v,\eta , w,\varphi ;\epsilon )\) must have a smooth, \(\mathcal {O}(1)\) limit as \(\epsilon \rightarrow 0.\) We, therefore, must require the function
$$\begin{aligned}&P_{2}\left( x,v,\eta ,w,\varphi ;\epsilon \right) \\&\quad =\epsilon M_{2}^{-1}\left( x,\eta ,\varphi ;\epsilon \right) Q_{2}\left( x,v,\eta ,w,\varphi ;\epsilon \right) \end{aligned}$$
to have a smooth limit at \(\epsilon =0\), defined by a smooth function
$$\begin{aligned} P_{2}(x,v,\eta ,w,\varphi ;0):=\lim _{\epsilon \rightarrow 0}P_{2}\left( x,v,\eta ,w,\varphi ;\epsilon \right) \end{aligned}$$
(67)
on an open and bounded subset of the extended phase space \(\mathcal {P}\). In order to be able to carry out a perturbation argument from this limit, we also require that
$$\begin{aligned} P_{1}\left( x,v,\eta ,w,\varphi ;\epsilon \right) =M_{1}^{-1}\left( x,\eta ,\varphi ;\epsilon \right) Q_{1}\left( x,v,\eta ,w,\varphi ;\epsilon \right) \end{aligned}$$
has a similar smooth limit at \(\epsilon =0\), defined as
$$\begin{aligned} P_{1}\left( x,v,\eta ,w,\varphi ;0\right) :=\lim _{\epsilon \rightarrow 0}P_{1}\left( x,v,\eta ,w,\varphi ;\epsilon \right) . \end{aligned}$$
With these quantities and assumptions, (66) becomes
$$\begin{aligned}&x^{\prime } = \epsilon v,\nonumber \\&v^{\prime } = \epsilon P_{1}\left( x,v,\eta ,w,\varphi ;\epsilon \right) ,\nonumber \\&\varphi ^{\prime } = \epsilon \omega ,\nonumber \\&\eta ^{\prime } = w,\nonumber \\&w^{\prime } = P_{2}\left( x,v,\eta ,w,\varphi ;\epsilon \right) . \end{aligned}$$
(68)
1.3 Existence of a critical manifold
We want to ensure the existence of a reduced-order model in which the \((\eta (t),w(t))\) dynamics can be uniquely expressed, at least for large enough times, as a function of the (x(t), v(t)) dynamics and the time t. In geometric terms, this amounts to the existence of an invariant manifold \(\mathcal {M}_{\epsilon }\) that is a graph over the (x, v, t) variables and attracts all nearby solutions of the full system.
We require our reduced model to be smooth in \(\epsilon \), which is equivalent to requiring a smooth limit \(\mathcal {M}_{0}=\lim _{\epsilon \rightarrow 0}\mathcal {M}_{\epsilon }\) for the invariant manifold in the \(\epsilon =0\) limit of system (68). This limiting system can be written as
$$\begin{aligned}&x^{\prime } = 0,\nonumber \\&v^{\prime } = 0,\nonumber \\&\varphi ^{\prime } = 0,\nonumber \\&\eta ^{\prime } = w,\nonumber \\&w^{\prime } = P_{2}\left( x,v,\eta ,w,\varphi ;0\right) . \end{aligned}$$
(69)
In this limit, therefore, \((x,v,\varphi )\equiv (x_{0},v_{0},\varphi _{0})\) plays the role of a constant parameter vector. Any trajectory of the fast dynamics
$$\begin{aligned}&\eta ^{\prime } = w,\nonumber \\&w^{\prime } = P_{2}\left( x_{0},v_{0},\eta ,w,\varphi _{0};0\right) , \end{aligned}$$
(70)
therefore, gives rise to a \((2s+1)\) -dimensional invariant manifold for the full system. Along nontrivial trajectories of (70), however, the fast \((\eta ,v)\) variables change and hence are not uniquely enslaved to \((x_{0},v_{0},\varphi _{0})\), as required for the smooth limit of a reduced-order model. Consequently, only invariant manifolds arising from fixed points of (70) can be considered as limits of reduced-order models.
Such fixed points of (70) form a set
$$\begin{aligned} \mathcal {M}_{0}= & {} \left\{ (x,v,\eta ,w,\varphi )\in \mathcal {P}: w=0,\right. \\&\quad \left. P_{2}\left( x,v,\eta ,w,\varphi ;0\right) =0\right\} . \end{aligned}$$
To be a limit of a slow manifold carrying a reduced-order model, \(\mathcal {M}_{0}\) must be a smooth graph over an open domain \(\mathcal {D}_{0}\subset \mathbb {R}^{m}\times \mathbb {R}^{m}\times \mathcal {C}\) of the space (x, v, t) variables. By the implicit function theorem, this is equivalent to the requirement that
$$\begin{aligned} \det \left[ \partial _{\eta }P_{2}\left( x,v,\eta ,0,\varphi ;0\right) \right] \ne 0, \end{aligned}$$
(71)
should hold at all points \((x,v,\eta ,w,\varphi )\in \mathcal {M}_{0}\). This condition ensures that if \(\mathcal {M}_{0}\) is nonempty, then it is a \(2s+1\) dimensional differentiable manifold that can locally be expressed as a smooth graph
$$\begin{aligned} \left( \begin{array}{c} \eta \\ w \end{array}\right) =\left( \begin{array}{c} G_{0}(x,v,\varphi )\\ 0 \end{array}\right) ,\quad (x,v,\varphi )\in \mathcal {D}_{0} \end{aligned}$$
(72)
with the function \(G_{0}:\mathcal {D}_{0}\rightarrow \mathbb {R}^{f}\) satisfying the identity
$$\begin{aligned} P_{2}\left( x,v,G_{0}(x,v,\varphi ),0,\varphi ;0\right) =0. \end{aligned}$$
(73)
We refer to the part of \(\mathcal {M}_{0}\) satisfying (71) as the critical manifold associated with the limiting system (69). In our discussion of assumption (A2), we use the term critical manifold for the \(t=const.\) times slice \(\mathcal {M}_{0}(t)\) of \(\mathcal {M}_{0}\).
1.4 Stability of \(\mathcal {M}_{0}\)
The critical manifold must be normally attracting to persist as an attracting invariant slow manifold in the full system (68). The stability type of \(\mathcal {M}_{0}\) can be identified by analyzing the linearization of the fast flow (70) at the fixed points forming \(\mathcal {M}_{0}\).
The stability of the manifold \(\mathcal {M}_{0}\) at the fixed point family \((\eta _{0},w_{0})=(G(x_{0},v_{0},\varphi _{0}),0)\) of the decoupled equations is governed by the eigenvalues of the Jacobian
$$\begin{aligned} J=\left[ \begin{array}{cc} 0 &{}\quad I\\ \partial _{\eta }P_{2} &{}\quad \partial _{w}P_{2} \end{array}\right] _{\left( x,v,\eta ,w,t;\epsilon \right) =\left( x_{0},v_{0},G_{0}(y_{0},w_{0},\varphi _{0}),0,\varphi _{0};0\right) }.\nonumber \\ \end{aligned}$$
(74)
The matrix J has eigenvalues with strictly negative real parts precisely when the fixed point of the linear vibratory system
$$\begin{aligned}&u^{\prime \prime }-\partial _{w}P_{2}\left( x_{0},v_{0},G_{0}(x_{0},v_{0},t_{0}),0,\varphi _{0};0\right) u^{\prime }\nonumber \\&\quad -\,\partial _{\eta }P_{2}\left( x_{0},v_{0},G_{0}(x_{0},v_{0},\varphi _{0}),0,\varphi _{0};0\right) u=0\nonumber \\ \end{aligned}$$
(75)
is asymptotically stable for the parameter values \(\left( x_{0},v_{0},\varphi _{0}\right) \in \mathcal {D}_{0}\), which is guaranteed by assumption (A3). In that case, a compact subset of the critical manifold \(\mathcal {M}_{0}\) is a compact normally hyperbolic invariant manifold with boundary when \((x_{0},v_{0},\varphi _{0})\) is restricted to a domain with a smooth boundary. (In case of \(\mathcal {C}=[a,b]\), one has to select a and b as smooth functions of \((y_{0},t_{0})\) to eliminate non-smooth corners in \(\partial \mathcal {M}_{0}\). This can always be done without loss of generality.)
