Nonlinear normal modes and spectral submanifolds: existence, uniqueness and use in model reduction

Abstract

We propose a unified approach to nonlinear modal analysis in dissipative oscillatory systems. This approach eliminates conflicting definitions, covers both autonomous and time-dependent systems and provides exact mathematical existence, uniqueness and robustness results. In this setting, a nonlinear normal mode (NNM) is a set filled with small-amplitude recurrent motions: a fixed point, a periodic orbit or the closure of a quasiperiodic orbit. In contrast, a spectral submanifold (SSM) is an invariant manifold asymptotic to a NNM, serving as the smoothest nonlinear continuation of a spectral subspace of the linearized system along the NNM. The existence and uniqueness of SSMs turns out to depend on a spectral quotient computed from the real part of the spectrum of the linearized system. This quotient may well be large even for small dissipation; thus, the inclusion of damping is essential for firm conclusions about NNMs, SSMs and the reduced-order models they yield.

Appendices

Appendix: Existence, uniqueness and analyticity issues for invariant manifolds tangent to eigenspaces

1.1 Modified Euler example of a non-analytic but $C^{\infty }$ center manifold

For the system (1), the origin is a fixed point with eigenvalues $\lambda _{1}=0$ and $\lambda _{2}=-1$ and corresponding eigenvectors $e_{1}=(1,1)$ and $e_{2}=(0,1)$. Therefore, the classic center manifold theorem (see, e.g., Guckenheimer and Holmes [16]) guarantees the existence of a center manifold $W^{c}(0)$, tangent to the x-axis at the origin. We seek $W^{c}(0)$ in the form of a Taylor expansion

$$\begin{aligned} y=h(x)=x+\sum _{j=2}^{\infty }a_{j}x^{j}, \end{aligned}$$

which we differentiate in time to obtain

$$\begin{aligned} \dot{y}= & {} \left( 1+\sum _{j=2}^{\infty }ja_{j}x^{j-1}\right) \dot{x}\!=\!-\left( 1+\sum _{j=2}^{\infty }ja_{j}x^{j-1}\right) \nonumber \\ x^{2}= & {} -x^{2}-\sum _{j=2}^{\infty }ja_{j}x^{j+1}=-\sum _{j=2}^{\infty }(j-1)a_{j-1}x^{j},\nonumber \\ \end{aligned}$$

(74)

where we have let $a_{1}=1.$ At the same, we evaluate the second equation in (1) on the manifold $W^{c}(0)$ to obtain

$$\begin{aligned} \dot{y}=-h(x)+x=-\sum _{j=2}^{\infty }a_{j}x^{j}. \end{aligned}$$

(75)

Equating (74) and (75) gives the recursion $a_{j}=(j-1)a_{j-1}$ with $a_{1}=1,$ which implies $a_{j}=(j-1)!.$ We therefore obtain the explicit form

$$\begin{aligned} h(x)=\sum _{j=1}^{\infty }(j-1)!x^{j} \end{aligned}$$

(76)

as a formal expansion of the center manifold, as stated in the Introduction. The formal series $h(x)=\sum _{j=1}^{\infty }(j-1)!x^{j}$, however, diverges for any $x\ne 0$; thus, the center manifold is $C^{\infty }$ but not analytic in any open neighborhood of the origin.

1.2 Uniqueness and analyticity issues for invariant manifolds in linear systems

Any invariant manifold through the origin of the linearized system (12) is locally a graph over q of the elements of the vector y. Such a graph is of the general form

$$\begin{aligned} y_{l}=f_{l}(y_{j_{1}},\ldots ,y_{j_{q}}),\quad l\notin \left\{ j_{1},\ldots ,j_{q}\right\} . \end{aligned}$$

(77)

By the invariance of these surfaces, one can substitute full trajectories into (77) and differentiate in time to obtain the PDE

$$\begin{aligned} \lambda _{l}f_{l}=\sum _{i=1}^{q}\lambda _{j_{i}}y_{j_{i}}\partial _{y_{j_{i}}}f_{l},\quad l\notin \left\{ j_{1},\ldots ,j_{q}\right\} . \end{aligned}$$

(78)

This linear PDE can be solved locally by the method of characteristics (see, e.g., Evans [14]), once we prescribe the value of $f_{l}$ along an appropriate codimension-one set $\Gamma (s_{1},\ldots ,s_{q-1})$ of the spectral subspace $E_{j_{1},\ldots ,j_{q}}$. Here the real variables $s=(s_{1},\ldots ,s_{q-1})$ parametrize the surface $\Gamma$. For instance, $\Gamma$ can be selected as a $q-1$ dimensional sphere in $E_{j_{1},\ldots ,j_{q}}$ that surrounds the origin.