1.5 Existence of a slow manifold
Under the above conditions, the results of Fenichel [7] guarantee for full system (68) the existence of an attracting slow manifold \(\mathcal {M}_{\epsilon }\) that is \(\mathcal {O}(\epsilon )\)
\(C^{r}\) -close to \(\mathcal {M}_{0}\), and hence continues to be a graph of the form
$$\begin{aligned} \left( \begin{array}{c} \eta \\ w \end{array}\right)= & {} \left( \begin{array}{c} G_{\epsilon }(x,v,\varphi )\\ \epsilon H_{\epsilon }(x,v,\varphi ) \end{array}\right) \\= & {} \left( \begin{array}{c} G_{0}(x,v,\varphi )+\epsilon G_{1}(x,w,\varphi )+\mathcal {O}(\epsilon ^{2})\\ \epsilon H_{0}(x,v,\varphi )+\epsilon ^{2}H_{1}(x,v,\varphi )+\mathcal {O}(\epsilon ^{3}) \end{array}\right) ,\nonumber \\&(x,v,\varphi )\in \mathcal {D}_{0}, \end{aligned}$$
with appropriate smooth functions \(G_{\epsilon }\) and \(H_{\epsilon }\). The relation \(\eta ^{\prime }=w\) in (68) imposes the relationships
$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}\tau }\left[ G_{0}(x,v,\varphi )+\epsilon G_{1}(x,v,\varphi )+\mathcal {O}(\epsilon ^{2})\right] \\&\quad = \epsilon H_{0}(x,v,\varphi )+\epsilon ^{2}H_{1}(x,v,\varphi )+\mathcal {O}(\epsilon ^{3}),\\&\frac{\mathrm{d}}{\mathrm{d}\tau }\left[ \epsilon H_{0}(x,v,\varphi )+\epsilon ^{2}H_{1}(x,v,\varphi )+\mathcal {O}(\epsilon ^{3})\right] \\&\quad = P_{2}\left( x,v,G_{\epsilon }(x,v,\varphi ),\epsilon H_{\epsilon }(x,v,\varphi ),\varphi ;\epsilon \right) . \end{aligned}$$
Carrying out the differentiation in these two equations gives
$$\begin{aligned}&\left( \epsilon \partial _{x}G_{0}+\epsilon ^{2}\partial _{x}G_{1}\right) v\\&\quad +\left( \epsilon \partial _{v}G_{0}+\epsilon ^{2}\partial _{v}G_{1}\right) P_{1}\left( x,v,G_{\epsilon },\epsilon H_{\epsilon },\varphi ;\epsilon \right) \\&\quad +\left( \epsilon \omega \partial _{\varphi }G_{0}+\epsilon ^{2}\omega \partial _{\varphi }G_{1}\right) +\mathcal {O}(\epsilon ^{3})\\&\quad \quad =\epsilon H_{0}+\epsilon ^{2}H_{1}+\mathcal {O}(\epsilon ^{3}),\\&\left( \epsilon ^{2}\partial _{x}H_{0}+\epsilon ^{3}\partial _{x}H_{1}\right) v\\&\quad +\left( \epsilon ^{2}\partial _{v}H_{0}+\epsilon ^{3}\partial _{v}H_{1}\right) P_{1}\left( x,v,G_{\epsilon },\epsilon H_{\epsilon },\varphi ;\epsilon \right) \\&\quad +\left( \epsilon ^{2}\omega \partial _{\varphi }H_{0}+\epsilon ^{3}\omega \partial _{\varphi }H_{1}\right) +\mathcal {O}(\epsilon ^{4})\\&\quad \quad =P_{2}\left( x,v,G_{\epsilon }(x,v,\varphi ),\epsilon H_{\epsilon }(x,v,\varphi ),\varphi ;\epsilon \right) . \end{aligned}$$
We Taylor-expand these two equations, then equate the \(\mathcal {O}(\epsilon )\) and \(\mathcal {O}(\epsilon ^{2})\) terms in the first equation, as well as \(\mathcal {O}(\epsilon )\) terms in the second equation, to obtain
$$\begin{aligned}&H_{0}(x,v,\varphi ) = \partial _{x}G_{0}(x,v,\varphi )v\\&\quad +\,\partial _{v}G_{0}(x,v,\varphi )P_{1}\left( x,v,G_{0}(x,v,\varphi ),0,\varphi ;0\right) \\&\quad +\,\omega \partial _{\varphi }G_{0}(x,v,\varphi ),\\&H_{1}(x,w,\varphi ) = \partial _{x}G_{1}(x,v,\varphi )v\\&\quad +\,\partial _{v}G_{1}(x,v,\varphi )P_{1}\left( x,v,G_{0}(x,v,\varphi ),0,\varphi ;0\right) \\&\quad +\,\omega \partial _{\varphi }G_{1}(x,v,\varphi )\\&\quad +\,\partial _{v}G_{0}(x,v,\varphi )\left( \partial _{\eta }P_{1}\left( x,v,G_{0}(x,v,\varphi ),0,\varphi ;0\right) \right. \\&\quad \times \,G_{1}(x,v,\varphi )\\&\quad +\,\partial _{w}P_{1}\left( x,v,G_{0}(x,v,\varphi ),0,\varphi ;0\right) H_{0}(x,v,\varphi )\\&\left. \quad +\,\partial _{\epsilon }P_{1}\left( x,v,G_{0}(x,v,\varphi ),0,\varphi ;0\right) \right) ,\\&G_{1}(x,v,\varphi ) = -\left[ D_{\eta }P_{2}\left( x,v,G_{0}(x,v,\varphi ),0,\varphi ;0\right) \right] ^{-1}\\&\quad \times \, D_{w}P_{2}\left( x,v,G_{0}(x,v,\varphi ),0,\varphi ;0\right) H_{0}(x,v,\varphi )\\&\quad -\,\left[ D_{\eta }P_{2}\left( x,v,G_{0}(x,v,\varphi ),0,\varphi ;0\right) \right] ^{-1}\\&\quad \times \, D_{\epsilon }P_{2}\left( x,v,G_{0}(x,v,\varphi ),0,\varphi ;0\right) . \end{aligned}$$
In terms of the original variables, therefore, the slow manifold satisfies
$$\begin{aligned} y= & {} \epsilon G_{0}(x,\dot{x},t)+\epsilon ^{2}G_{1}(x,\dot{x},t)+\mathcal {O}(\epsilon ^{3}),\\ \dot{y}= & {} \epsilon H_{0}(x,\dot{x},t)+\epsilon ^{2}H_{1}(x,\dot{x},t)+\mathcal {O}(\epsilon ^{3}), \end{aligned}$$
where the functions \(H_{0},G_{1}\) and \(H_{1}\) are those listed in (17).
1.6 The reduced flow on the slow manifold
The slow manifold \(\mathcal {M}_{\epsilon }\) attracts all nearby solutions; thus, the reduced flow on \(\mathcal {M}_{\epsilon }\) will serve as the type of reduced-order model we have been seeking to construct (cf. requirement (R1) in the Introduction). The reduced equations on \(\mathcal {M}_{\epsilon }\) can be written by restricting the \((x,v,\phi )\) components of our system (68) to \(\mathcal {M}_{\epsilon }\), which yields
$$\begin{aligned} x^{\prime }= & {} \epsilon v,\\ v^{\prime }= & {} \epsilon P_{1}\left( x,v,G_{0}(x,v,\varphi ),0,\varphi ;0\right) \\&+\,\epsilon ^{2}\left[ D_{\eta }P_{1}\left( x,v,G_{0}(x,v,\varphi ),0,\varphi ;0\right) G_{1}(x,v,\varphi )\right. \\&+\,D_{w}P_{1}\left( x,v,G_{0}(x,v,\varphi ),0,\varphi ;0\right) H_{0}(x,v,\varphi )\\&\left. +\,D_{\epsilon }P_{2}\left( x,v,G_{0}(x,v,\varphi ),0,\varphi ;0\right) \right] +\mathcal {O}(\epsilon ^{2}),\\ \varphi ^{\prime }= & {} \epsilon \omega . \end{aligned}$$
In the original set of coordinates, this reduced flow can be written as in Eq. (18).
Using the definition of \(P_{1}\), we find that if \(M_{1}(x,G_{0}(x,\dot{x},t),t)\) has a smooth limit at \(\epsilon =0\), then the reduced equation can be multiplied by \(M_{1}(x,G_{0}(x,\dot{x},t),t)\) to yield the leading-order equivalent form of (18) as given in Eq. (19). When necessary, the \(\mathcal {O}(\epsilon )\) terms in (19) can also be computed from the formulas we have given above.
1.7 Convergence to the reduced trajectories
By the invariant foliation results of Fenichel [7], for small enough \(\epsilon \) and for motions close enough to the critical manifold, the y(t) component of all solutions of Eq. (1) synchronizes exponentially fast with solutions of reduced-order model (18).
Specifically, the local stable manifold \(W_{loc}^{s}(\mathcal {M}_{\epsilon })\) is foliated by an invariant family of class \(C^{r}\)
stable fibers
\(f^{s}(p)\). This \(\left( 2s+\dim \mathcal {C}\right) \)-parameter fiber family is parametrized by the base points
\(p\in \mathcal {M}_{\epsilon }\) of the fibers. Each fiber is a class \(C^{r-1}\) manifold whose dimension is 2f. The invariance of the fiber family means that for the flow map \(F^{\tau }:\mathcal {P}\rightarrow \mathcal {P}\) of system (68), we have
$$\begin{aligned} F^{\tau }\left( f^{s}(p)\right) \subset f^{s}(F^{\tau }(p)) \end{aligned}$$
for all \(\tau >0.\) Furthermore, the trajectory of the reduced flow through a fiber base point p attracts exponentially all trajectories that cross the fiber \(f^{s}(p)\). Specifically, if \(p=\left( x_{R}(\tau _{0}),v_{R}(\tau _{0}),\varphi _{R}(\tau _{0})\right) \) and \(\left( x(\tau _{0}),v(\tau _{0}),\varphi (\tau _{0}),\eta (\tau _{0}),w(\tau _{0})\right) \in f^{s}(p)\), then for all \(\tau \) values satisfying
$$\begin{aligned} \left( x(\tau ),v(\tau ),\varphi (\tau ),\eta (\tau ),w(\tau )\right) \in W_{loc}^{s}(\mathcal {M}_{\epsilon }), \end{aligned}$$
we have the estimate
$$\begin{aligned}&\left| \left( \begin{array}{c} x(\tau )-x_{R}(\tau )\\ v(\tau )-v_{R}(\tau )\\ \varphi (\tau )-\varphi _{R}(\tau )\\ \eta (\tau )-\eta _{R}(\tau )\\ w(\tau )-w_{R}(\tau ) \end{array}\right) \right| \nonumber \\&\quad \le C\left| \left( \begin{array}{c} x(\tau _{0})-x_{R}(\tau _{0})\\ v(\tau _{0})-v_{R}(\tau _{0})\\ \varphi (\tau _{0})-\varphi _{R}(\tau _{0})\\ \eta (\tau _{0})-\eta _{R}(\tau _{0})\\ w(\tau _{0})-w_{R}(\tau _{0}) \end{array}\right) \right| e^{-\varLambda (\tau -\tau _{0})},\nonumber \\&\tau >\tau _{0}. \end{aligned}$$
(76)
Here \(\varLambda >0\) can be selected as any constant satisfying
$$\begin{aligned} \max _{j\in [1,2f],\,(x,v,\varphi )\in \mathcal {D}_{0}}\mathrm {Re}\,\lambda _{j}(x,v,\varphi )<-\varLambda <0, \end{aligned}$$
with \(\lambda _{j}(x,v,\varphi ), j=1,\ldots ,2f\), denoting the eigenvalues of the Jacobian J, or equivalently, of the associated linear system (75). The constant \(C>0\) depends on \(\varLambda \) but is independent of the choice of the fiber base point p and the times \(\tau \) and \(\tau _{0}\).
By the form of system (68), we have \(\left| \varphi (\tau )-\varphi _{R}(\tau )\right| =\left| \varphi (\tau _{0})-\varphi _{R}(\tau _{0})\right| \). This is only consistent with (76), if \(\varphi (\tau _{0})\equiv \varphi _{R}(\tau _{0})\), which implies that the fibers \(f^{s}(p)\) are necessarily flat (i.e, constant) in the coordinate \(\varphi .\) Using this fact in (76) and passing back to the original coordinates gives
$$\begin{aligned}&\left| \left( \begin{array}{c} x(t)-x_{R}(t)\\ \dot{x}(t)-\dot{x}_{R}(t)\\ \frac{1}{\epsilon }y(t)-\frac{1}{\epsilon }y_{R}(t)\\ \dot{y}(t)-\dot{y}(t) \end{array}\right) \right| \\&\quad \le C\left| \left( \begin{array}{c} x(t_{0})-x_{R}(t_{0})\\ \dot{x}(t_{0})-\dot{x}_{R}(t_{0})\\ \frac{1}{\epsilon }y(t_{0})-\frac{1}{\epsilon }y_{R}(t_{0})\\ \dot{y}(t_{0})-\dot{y}_{R}(t_{0}) \end{array}\right) \right| e^{-\frac{\varLambda }{\epsilon }(t-t_{0})},\quad \tau >\tau _{0}. \end{aligned}$$
Along the reduced flow on the slow manifold \(\mathcal {M}_{\epsilon }\), the \((y,\dot{y})\) variables are enslaved to the (x, v, t) variables; thus, we can further rewrite this last inequality as
$$\begin{aligned}&\left| \left( \begin{array}{c} x(t)-x_{R}(t)\\ \dot{x}(t)-\dot{x}_{R}(t)\\ \frac{1}{\epsilon }y(t)-G_{\epsilon }\left( x_{R}(t),\dot{x}_{R}(t),t\right) \\ \dot{y}(t)-\epsilon H_{\epsilon }\left( x_{R}(t),\dot{x}_{R}(t),t\right) \end{array}\right) \right| \\&\quad \le C\left| \left( \begin{array}{c} x(t_{0})-x_{R}(t_{0})\\ \dot{x}(t_{0})-\dot{x}_{R}(t_{0})\\ \frac{1}{\epsilon }y(t_{0})-G_{\epsilon }\left( x_{R}(t_{0}),\dot{x}_{R}(t_{0}),t\right) \\ \dot{y}(t_{0})-\epsilon H_{\epsilon }\left( x_{R}(t_{0}),\dot{x}_{R}(t_{0}),t\right) \end{array}\right) \right| e^{-\frac{\varLambda }{\epsilon }(t-t_{0})},\\&\quad \tau >\tau _{0}. \end{aligned}$$
Applying the triangle inequality to the left-hand side and using the definition of \(G_{\epsilon }\) and \(H_{\epsilon }\) on the right-hand side of this inequality proves formula (21).
1.8 The validity of requirements (R1) and (R2)
First, the persistence of inflowing, normally attracting, compact invariant manifolds with boundary (cf. Fenichel [7]) together with statement (ii) of the Theorem prove that requirement (R1) is satisfied.
Second, formula (21) shows that solutions decay to the slow manifold \(\mathcal {M}_{\epsilon }(t)\) at an exponential rate of at least by \(-\varLambda /\epsilon \). In contrast, any possible exponential attraction rate generated along \(\mathcal {M}_{\epsilon }(t)\) may at most have an exponent of the order \(-K+\mathcal {O}(\epsilon )\) for some constant \(K>0\). This follows because Eq. (18) is \(\epsilon \)-independent at leading order, and its higher-order terms are smooth in \(\epsilon \). For small enough epsilon, we have
$$\begin{aligned} -\varLambda /\epsilon \ll -K+\mathcal {O}(\epsilon ), \end{aligned}$$
and hence the transverse decay rates to the manifold are much larger than any possible tangential decay rate along the manifold. This proves that requirement (R2) holds for \(\mathcal {M}_{\epsilon }(t)\) for \(\epsilon >0\) small enough, completing the proof of statement (i) of the Theorem.