Fixing a boundary condition

$$\begin{aligned} f_{l}(\Gamma (s_{1},\ldots ,s_{q-1}))=f_{l}^{0}(s_{1},\ldots ,s_{q-1}) \end{aligned}$$

(79)

gives the equation for characteristics:

$$\begin{aligned}&y_{j_{i}}(t)=\Gamma _{i}(s_{1},\ldots ,s_{q-1})e^{\lambda _{j_{i}}t},\quad i=1,\ldots ,q.\nonumber \\\end{aligned}$$

(80)

$$\begin{aligned}&f_{l}(y_{j_{1}}(t),\ldots ,y_{j_{q}}(t))=f_{l}^{0}(s_{1},\ldots ,s_{q-1})e^{\lambda _{p}t}.\nonumber \\ \end{aligned}$$

(81)

Then, the strategy to obtain a solution for the PDE (78) is the following: express the variables $(s_{1},\ldots ,s_{q-1},t)$ as a function of $(y_{j_{1}},\ldots ,y_{j_{q}})=(y_{j_{1}}(t),\ldots ,y_{j_{q}}(t))$ from the q algebraic equations (80) in the vicinity of $\Gamma$ and substitute the result into (81) to obtain a solution $f_{l}(y_{j_{1}},\ldots ,y_{j_{q}})$ to (78) that satisfies the boundary condition (79).

To this end, we rewrite (80) as

$$\begin{aligned} \Gamma _{i}(s_{1},\ldots ,s_{q-1})e^{\lambda _{j_{i}}t}-y_{j_{i}}=0,\quad i=1,\ldots ,q,\nonumber \\ \end{aligned}$$

(82)

and observe that this system of q algebraic equations is solved by $t=0$ and $y_{j_{i}}^{0}=y_{j_{i}}(0)=\Gamma _{i}(s_{1},\ldots ,s_{q-1})$. By the implicit function theorem, the variables $(s_{1},\ldots ,s_{q-1},t)$ can be expressed from (82) near $\Gamma$ as a function of $y_{j_{i}}$ if the Jacobian

$$\begin{aligned}&D_{s_{1},\ldots ,s_{q-1},t}\left[ \begin{array}{c} \Gamma _{1}(s_{1},\ldots ,s_{q-1})e^{\lambda _{j_{1}}t}-y_{j_{1}}\\ \vdots \\ \Gamma _{q}(s_{1},\ldots ,s_{q-1})e^{\lambda _{j_{q}}t}-y_{j_{q}} \end{array}\right] _{(y_{j_{i}}=y_{j_{i}}^{0},t=0)}\nonumber \\&\quad =\left[ D_{s}\Gamma ,\,\,-\Lambda y\vert _{E_{j_{1},\ldots ,j_{q}}}\right] , \end{aligned}$$

(83)

is non-degenerate. In other words, along the surface $\Gamma$, all tangent vectors of $\Gamma$ should be linearly independent of the vector field $\Lambda y$ restricted to its invariant subspace $E_{j_{1},\ldots ,j_{q}}$. In the language of linear PDEs, the boundary surface $\Gamma$ should be a non-characteristic surface for a unique, local solution to exist near $\Gamma$ for any boundary condition posed over $\Gamma$. This argument just reproduces the classic local existence and uniqueness result for linear first-order PDEs (see, e.g., Evans [14]).

Under these conditions, therefore, we have a unique, local solution for any initial function $f_{l}^{0}(s_{1},\ldots ,s_{q-1})$ defined on $\Gamma$. There are infinitely many different choices both for the surface $\Gamma$ and the boundary values $f_{l}^{0}$. Since the Jacobian (83) is non-degenerate for any $y\ne 0$, each of these infinitely many choices leads to a local invariant surface satisfying (78) in the vicinity of $\Gamma$, which in turn can be propagated all the day to the $y=0$ fixed point along characteristics of the PDE. Accordingly, we obtain infinitely many invariant surfaces tangent to the spectral subspace $E_{j_{1},\ldots ,j_{q}}$ in the linearized system (12). Applying the more general Theorem 3 in the current linear setting, however, we obtain that only one analytic solution exists to the PDE (6) for any fixed subspace $E_{j_{1},\ldots ,j_{q}}$ under the nonresonance conditions detailed in Theorem 3. Since $f_{l}(y_{j_{1}},\ldots ,y_{j_{q}})\equiv 0$ is analytic, this flat solution must be the unique analytic solution of (78). All other solutions are only finitely many times differentiable and hence are not even $C^{\infty }$.

1.3 Uniqueness issues for invariant manifolds obtained from numerical solutions of PDEs

The PDE approach we described in Sect. 1 is broadly used in the literature to compute Shaw–Pierre-type invariant surfaces for nonlinear systems. This approach was originally suggested by Shaw and Pierre [39], explored first in detail first by Peschek et al. [32], then developed and applied further by various authors (see Renson et al. [37] for a recent review). Interestingly, none of these studies report or discuss non-uniqueness of solutions, which appears to be in contradiction with our conclusions in Sect. 1. Here we take a closer look to understand the reason behind this paradox.

In the simplified setting of Sect. 1, one may seek invariant manifolds of the form $y_{l}=f_{l}(y_{j_{1}},\ldots ,y_{j_{q}}), l\notin \left\{ j_{1},\ldots ,j_{q}\right\}$ in a nonlinear system

$$\begin{aligned}&\dot{y}=\Lambda y+g(y),\quad \Lambda =\mathrm {diag}\left( \lambda _{1},\ldots ,\lambda _{N}\right) ,\nonumber \\&g(y)=\mathbb {\mathcal {O}}\left( \left| y\right| ^{2}\right) , \end{aligned}$$

(84)

over a spectral subspace $E_{j_{1},\ldots ,j_{q}}$ of the operator A. The same argument we used in the linear case now leads to a quasilinear version of the linear system of PDEs (78). This quasilinear system of PDEs is of the form

$$\begin{aligned} \lambda _{l}f_{l}+g_{l}(y_{j},f)= & {} \sum _{i=1}^{q}\left[ \lambda _{j_{i}}y_{j_{i}}+g_{j_{i}}(y_{j},f)\right] \partial _{y_{j_{i}}}f_{l},\nonumber \\&l\notin \left\{ j_{1},\ldots ,j_{q}\right\} , \end{aligned}$$

(85)

with $y_{j}=(y_{j_{1}},\ldots ,y_{j_{q}})$ and f denoting the vector of the $f_{l}$ functions.

The local existence and uniqueness theory relevant to this PDE is identical to that for its linear counterpart (cf. Evans [14]). Specifically, as in Sect. 1, boundary conditions

$$\begin{aligned} f_{l}(\Gamma (s_{1},\ldots ,s_{q-1}))=f_{l}^{0}(s_{1},\ldots ,s_{q-1}), \end{aligned}$$

(86)

must be posed on a non-characteristic, codimension-one boundary surface $\Gamma$ inside the subspace $E_{j_{1},\ldots ,j_{q}}$ for the PDE (85) to have a unique local solution near $\Gamma$. Here the required non-characteristic property of $\Gamma$ is that the projected vector field $\dot{y}_{j}=\left[ \Lambda y+g(y)\right] _{j}$ over $E_{j_{1},\ldots ,j_{q}}$ should be transverse to $\Gamma$ at all points. Since this boundary condition is arbitrary, one again obtains infinitely many local Shaw–Pierre-type invariant manifolds near the boundary surface $\Gamma$ for the nonlinear problem (84): one for any boundary condition posed over any non-characteristic surface $\Gamma .$ In the general case, all of these are also global solutions that extend smoothly to the origin and give a smooth solution to the PDE (85) in a whole neighborhood of the fixed point. The only exception is when the invariant manifold is sought as a graph over the q fastest modes. In this case, the strong stable manifold theorem (Hirsch et al. [20]) guarantees the existence of a unique invariant manifold. In this case, while infinitely many local solutions still exist near a non-characteristic boundary surface $\Gamma$, these local solutions do not extend smoothly to the origin.