Appendix 2: Proof of Proposition 1
We start by noting that, as a consequence of assumption (30), the graph
$$\begin{aligned} \eta =G_{0}(x,t) \end{aligned}$$
of the critical manifold \(\mathcal {M}_{0}\) depends only on the slow positions x and the time t. Near the unperturbed equilibrium, \(\mathcal {M}_{0}(t)\) can therefore be approximated by its Taylor expansion with respect to x. Specifically, we have
$$\begin{aligned} \eta= & {} G_{0}(x,t) \nonumber \\= & {} G_{0}(0,t)+\partial _{x}G_{0}(0,t)x\nonumber \\&+\frac{1}{2}\left( \partial _{xx}^{2}G_{0}(0,t)x\right) x+\mathcal {O}\left( \left| x\right| ^{3}\right) . \end{aligned}$$
(77)
Differentiation of the implicit equation \(P_{2}(x,G_{0}(x,t),0,t;0)=0\) with respect to x gives
$$\begin{aligned} \partial _{x}P_{2}+\partial _{\eta }P_{2}\partial _{x}G_{0}=0. \end{aligned}$$
(78)
Substitution of (77) into(78) and setting \(x=0\) yields
$$\begin{aligned} \partial _{x}G_{0}(0,t)=-\left. \left[ \partial _{\eta }P_{2}\right] ^{-1}\partial _{x}P_{2}\right| _{x=0,\eta =G(0,t),\dot{y}=0,\epsilon =0}, \end{aligned}$$
where the inverse of \(\partial _{\eta }P_{2}(x,G_{0}(x,t),0,t;0)\) is guaranteed to exist by assumption (A3). Differentiating (78) once more in x gives
$$\begin{aligned}&\partial _{xx}^{2}P_{2}+\left( 2\partial _{x\eta }^{2}P_{2}+\partial _{\eta \eta }^{2}P_{2}\partial _{x}G_{0}\right) \partial _{x}G_{0}\\&\quad +\,\partial _{\eta }P_{2}\partial _{x}^{2}G_{0}=0, \end{aligned}$$
enabling us to express the three-tensor \(\partial _{xx}^{2}G_{0}(0,t)\) as
$$\begin{aligned}&\partial _{xx}^{2}G_{0}(0,t)=-\left[ \partial _{\eta }P_{2}\right] ^{-1}\nonumber \\&\quad \times \,\left. \left[ \partial _{xx}^{2}P_{2}+\left( 2\partial _{x\eta }^{2}P_{2}+\partial _{\eta \eta }^{2}P_{2}\partial _{x}G_{0}\right) \partial _{x}G_{0}\right] \right| _{x=0,\eta =G(0,t),\dot{y}=0,\epsilon =0}.\nonumber \\ \end{aligned}$$
(79)
Therefore, with the help of formulas (30), the critical manifold \(\mathcal {M}_{0}\) can be written near the origin as a smooth, codimension-2f graph of the form
$$\begin{aligned}&\mathcal {M}_{0}(t)=\left\{ (x,\dot{x},\eta ,\dot{y},t)\in \mathcal {P}: \eta =G_{0}(x,t)=\varGamma (t)\right. \nonumber \\&\quad \left. +\,\varPhi (t)x+\left( \varTheta (t)x\right) x+\mathcal {O}\left( \left| x\right| ^{3}\right) ,\quad \dot{y}=0\right\} , \end{aligned}$$
(80)
where
$$\begin{aligned}&P_{2}(0,\varGamma (t),0,t;0) = 0,\nonumber \\&\varPhi (t) = -\left. \left[ \partial _{\eta }P_{2}\right] ^{-1}\partial _{x}P_{2}\right| _{x=0,\eta =\varGamma (t),\dot{y}=0,\epsilon =0}, \nonumber \\&\varTheta (t) = -\frac{1}{2}\left[ \partial _{\eta }P_{2}\right] ^{-1}\left[ \partial _{xx}^{2}P_{2}\right. \nonumber \\&\quad \left. \left. +\left( 2\partial _{x\eta }^{2}P_{2}+\partial _{\eta \eta }^{2}P_{2}\varPhi (t)\right) \varPhi (t)\right] \right| _{x=0,\eta =\varGamma (t),\dot{y}=0,\epsilon =0},\nonumber \\ \end{aligned}$$
(81)
as claimed in statement (i) of the Proposition. These expressions in (81) can then be used in the reduced-order models (18)–(19) to obtain more specific local approximations to the reduced dynamics, in case a global expression for the critical manifold is not explicitly available. Specifically, (18) can be localized near \(x=0\) as
$$\begin{aligned}&\ddot{x}-P_{1}\left( x,\dot{x},\left[ \varGamma (t)+\varPhi (t)x+\left( \varTheta (t)x\right) x\right] ,0,t;0\right) \nonumber \\&+\,\mathcal {O}(\epsilon ,\left| x\right| ^{3}) = 0. \end{aligned}$$
(82)
Under the further assumptions in statement (ii) of the Proposition, we have the following simplifications in formulas (81):
$$\begin{aligned}&\varGamma (t)\equiv 0,\qquad \varPhi (t)\equiv 0,\nonumber \\&\varTheta (t)\equiv -\frac{1}{2}\left[ \partial _{\eta }P_{2}(0,0,0;0)\right] ^{-1}\partial _{xx}^{2}P_{2}(0,0,0;0).\nonumber \\ \end{aligned}$$
(83)
Substituting these quantities into (82) and truncating the expression for \(\mathcal {M}_{0}(t)\) at linear and then at quadratic order proves the leading-order forms of the reduced equations in statements (iii) and (iv) of the Proposition, respectively. To obtain the order of the error terms in these equations, note that if x and y are modal coordinates of the linearized system, then we have
$$\begin{aligned}&P_{1}(x,\dot{x},\eta ,\dot{y},t;\epsilon ) = P_{1}(x,\dot{x},\eta ,0,t;0)+\mathcal {\mathcal {O}}(\epsilon )\nonumber \\&\quad = P_{1}(x,\dot{x},0,0,t;0)+\mathcal {\mathcal {O}}(\left| x\right| \left| \eta \right| )+\mathcal {\mathcal {O}}(\epsilon ). \end{aligned}$$
(84)
Substitution of \(\eta =0+\mathcal {O}(\left| x\right| ^{2})\) and \(\eta =\left( \varTheta (t)x\right) x+\mathcal {O}(\left| x\right| ^{3})\), respectively, into the \(\mathcal {\mathcal {O}}(\left| x\right| \left| \eta \right| )\) term in (84) then proves the order of the higher-order terms, as listed in statements (iii) and (iv) of the Proposition.
Appendix 3: Details for Example 8
For the system
$$\begin{aligned}&\ddot{x}\!+\!\left( c_{1}\!+\!\mu _{1}x^{2}\right) \dot{x}\!+\!k_{1}x+axy+bx^{3} = 0,\quad x\in \mathbb {R},\nonumber \\&\ddot{y}+c_{2}\dot{y}+k_{2}y+cx^{2} = 0,\quad y\in \mathbb {R}, \end{aligned}$$
(85)
we consider reduction by static condensation via the linear change of variables
$$\begin{aligned} \left( \begin{array}{c} x\\ y \end{array}\right) =U\hat{x},\qquad U=\left( \begin{array}{c} 1\\ 0 \end{array}\right) ,\quad \hat{x}\in \mathbb {R}. \end{aligned}$$
(86)
Dropping the tilde from \(\hat{x}\) and substituting \(y=0\) from (86) into the first equation of (44) gives statically condensed model (45).
Next, applying the idea of modal derivatives, we seek a quadratic invariant manifold of form (32), with the coefficients computed in the unscaled variables as
$$\begin{aligned} \varPhi= & {} 0,\nonumber \\ \varTheta= & {} -\frac{1}{2}\left[ \partial _{y}\left( c_{2}\dot{y}+k_{2}y+cx^{2}\right) \right] ^{-1}\nonumber \\&\left. \times \left[ \partial _{xx}^{2}\left( c_{2}\dot{y}+k_{2}y+cx^{2}\right) \right] \right| _{x=0,y=0,\dot{y}=0}\nonumber \\= & {} -\frac{c}{k_{2}}. \end{aligned}$$
(87)
Substitution of \(y=\varTheta x^{2}\) into the first equation of (44) gives the modal-derivatives-based reduced-order model (46), representing only a slight correction to (45) at cubic order. All this appears reasonable at this point, with the statically condensed system (45) offering a leading-order model that is subsequently refined at cubic order by the modal derivatives approach in (46).