Surprisingly, all available numerical algorithms aiming to solve (85) in the nonlinear normal modes literature ignore this non-uniqueness issue. They are typically validated or illustrated on the computation of two-dimensional invariant manifolds tangent to the single, slowest decaying spectral subspace ($q=1, \dim E_{1}=2$). Already in this simplest case, the high degree of non-uniqueness illustrated in Fig. 2 definitely applies. This raises the question: How do these studies obtain a unique invariant manifold? There are different reasons for each numerical algorithm, as we review next.

Peschek et al. [32] consider a spectral subspace $E_{1}$ corresponding to a simple, complex conjugate pair of eigenvalues. They pass to amplitude-phase variables $(a,\varphi )$ by letting $y_{j_{1}}=ae^{i\varphi }$ and reconsider the quasilinear PDE (85) posed for the unknown functions $f_{l}(a,\varphi )$. As domain boundary $\Gamma$, they then consider the $a=0$ axis, over which they prescribe $f_{l}(0,\varphi )=0$ and $\partial _{a}f_{l}(0,\varphi )=0.$ This is consistent with the fact that the origin $y_{j_{1}}=0$ is mapped, due to the singularity of the polar coordinate change, to the $a=0$ of the $(a,\phi )$ coordinate space, and hence, the surface should have a quadratic tangency with this line. However, the $a=0$ line is invariant under the transformed nonlinear vector field $(\dot{a},\dot{\varphi )}$, given that it is the image of the fixed point of the original nonlinear system, which satisfies $\dot{a}=0$. As a consequence, $\Gamma$ is a characteristic surface, and hence, local existence and uniqueness are not guaranteed for the quasilinear PDE (85) with this boundary condition. As we discussed above, the PDE is in fact known to have infinitely many solutions, all of which have a quadratic tangency with the origin and hence satisfy the singular boundary conditions $f_{l}(0,\phi )=0$ and $\partial _{a}f_{l}(0,\phi )=0$ in polar coordinates. Therefore, the problem considered by Peschek et al. [32] only has a unique solution for invariant manifolds over the fast modes, but not over the slow or intermediate modes. The same holds true for all other studies utilizing the approach developed by Peschek et al. [32].

Renson et al. [36] solve the same quasilinear PDE (85) in the setting of Peschek et al. [32] (autonomous system with $q=1$ and with $\dim E_{1}=2$). In the conservative case, they seek to construct solutions using a closed boundary curve $\Gamma$ to which the nonlinear vector field $\dot{y}_{j}$ is tangent at each point. For damped systems, they solve the PDE outward from the equilibrium, first over an elliptic domain and then gradually outward over a nested sequence of annuli. The boundaries of all these domains are selected as non-characteristic curves; thus, a unique solution can be constructed over each domain in the nested sequence. Over the initial (elliptic) domain boundary, however, the spectral subspace itself is chosen as initial condition ($f_{l}^{0}(\Gamma )=0$ for all $l>2$), which singles out one special solution out of the arbitrarily many. The perceived uniqueness is, therefore, the artifact of the numerical procedure.

Finally, Blanc et al. [6] start out by correctly selecting a non-characteristic boundary curve $\Gamma$ in the amplitude–phase–coordinate setting of Peschek et al. [32] discussed above. This curve is just the $\varphi =0$ line of the $(a,\varphi )$ coordinate plane, to which the characteristics of the PDE are transverse in a neighborhood of the origin, as required for the local existence and uniqueness of solutions near $\Gamma$. In this case, any initial profile $f_{l}(a,0)=f_{l}^{0}(a)$ with $f_{l}^{0}(0)=0$ and $f_{l}^{0\prime }(0)=0$ would lead to a Shaw–Pierre-type invariant manifold, thereby revealing the inherent non-uniqueness of this numerical approach. Instead of realizing this, Blanc et al. [6] assert that there is a single correct boundary condition that they need to find by an optimization process.

In this optimization process, Blanc et al. [6] modify the initial boundary condition iteratively so that the computed PDE solution along the line $\varphi =2\pi$, given by $f_{l}(a,2\pi )$, is as close to $f_{l}(a,0)=f_{l}^{0}(a)$ as possible in the $L^{2}$ norm. Should they enforce the exact periodicity of the solution of the PDE on the periodic domain $(a,\varphi )\in [0,a_{max}]\times [0,2\pi ]$ (say, by a spectral method), they would always have $f_{l}(a,\varphi )\equiv f_{l}(a,0)$ on any solution, so minimizing the error in this identity would lead to a vacuous process. In other words, the seemingly unique solution in this approach is the surface along which the error arising from an inaccurate handling of the periodic boundary conditions is minimal in a particular norm.