At the same time, there exists a slow spectral submanifold (SSM), the unique smoothest, nonlinear continuation of the \(y=0\) modal subspace of the equilibrium. This unique, two-dimensional analytic invariant manifold is tangent to the modal subspace of the x-degree of freedom at the origin (cf. Haller and Ponsioen [15]). The slow SSM, therefore, can locally be written as a two-dimensional invariant graph \((y,\dot{y})=\left( g_{1}(x,\dot{x}),g_{2}(x,\dot{x})\right) =\mathcal {O}\left( x^{2},x\dot{x},\dot{x}^{2}\right) \) over \((x,\dot{x})\), as originally envisioned by Shaw and Pierre [25]. Differentiating the general form
$$\begin{aligned} y=g_{1}(x,\dot{x})=\alpha x^{2}+\beta x\dot{x}+\gamma \dot{x}^{2}+\mathcal {O}\left( 3\right) \end{aligned}$$
(88)
of such an invariant graph twice in time, with \(\ddot{x}\) substituted from the first equation of system (44), we obtain
$$\begin{aligned} \ddot{y}= & {} -k_{1}(2\alpha -2\gamma k_{1}-\beta c_{1})x^{2}\\&-[2\beta k_{1}+c_{1}\left( 2\alpha -2\gamma k_{1}-\beta c_{1}\right) \\&+2\left( \beta -2\gamma c_{1}\right) k_{1}]x\dot{x}\\&+\left[ \left( 2\alpha -2\gamma k_{1}-\beta c_{1}\right) -2c_{1}\left( \beta -2\gamma c_{1}\right) \right] \dot{x}^{2}\\&+\,\mathcal {O}\left( 3\right) . \end{aligned}$$
A comparison of this differential equation with the second equation of system (44), with y and \(\dot{y}\) substituted from (88), leads to the linear system of algebraic equations
$$\begin{aligned}&\left( \begin{array}{ccc} k_{2}-2k_{1} &{}\quad k_{1}\left( c_{1}-c_{2}\right) &{}\quad 2k_{1}^{2}\\ 2\left( c_{2}-c_{1}\right) &{}\quad k_{2}-4k_{1}+c_{1}^{2}-c_{1}c_{2} &{}\quad 2k_{1}\left( 3c_{1}-c_{2}\right) \\ 2 &{}\quad c_{2}-3c_{1} &{}\quad k_{2}-2k_{1}+4c_{1}^{2}-2c_{1}c_{2} \end{array}\right) \\&\quad \left( \begin{array}{c} \alpha \\ \beta \\ \gamma \end{array}\right) =-\left( \begin{array}{c} c\\ 0\\ 0 \end{array}\right) \end{aligned}$$
for the unknown coefficients \(\alpha , \beta \) and \(\gamma \) in expression (88) of the slow SSM. The solution of this system of equations is given by
$$\begin{aligned} \alpha= & {} -\frac{c}{D}\left( 4c_{1}^{4}-6c{}_{1}^{3}c_{2}+2c{}_{1}^{2}c{}_{2}^{2}+5c{}_{1}^{2}k_{2}-c_{1}c_{2}\left( 2k_{1}+3k_{2}\right) \right. \nonumber \\&\left. +\,2c{}_{2}^{2}k_{1}+8k_{1}^{2}-6k_{1}k_{2}+k{}_{2}^{2}\right) ,\nonumber \\ \beta= & {} -\frac{2c}{D}\left( 4c_{1}k_{1}+k_{2}\left( c_{1}-c_{2}\right) +2c_{1}c_{2}^{2}-6c_{1}^{2}c_{2}+4c_{1}^{3}\right) ,\nonumber \\ \gamma= & {} -\frac{2c}{D}\left( 2c_{1}^{2}-3c_{1}c_{2}+c_{2}^{2}+4k_{1}-k_{2}\right) , \end{aligned}$$
(89)
with
$$\begin{aligned} D= & {} \left( c_{1}^{2}-c_{1}c_{2}+k_{2}\right) \left( 4c_{1}^{2}k_{2}-8c_{1}c_{2}k_{1}-2c_{1}c_{2}k_{2}\right. \nonumber \\&\left. +\,4c_{2}^{2}k_{1}+16k_{1}^{2}-8k_{1}k_{2}+k_{2}^{2}\right) . \end{aligned}$$
(90)
With these coefficients, substitution of (88) into the first equation of system (44) gives the exact reduced system on the slow SSM, up to cubic order, in the form
$$\begin{aligned}&\ddot{x}+\left[ c_{1}+\left( \mu _{1}+a\beta \right) x^{2}\right] \dot{x}+\left( k_{1}+a\gamma \dot{x}^{2}\right) x\\&\quad +\left( b+a\alpha \right) x^{3}+\mathcal {O}\left( 4\right) =0. \end{aligned}$$
Substitution of formulas (89) into this last equation gives final form (47) of the exact reduced model on the SSM.
Appendix 4: Details for Sect. 6.1
For the parameter range described by scalings (59), we take the \(\epsilon \rightarrow 0\) limit in the expressions for \(P_{1}\) and \(P_{2}\) in (57)–(58). We then obtain
$$\begin{aligned}&P_{1}\left( x,v,\eta ,w,t;0\right) =\left[ \begin{array}{c} p_{1}^{0}\\ p_{2}^{0} \end{array}\right] ,\nonumber \\&p_{1}^{0} =\frac{1+\beta }{\delta ^{2}}\biggl (-\mu _{p}v_{\gamma }-\delta ^{2}\sin x_{\gamma }+\delta ^{2}G_{p}(t)\end{aligned}$$
(91)
$$\begin{aligned}&\quad +\,\frac{\delta \sin x_{\gamma }}{1+\beta }\bigl [\beta \delta \cos x_{\gamma }v_{\gamma }^{2}-\mu _{h}w-\varOmega _{h}^{2}\eta -\alpha _{h}\eta ^{3}\nonumber \\&\quad +\,(1+\beta )\delta +F_{h}(t)\delta -F_{p}(t)\delta \sin x_{\gamma }\bigr ]\biggr )\nonumber \\&\quad -\,\frac{\phi }{\delta }\cos x_{\gamma }\left( \beta \frac{\delta }{\phi }\sin x_{\gamma }v_{\gamma }^{2}\right. \nonumber \\&\quad -\,\mu _{d}v_{d}-\varOmega _{d}^{2}\left( 1+x_{d}\right) Q^{0}(x_{d})+F_{d}(t)\frac{\delta }{\phi }\nonumber \\&\quad \left. +\,F_{p}(t)\frac{\delta }{\phi }\cos x_{\gamma }\right) , \nonumber \\&p_{2}^{0} =-\frac{\beta }{\delta \phi }\cos x_{\gamma }\biggl (-\mu _{p}v_{\gamma }-\delta ^{2}\sin x_{\gamma }+\delta ^{2}G_{p}(t)\nonumber \\&\quad +\,\frac{\delta \sin x_{\gamma }}{1+\beta }\bigl [\beta \delta \cos x_{\gamma }v_{\gamma }^{2}-\mu _{h}w-\varOmega _{h}^{2}\eta -\alpha _{h}\eta ^{3}\nonumber \\&\quad +\,(1+\beta )\delta +F_{h}(t)\delta -F_{p}(t)\delta \sin x_{\gamma }\bigr ]\biggr )\nonumber \\&\quad +\,\frac{1}{1+\beta }\left( 1+\beta \cos ^{2}x_{\gamma }\right) \left( \beta \frac{\delta }{\phi }\sin x_{\gamma }v_{\gamma }^{2}\right. \nonumber \\&\quad -\,\mu _{d}v_{d}-\varOmega _{d}^{2}\left( 1+x_{d}\right) Q^{0}(x_{d})+F_{d}(t)\frac{\delta }{\phi }\nonumber \\&\quad \left. +\,F_{p}(t)\frac{\delta }{\phi }\cos x_{\gamma }\right) ,\\&P_{2}\left( x,v,\eta ,w,t;0\right) = \left( \frac{1+\beta \sin ^{2}x_{\gamma }}{1+\beta }\right) \biggl (\beta \delta \cos x_{\gamma }v_{\gamma }^{2}\nonumber \\&\quad -\,\mu _{h}w-\varOmega _{h}^{2}\eta -\alpha _{h}\eta ^{3}+(1+\beta )\delta \nonumber \\&\quad +\,F_{h}(t)\delta -F_{p}(t)\delta \sin x_{\gamma }\nonumber \\&\quad +\,\frac{\left( 1+\beta \right) \beta \sin x_{\gamma }}{\delta \left( 1+\beta \sin ^{2}x_{\gamma }\right) }\left[ -\mu _{p}v_{\gamma }-\delta ^{2}\sin x_{\gamma }+\delta ^{2}G_{p}(t)\right] \nonumber \\&\quad -\,\frac{\beta \phi \sin x_{\gamma }\cos x_{\gamma }}{1+\beta \sin ^{2}x_{\gamma }}\biggl [\beta \frac{\delta }{\phi }\sin x_{\gamma }v_{\gamma }^{2}\nonumber \\&\quad -\,\mu _{d}v_{d}-\varOmega _{d}^{2}\left( 1+x_{d}\right) Q^{0}(x_{d})+F_{d}(t)\frac{\delta }{\phi }\nonumber \\&\quad +\,F_{p}(t)\frac{\delta }{\phi }\cos x_{\gamma }\biggr ]\biggr ),\nonumber \end{aligned}$$
(92)
where \(Q^{0}(x_{d})\) is defined as
$$\begin{aligned} Q^{0}(x_{d})=\left( 1-\frac{1}{1+x_{d}}\right) ,\quad x_{d}>-1. \end{aligned}$$
We observe that both \(P_{1}\) and \(P_{2}\) continue to be smooth in \(\epsilon \) at the \(\epsilon =0\) limit, thereby satisfying assumption (A1).
For the critical manifold defined through the relationship \(\eta =G_{0}(x,v,t)\) in assumption (A2), we have the equation
$$\begin{aligned}&P_{2}\left( x,v,\eta ,0,t;0\right) =0\quad \Longleftrightarrow \quad \varOmega _{h}^{2}\eta \\&\quad +\,\alpha _{h}\eta ^{3}=T(x,v,t), \end{aligned}$$
where
$$\begin{aligned}&T(x,v,t) = \beta \delta \cos x_{\gamma }v_{\gamma }^{2}+(1+\beta )\delta \\&\quad +\,F_{h}(t)\delta -F_{p}(t)\delta \sin x_{\gamma } +\,\frac{\left( 1+\beta \right) \beta \sin x_{\gamma }}{\delta \left( 1+\beta \sin ^{2}x_{\gamma }\right) } \\&\quad \times \left[ -\mu _{p}v_{\gamma }-\delta ^{2}\sin x_{\gamma }+\delta ^{2}G_{p}(t)\right] \\&\quad -\,\frac{\beta \phi \sin x_{\gamma }\cos x_{\gamma }}{1+\beta \sin ^{2}x_{\gamma }}\biggl [\beta \frac{\delta }{\phi }\sin x_{\gamma }v_{\gamma }^{2}-\mu _{d}v_{d}\\&\quad -\,\varOmega _{d}^{2}\left( 1\!+\!x_{d}\right) Q^{0}(x_{d})\!+\! F_{d}(t)\frac{\delta }{\phi }\!+\!F_{p}(t)\frac{\delta }{\phi }\cos x_{\gamma }\biggr ]. \end{aligned}$$
Using the cubic formula, the real root of this equation can be expressed explicitly as
$$\begin{aligned} \eta= & {} G_{0}(x,v,t)\\= & {} \root 3 \of {\frac{T(x,v,t)}{2\alpha _{h}}+\sqrt{\frac{T^{2}(x,v,t)}{4\alpha _{h}^{2}}+\frac{\varOmega _{h}^{6}}{27\alpha _{h}^{3}}}}\\&-\root 3 \of {-\frac{T(x,v,t)}{2\alpha }+\sqrt{\frac{T^{2}(x,v,t)}{4\alpha _{h}^{2}}+\frac{\varOmega _{h}^{6}}{27\alpha _{h}^{3}}}}, \end{aligned}$$
assuming that \(\varOmega _{h}^{2}\) and \(\alpha _{h}\) are greater than zero.
Oscillatory system (9) determining the stability of the critical manifold takes the specific form
$$\begin{aligned}&A(x,v,t) = -\partial _{w}P_{2}\left( x,v,G_{0}(x,v,t),0,t;0\right) \nonumber \\&\quad =\left( \frac{1+\beta \sin ^{2}x_{\gamma }}{1+\beta }\right) \mu _{h}, \end{aligned}$$
(93)
$$\begin{aligned}&B(x,v,t) = -\partial _{\eta }P_{2}\left( x,v,G_{0}(x,v,t),0,t;0\right) \nonumber \\&\quad =\left( \frac{1+\beta \sin ^{2}x_{\gamma }}{1+\beta }\right) \left( \varOmega _{h}^{2}+3\alpha _{h}G_{0}^{2}(x,v,t)\right) .\nonumber \\ \end{aligned}$$
(94)
The equilibrium solution of the unforced linear oscillatory system (9) is, therefore, always asymptotically stable, given that
$$\begin{aligned} \mu _{h}>0,\quad \beta>0,\quad \varOmega _{h}^{2}>0,\quad \alpha _{h}>0. \end{aligned}$$
We conclude that assumptions (A1)–(A3) hold, and hence a global reduced-order model (18) exists over the slow variables in the specific form
$$\begin{aligned} \ddot{x}=&\left[ \begin{array}{l} \frac{1+\beta }{\delta ^{2}\left( 1+\beta \sin ^{2}x_{\gamma }\right) }\mathcal {A}-\frac{\phi \cos x_{\gamma }}{\delta \left( 1+\beta \sin ^{2}x_{\gamma }\right) }\mathcal {B}\\ \frac{1}{1+\beta \sin ^{2}x_{\gamma }}\mathcal {B}-\frac{\beta \cos x_{\gamma }}{\phi \delta \left( 1+\beta \sin ^{2}x_{\gamma }\right) }\mathcal {A} \end{array}\right] +\mathcal {O}(\epsilon ), \end{aligned}$$
where
$$\begin{aligned} \mathcal {A}(x_{\gamma },\dot{x}_{\gamma })= & {} -\mu _{p}\dot{x}_{\gamma }-\delta ^{2}\sin x_{\gamma }+\delta ^{2}G_{p}(t),\\ \mathcal {B}(x_{\gamma },x_{d},\dot{x}_{\gamma })= & {} \beta \frac{\delta }{\phi }\sin x_{\gamma }\dot{x}_{\gamma }^{2}-\mu _{d}\dot{x}_{\gamma }-\varOmega _{d}^{2}x_{d}\\&+\,F_{d}(t)\frac{\delta }{\phi }+F_{p}(t)\frac{\delta }{\phi }\cos x_{\gamma }. \end{aligned}$$
Scaling back to the original time and substituting the physical parameters back into the non-dimensionalized equations, we obtain that the exact reduced-order model on the slow manifold of form (60)–(61).
Appendix 5: Details for Sect. 6.2
Here we verify assumptions (A1)–(A3) in detail for the fast–fast–slow setting treated in Sect. 6.2. To make the horizontal spring stiff, we choose its length as \(D=L\), so that the original equations of motion (54) now become
$$\begin{aligned}&ml^{2}\ddot{\gamma }-ml\sin \gamma \ddot{h}+ml\cos \gamma \ddot{d}+c_{p}\dot{\gamma }\\&\quad +\,mgl\sin \gamma = f_{p}(t)l,\\&(M+m)\ddot{h}-ml\sin \gamma \ddot{\gamma }-ml\cos \gamma \dot{\gamma }^{2}+C_{h}\dot{h}\\&\quad +\,K_{h}h+K_{d}Q(d,h)h+\varGamma _{h}h^{3} = (M+m)g\\&\quad +\,f_{h}(t)-f_{p}(t)\sin \gamma ,\\&(M+m)\ddot{d}+ml\cos \gamma \ddot{\gamma }-ml\sin \gamma \dot{\gamma }^{2}+C_{d}\dot{d}\\&\quad +\,K_{d}\left( L+d\right) Q(d,h) = f_{d}(t)+f_{p}(t)\cos \gamma , \end{aligned}$$
with
$$\begin{aligned} Q(d,h)=\left( 1-\frac{L}{\sqrt{\left( L+d\right) ^{2}+h^{2}}}\right) . \end{aligned}$$
(95)
The linearized oscillation frequencies of the uncoupled springs and pendulum remain the same as in (55). We adopt the same scaling as in Sect. 6.1, except that we now scale the d coordinate with the unstretched length L of the vertical spring. Denoting differentiation with respect to the new time \(\tilde{t}\) still by a dot, then dropping all the tildes, we obtain the non-dimensionalized equations of motions
$$\begin{aligned}&\varDelta ^{2}\ddot{\gamma }-\varDelta \sin \gamma \ddot{h}+\varDelta \cos \gamma \ddot{d}+\pi _{p}\dot{\gamma }\\&\quad +\,\varDelta ^{2}\sin \gamma =\varDelta ^{2}G_{p}(t),\\&(1+\beta )\ddot{h}-\beta \varDelta \sin \gamma \ddot{\gamma }-\beta \varDelta \cos \gamma \dot{\gamma }^{2}+\pi _{h}\dot{h}+q_{h}h\\&\quad +\,q_{d}hQ(d,h)+a_{h}h^{3} = (1+\beta )\varDelta +F_{h}(t)\varDelta \\&\quad -\,F_{p}(t)\varDelta \sin \gamma ,\\&(1+\beta )\ddot{d}+\beta \varDelta \cos \gamma \ddot{\gamma }-\beta \varDelta \sin \gamma \dot{\gamma }^{2}+\pi _{d}\dot{d}\\&\quad +\,q_{d}\left( 1+d\right) Q(d,h) = F_{d}(t)\varDelta +F_{p}(t)\varDelta \cos \gamma . \end{aligned}$$
In the notation used for system (1), we now have
$$\begin{aligned}&M(q,t;\epsilon ) = \left( \begin{array}{ccc} \varDelta ^{2} &{}\quad -\varDelta \sin x &{}\quad \varDelta \cos x\\ -\beta \varDelta \sin x &{}\quad 1+\beta &{}\quad 0\\ \beta \varDelta \cos x &{}\quad 0 &{}\quad 1+\beta \end{array}\right) ,\\&F(q,\dot{q},t;\epsilon ) \\&\quad = \left( \begin{array}{l} -\pi _{p}\dot{x}-\varDelta ^{2}\sin x+\varDelta ^{2}G_{p}(t)\\ \beta \varDelta \cos x\dot{x}^{2}-\pi _{h}\dot{y}_{h}-q_{h}\epsilon \frac{y_{h}}{\epsilon }-q_{d}\epsilon \frac{y_{h}}{\epsilon }Q(\frac{y_{d}}{\epsilon },\frac{y_{h}}{\epsilon })-a_{h}\epsilon ^{3}\left( \frac{y_{h}}{\epsilon }\right) ^{3}\\ \quad +(1+\beta )\varDelta +F_{h}(t)\varDelta -F_{p}(t)\varDelta \sin x\\ \beta \varDelta \sin x\dot{x}^{2}-\pi _{d}\dot{y}_{d}-q_{d}\left( 1+\epsilon \frac{y_{d}}{\epsilon }\right) Q(\frac{y_{d}}{\epsilon },\frac{y_{h}}{\epsilon })+F_{d}(t)\varDelta +F_{p}(t)\varDelta \cos x \end{array}\right) , \end{aligned}$$
with the parameter \(\epsilon >0\) yet to be determined based on the assumptions of the SFD approach. Note that the mass matrix above is not symmetric due to the scalings we have employed, but it is, nevertheless, nonsingular, as we generally assume in this paper.