Existence, uniqueness and persistence of non-autonomous NNMs

We rewrite system (5) in the form of a $(N+k)$-dimensional autonomous system

$$\begin{aligned} \dot{x}= & {} Ax+f_{0}(x)+\epsilon f_{1}(x,\phi ;\epsilon ),\nonumber \\ \dot{\phi }= & {} \Omega , \end{aligned}$$

(87)

defined on the phase space $\mathcal {P}=\mathcal {U}\times \mathbb {T}^{k}$. For $\epsilon =0$, the trivial normal mode $x=0$ now appears as an invariant, k-dimensional torus

$$\begin{aligned} T_{0}=\left\{ (x,\phi )\in \mathcal {P}\,:\,x=0,\,\,\phi \in \mathbb {T}^{k}\right\} \end{aligned}$$

for system (87).

Assume that all eigenvalues of A satisfy the condition $\mathrm {Re}\lambda _{i}\ne 0.$ This means that all possible exponential contraction and expansion rates transverse to $T_{0}$ dominate (the zero) expansion and contraction rates in directions tangent to $T_{0}$, along the $\phi$ coordinates. In the language of the theory of normally hyperbolic invariant manifolds, the torus $T_{0}$ is a compact, r-normally hyperbolic invariant manifold for any integer $r\ge 1$ (Fenichel [15]).

Fenichel’s general result on invariant manifolds does not allow, however, to conclude the persistence of $C^{0}, C^{\infty }$ or $C^{a}$ normally hyperbolic invariant manifolds. Instead, such persistence is established by Haro and de la Llave [18], who specifically study persistence of invariant tori in systems of the form of (87).

Existence, uniqueness and persistence for autonomous SSMs ($k=0$)

First, we recall a more abstract result of Cabré et al. [8] on mappings in Banach spaces, which we subsequently apply to our setting.

1.1 Spectral submanifolds for mappings on complex Banach spaces

We denote by $\mathcal {P}$ a real or complex Banach space and by $\mathcal {U}\subset \mathcal {P}$ an open set. We let $C^{r}(\mathcal {U},Y)$ denote the set of functions $f:U\rightarrow Y$ that have continuous and bounded derivatives up to order r in $\mathcal {U}$. Let the space $C^{\infty }(\mathcal {U},Y)$ denote the set of those functions f that are in the class $C^{r}(\mathcal {U},Y)$ for every $r\in \mathbb {N}$, and let $C^{a}(\mathcal {U},Y)$ denote the set of functions f that are bounded and analytic in U.

Let $0\in \mathcal {U}$ be a fixed point for a $C^{r}$ map $\mathcal {\mathcal {F}:}\mathcal {U}\rightarrow \mathcal {P},$ where $r\in \mathbb {N}\cup \{\infty ,a\}.$ We denote the linearized map at the fixed point by $\mathcal {A}=D\mathcal {F}(0)$ and its spectrum by $\mathrm {spec}(\mathcal {A}).$

We also assume a direct sum decomposition $\mathcal {P}=\mathcal {P}_{1}\oplus \mathcal {P}_{2}$, with the subspaces $\mathcal {P}_{1}$ and $\mathcal {P}_{2}$ to be described shortly in terms of the spectral properties of $\mathcal {A}$. We denote the projections from the full space $\mathcal {P}$ onto these two subspaces by $\pi _{1}:\mathcal {P}\rightarrow \mathcal {P}_{1}$ and $\pi _{2}:\mathcal {P}\rightarrow \mathcal {P}_{2},$ and assume that both projections are bounded. Finally, for any set S and positive integer k, we will use the notation

$$\begin{aligned} S^{k}=\underbrace{S\times \ldots \times S}_{k} \end{aligned}$$

for the k-fold direct product of S with itself.

Assume now that

• (0) $\mathcal {A}$ is invertible

• (1) The subspace $\mathcal {P}_{1}$ is invariant under the map $\mathcal {A}$, i.e.,

  $$\begin{aligned} \mathcal {A}\mathcal {P}_{1}\subset \mathcal {P}_{1}. \end{aligned}$$

  As a result, we have a representation of $\mathcal {A}$ with respect to above decomposition as

  $$\begin{aligned} \mathcal {A}=\left( \begin{array}{cc} \mathcal {A}_{1} &{} \mathcal {B}\\ 0 &{} \mathcal {A}_{2} \end{array}\right) , \end{aligned}$$

  (88)

  with the operators $\mathcal {A}_{1}=\pi _{1}\mathcal {A}\vert _{\mathcal {P}_{1}},$ $\mathcal {A}_{2}=\pi _{2}\mathcal {A}\vert _{\mathcal {P}_{2}},$ and $\mathcal {B}=\pi _{1}\mathcal {A}\vert _{\mathcal {P}_{2}}.$ If $\mathcal {P}_{2}$ is also an invariant subspace for $\mathcal {A}$, then we have $\mathcal {B}=0$.

• (2) The spectrum of $\mathcal {A}_{1}$ lies strictly inside the complex unit circle, i.e., $\mathrm {Spect}(\mathcal {A}_{1})\!\subset \!\left\{ z\in \mathbb {C}\,:\,\left| z\right| \!<\!1\right\}$.

• (3) The spectrum of $\mathcal {A}_{2}$ does not contain zero, i.e., $0\notin \mathrm {Spect}(\mathcal {A}_{2})$.

• (4) For the smallest integer $L\ge 1$ satisfying

  $$\begin{aligned} \left[ \mathrm {Spect}(\mathcal {A}_{1})\right] ^{L+1}\mathrm {Spect}(\mathcal {A}_{2}^{-1})\subset \left\{ z\in \mathbb {C}\,:\,\left| z\right| <1\right\} , \end{aligned}$$

  (89)

  we have

  $$\begin{aligned} \left[ \mathrm {Spect}(\mathcal {A}_{1})\right] ^{i}\cap \mathrm {Spect}(\mathcal {A}_{2})=\emptyset \end{aligned}$$

  (90)

  for every integer $i\in [2,L]$ (in case $L\ge 2)$.

• (5) $L+1\le r.$

We then have the following result:

Theorem 5

(Theorems 1.1 and 1.2, Cabré, Fontich and de la Llave [8]) Under assumptions (0–5):

• (i) There exists a $C^{r}$ manifold $\mathcal {M}_{1}$ that is invariant under $\mathcal {F}$ and tangent to the subspace $\mathcal {P}_{1}$ at 0.

• (ii) The invariant manifold $\mathcal {M}_{1}$ is unique among all $C^{L+1}$ invariant manifolds of $\mathcal {F}$ that are tangent to the subspace $\mathcal {P}_{1}$ at 0. That is, every two $C^{L+1}$ invariant manifolds with this tangency property will coincide in a neighborhood of 0.

• (iii) There exists a polynomial map $R:\mathcal {P}_{1}\rightarrow \mathcal {P}_{1}$ of degree not larger than L and a $C^{r}$ map $K:\mathcal {U}_{1}\subset \mathcal {P}_{1}\rightarrow \mathcal {P}$, defined over an open neighborhood $\mathcal {U}_{1}$ of 0, satisfying

  $$\begin{aligned}&R(0)=0,\quad DR(0)=\mathcal {A}_{1},\quad K(0)=0,\\&\pi _{1}DK(0)=I,\quad \pi _{2}DK(0)=0, \end{aligned}$$

  such that K serves as an embedding of $\mathcal {M}_{1}$ from $\mathcal {P}_{1}$ to $\mathcal {P}$, and R represents the pull-back of the dynamics on $\mathcal {M}_{1}$ to $\mathcal {U}_{1}$ under this embedding. Specifically, we have

  $$\begin{aligned} \mathcal {F}\circ K=K\circ R. \end{aligned}$$

• (iv) If, furthermore, $\left[ \mathrm {Spec}(\mathcal {A}_{1})\right] ^{i}\cap \mathrm {Spec}(\mathcal {A}_{1})=\emptyset$ holds for every integer $i\in [L_{-},L],$ then R can be chosen to be a polynomial of degree not larger than $L_{-}-1.$

• (v) Dependence on parameters: If $\mathcal {F}$