With the above quantities at hand, we obtain the modified mass matrices \(M_{i}\) and the forcing terms \(Q_{i}\) defined in (5) in the specific form
$$\begin{aligned} M_{1}= & {} M_{11}-M_{12}M_{22}^{-1}M_{21}=\frac{\varDelta ^{2}}{1+\beta },\\ M_{2}= & {} M_{22}-M_{21}M_{11}^{-1}M_{12}\\= & {} \left( \begin{array}{cc} 1+\beta \cos ^{2}x &{}\quad \beta \sin x\cos x\\ \beta \sin x\cos x &{}\quad 1+\beta \sin ^{2}x \end{array}\right) ,\\ Q_{1}= & {} F_{1}-M_{12}M_{22}^{-1}F_{2}=-\pi _{p}\dot{x}-\varDelta ^{2}\sin x\\&+\,\varDelta ^{2}G_{p}(t)+\frac{\varDelta }{1+\beta }\sin x\biggl [\beta \varDelta \cos x\dot{x}^{2}-\pi _{h}\dot{y}_{h}\\&-\,q_{h}\epsilon \frac{y_{h}}{\epsilon }-q_{d}\epsilon \frac{y_{h}}{\epsilon }Q\left( \frac{y_{d}}{\epsilon },\frac{y_{h}}{\epsilon }\right) -a_{h}\epsilon ^{3}\left( \frac{y_{h}}{\epsilon }\right) ^{3}\\&+\,(1+\beta )\varDelta +F_{h}(t)\varDelta -F_{p}(t)\varDelta \sin x\biggr ]\\&-\,\frac{\varDelta }{1+\beta }\cos x\biggl [\beta \varDelta \sin x\dot{x}^{2}-\pi _{d}\dot{y}_{d}\\&-\,q_{d}\left( 1+\epsilon \frac{y_{d}}{\epsilon }\right) Q\left( \frac{y_{d}}{\epsilon },\frac{y_{h}}{\epsilon }\right) +F_{d}(t)\varDelta \\&+\,F_{p}(t)\varDelta \cos x\biggr ], \end{aligned}$$
$$\begin{aligned} Q_{2} = F_{2}-M_{21}M_{11}^{-1}F_{1} = \left[ \begin{array}{l} \beta \varDelta \cos x\dot{x}^{2}-\pi _{h}\dot{y}_{h}-q_{h}\epsilon \frac{y_{h}}{\epsilon }-q_{d}\epsilon \frac{y_{h}}{\epsilon }Q(\frac{y_{d}}{\epsilon },\frac{y_{h}}{\epsilon })-a_{h}\epsilon ^{3}\left( \frac{y_{h}}{\epsilon }\right) ^{3}+(1+\beta )\varDelta \\ \quad +\,F_{h}(t)\varDelta -F_{p}(t)\varDelta \sin x-\frac{\beta }{\varDelta }\pi _{p}\sin x\dot{x}-\beta \varDelta \sin ^{2}x+\beta \varDelta \sin xG_{p}(t)\\ \beta \varDelta \sin x\dot{x}^{2}-\pi _{d}\dot{y}_{d}-q_{d}\left( 1+\epsilon \frac{y_{d}}{\epsilon }\right) Q(\frac{y_{d}}{\epsilon },\frac{y_{h}}{\epsilon })+F_{d}(t)\varDelta +F_{p}(t)\varDelta \cos x\\ \quad +\,\frac{\beta }{\varDelta }\pi _{p}\cos x\dot{x}+\beta \varDelta \sin x\cos x-\beta \varDelta \cos xG_{p}(t) \end{array}\right] . \end{aligned}$$
We therefore obtain
$$\begin{aligned}&P_{1}\left( x,v,\eta ,w,t;\epsilon \right) = \frac{1+\beta }{\varDelta ^{2}}\Biggl [-\pi _{p}v-\varDelta ^{2}\sin x\\&\quad +\,\varDelta ^{2}G_{p}(t)+\frac{\varDelta }{1+\beta }\sin x\biggl [\beta \varDelta \cos xv^{2}-\pi _{h}w_{h}\\&\quad -\,q_{h}\epsilon \eta _{h}-q_{d}\epsilon \eta _{h}Q(\eta _{d},\eta _{h})-a_{h}\epsilon ^{3}\eta _{h}^{3}\\&\quad +\,(1+\beta )\varDelta +F_{h}(t)\varDelta -F_{p}(t)\varDelta \sin x\biggr ]\\&\quad -\,\frac{\varDelta }{1+\beta }\cos x\biggl [\beta \varDelta \sin xv^{2}-\pi _{d}w_{d}\\&\quad -\,q_{d}\left( 1+\epsilon \eta _{d}\right) Q(\eta _{d},\eta _{h})+F_{d}(t)\varDelta +F_{p}(t)\varDelta \cos x\biggr ]\Biggr ], \end{aligned}$$
$$\begin{aligned} P_{2}\left( x,v,\eta ,w,t;\epsilon \right) = \epsilon M_{2}^{-1}\left[ \begin{array}{l} \beta \varDelta \cos xv^{2}-\pi _{h}w_{h}-q_{h}\epsilon \eta _{h}-q_{d}\epsilon \eta _{h}Q(\eta _{d},\eta _{h})-a_{h}\epsilon ^{3}\eta _{h}^{3}+(1+\beta )\varDelta \\ \quad +\,F_{h}(t)\varDelta -F_{p}(t)\varDelta \sin x-\frac{\beta }{\varDelta }\pi _{p}\sin xv-\beta \varDelta \sin ^{2}x+\beta \varDelta \sin xG_{p}(t)\\ \\ \beta \varDelta \sin xv^{2}-\pi _{d}w_{d}-q_{d}\left( 1+\epsilon \eta _{d}\right) Q(\eta _{d},\eta _{h})+F_{d}(t)\varDelta +F_{p}(t)\varDelta \cos x\\ \quad +\,\frac{\beta }{\varDelta }\pi _{p}\cos xv+\beta \varDelta \sin x\cos x-\beta \varDelta \cos xG_{p}(t) \end{array}\right] , \end{aligned}$$
where \(M_{2}^{-1}\) is equal to
$$\begin{aligned} M_{2}^{-1}=\frac{1}{1+\beta }\left[ \begin{array}{cc} 1+\beta \sin ^{2}x &{} -\beta \sin x\cos x\\ -\beta \sin x\cos x &{} 1+\beta \cos ^{2}x \end{array}\right] . \end{aligned}$$
Recall that \(\epsilon >0\) has been a completely arbitrary small parameter so far. We now need to define \(\epsilon \) in a way that assumptions (A1)–(A3) are satisfied. Since at present we have \(\lim _{\epsilon \rightarrow 0}P_{2}\left( x,v,\eta ,w,t;\epsilon \right) \equiv 0,\), these assumptions will not hold. We can only satisfy (A1)–(A3) by making the system parameters appropriate functions of \(\epsilon .\)
With the parameter choices listed in (62), we have
$$\begin{aligned}&P_{1}\left( x,v,\eta ,w,t;\epsilon \right) = \frac{1+\beta }{\delta ^{2}}\Biggl [-\mu _{p}v-\delta ^{2}\sin x\\&\quad +\,\delta ^{2}G_{p}(t)+\frac{\delta }{1+\beta }\sin x\biggl [\beta \delta \cos xv^{2}-\mu _{h}w_{h}\\&\quad -\,\varOmega _{h}^{2}\eta _{h}-\varOmega _{d}^{2}\eta _{h}Q(\eta _{d},\eta _{h})-\alpha _{h}\eta _{h}^{3}+(1+\beta )\delta \\&\quad +\,F_{h}(t)\delta -F_{p}(t)\delta \sin x\biggr ]\\&\quad -\,\frac{\delta }{1\!+\!\beta }\cos x\biggl [\beta \delta \sin xv^{2}\!-\!\mu _{d}w_{d} -\frac{\varOmega _{d}^{2}}{\epsilon }\left( 1+\epsilon \eta _{d}\right) \\&\quad \times \, Q(\eta _{d},\eta _{h})+F_{d}(t)\delta +F_{p}(t)\delta \cos x\biggr ]\Biggr ], \end{aligned}$$
$$\begin{aligned} P_{2}\left( x,v,\eta ,w,t;\epsilon \right) = M_{2}^{-1}\left[ \begin{array}{l} \beta \delta \cos xv^{2}-\mu _{h}w_{h}-\varOmega _{h}^{2}\eta _{h}-\varOmega _{d}^{2}\eta _{h}Q(\eta _{d},\eta _{h})-\alpha _{h}\eta _{h}^{3}+(1+\beta )\delta \\ +F_{h}(t)\delta -F_{p}(t)\delta \sin x-\frac{\beta }{\delta }\mu _{p}\sin xv-\beta \delta \sin ^{2}x+\beta \delta \sin xG_{p}(t)\\ \\ \beta \delta \sin xv^{2}-\mu _{d}w_{d}-\frac{\varOmega _{d}^{2}}{\epsilon }\left( 1+\epsilon \eta _{d}\right) Q(\eta _{d},\eta _{h})+F_{d}(t)\delta +F_{p}(t)\delta \cos x\\ +\frac{\beta }{\delta }\mu _{p}\cos xv+\beta \delta \sin x\cos x-\beta \delta \cos xG_{p}(t) \end{array}\right] . \end{aligned}$$
where \(M_{2}^{-1}\) remains unchanged.
Noting that
$$\begin{aligned}&\lim _{\epsilon \rightarrow 0}\frac{\varOmega _{d}^{2}}{\epsilon }\left( 1+\epsilon \eta _{d}\right) Q(\eta _{d},\eta _{h}) =\lim _{\epsilon \rightarrow 0}\frac{\varOmega _{d}^{2}}{\epsilon }\left( 1+\epsilon \eta _{d}\right) \left( 1-\frac{1}{\sqrt{\left( 1+\epsilon \eta _{d}\right) ^{2}+\left( \epsilon \eta _{h}\right) ^{2}}}\right) \\&\quad =\lim _{\epsilon \rightarrow 0}\frac{\varOmega _{d}^{2}\left( 1+\epsilon \eta _{d}\right) \left( \sqrt{\left( 1+\epsilon \eta _{d}\right) ^{2}+\left( \epsilon \eta _{h}\right) ^{2}}-1\right) }{\epsilon \sqrt{\left( 1+\epsilon \eta _{d}\right) ^{2}+\left( \epsilon \eta _{h}\right) ^{2}}} =\lim _{\epsilon \rightarrow 0}\frac{f(\epsilon )}{g(\epsilon )}=\lim _{\epsilon \rightarrow 0}\frac{\partial _{\epsilon }f(\epsilon )}{\partial _{\epsilon }g(\epsilon )}\\&\quad =\lim _{\epsilon \rightarrow 0}\frac{\varOmega _{d}^{2}\eta _{d}\left( \sqrt{\left( 1+\epsilon \eta _{d}\right) {}^{2}+\left( \epsilon \eta _{h}\right) ^{2}}-1\right) +\varOmega _{d}^{2}\left( 1+\epsilon \eta _{d}\right) \left( \left( (1+\epsilon \eta _{d})\eta _{d}+\epsilon \eta _{h}^{2}\right) \left( \left( 1+\epsilon \eta _{d}\right) ^{2}+\left( \epsilon \eta _{h}\right) ^{2}\right) ^{-\frac{1}{2}}\right) }{\sqrt{\left( 1+\epsilon \eta _{d}\right) ^{2}+\left( \epsilon \eta _{h}\right) ^{2}}+\epsilon \left( (1+\epsilon \eta _{d})\eta _{d}+\epsilon \eta _{h}^{2}\right) \left( \left( 1+\epsilon \eta _{d}\right) ^{2}+\left( \epsilon \eta _{h}\right) ^{2}\right) ^{-\frac{1}{2}}}\\&\quad =\varOmega _{d}^{2}\eta _{d}, \end{aligned}$$
we conclude that both \(P_{1}\) and \(P_{2}\) continue to be smooth in \(\epsilon \) at the \(\epsilon =0\) limit, thereby satisfying assumption (A1).
For the critical manifold defined through the relationship \(\eta =G_{0}(x,v,t)\) in assumption (A2), we have the equations
$$\begin{aligned} P_{2}\left( x,v,\eta ,0,t;0\right) = M_{2}^{-1}\left[ \begin{array}{l} \beta \delta \cos xv^{2}-\varOmega _{h}^{2}\eta _{h}-\alpha _{h}\eta _{h}^{3}+(1+\beta )\delta +F_{h}(t)\delta \\ \quad -F_{p}(t)\delta \sin x-\frac{\beta }{\delta }\mu _{p}\sin xv-\beta \delta \sin ^{2}x+\beta \delta \sin xG_{p}(t)\\ \\ \beta \delta \sin xv^{2}-\varOmega _{d}^{2}\eta _{d}+F_{d}(t)\delta +F_{p}(t)\delta \cos x\\ \quad +\frac{\beta }{\delta }\mu _{p}\cos xv+\beta \delta \sin x\cos x-\beta \delta \cos xG_{p}(t) \end{array}\right] =\left[ \begin{array}{c} 0\\ 0 \end{array}\right] . \end{aligned}$$
Since \(M_{2}^{-1}\) is invertible, the critical manifold can be found by solving the following equations for \(\eta _{h}\) and \(\eta _{d}\):
$$\begin{aligned}&\varOmega _{h}^{2}\eta _{h}+\alpha _{h}\eta _{h}^{3}=T_{h}(x,v,t)=\beta \delta \cos xv^{2}\\&\quad +\,(1+\beta )\delta +F_{h}(t)\delta -F_{p}(t)\delta \sin x\\&\quad -\,\frac{\beta }{\delta }\mu _{p}\sin xv-\beta \delta \sin ^{2}x+\beta \delta \sin xG_{p}(t),\\&\varOmega _{d}^{2}\eta _{d}=T_{d}(x,v,t)=\beta \delta \sin xv^{2}+F_{d}(t)\delta \\&\quad +\,F_{p}(t)\delta \cos x+\frac{\beta }{\delta }\mu _{p}\cos xv+\beta \delta \sin x\cos x\\&\quad -\,\beta \delta \cos xG_{p}(t). \end{aligned}$$
The real roots of these two equations can be expressed explicitly as
$$\begin{aligned} \eta _{h}= & {} \root 3 \of {\frac{T_{h}(x,v,t)}{2\alpha _{h}}+\sqrt{\frac{T_{h}^{2}(x,v,t)}{4\alpha _{h}^{2}}+\frac{\varOmega _{h}^{6}}{27\alpha _{h}^{3}}}}\\&-\root 3 \of {-\frac{T_{h}(x,v,t)}{2\alpha }+\sqrt{\frac{T_{h}^{2}(x,v,t)}{4\alpha _{h}^{2}}+\frac{\varOmega _{h}^{6}}{27\alpha _{h}^{3}}}},\\ \eta _{d}= & {} \frac{T_{d}(x,v,t)}{\varOmega _{d}^{2}}, \end{aligned}$$
assuming that \(\varOmega _{h}^{2}, \varOmega _{d}^{2}\) and \(\alpha _{h}\) are greater than zero. The stability of this critical manifold is determined by the associated oscillatory system (9), whose coefficient matrices now take the specific form
$$\begin{aligned}&A(x,v,t) = -\partial _{w}P_{2}\left( x,v,G_{0}(x,v,t),0,t;0\right) \\&\quad = \frac{1}{1+\beta }\left[ \begin{array}{cc} 1+\beta \sin ^{2}x &{} -\beta \sin x\cos x\\ -\beta \sin x\cos x &{} 1+\beta \cos ^{2}x \end{array}\right] \left[ \begin{array}{cc} \mu _{h} &{} 0\\ 0 &{} \mu _{d} \end{array}\right] ,\\&B(x,v,t) = -\partial _{\eta }P_{2}\left( x,v,G_{0}(x,v,t),0,t;0\right) \\&\quad = \frac{1}{1+\beta }\left[ \begin{array}{cc} 1+\beta \sin ^{2}x &{} -\beta \sin x\cos x\\ -\beta \sin x\cos x &{} 1+\beta \cos ^{2}x \end{array}\right] \\&\qquad \times \left[ \begin{array}{cc} \varOmega _{h}^{2}+3\alpha _{h}\eta _{h}^{2} &{} 0\\ 0 &{} \varOmega _{d}^{2} \end{array}\right] . \end{aligned}$$
Consequently, the equilibrium solution of the unforced linear oscillatory system (9) is always asymptotically stable, given that
$$\begin{aligned} \mu _{h}\!>\!0,\quad \!\mu _{d}\!>\!0\quad \!\beta>0,\quad \varOmega _{h}^{2}\!>\!0,\quad \!\varOmega _{d}^{2}\!>\!0,\quad \alpha _{h}>0. \end{aligned}$$
We conclude that assumptions (A1)–(A3) hold, and hence a global reduced-order model exists over the softer variables \((x,v,t)\in \mathcal {D}_{0}=\mathbb {R}\times \mathbb {R}\times S^{1}\).
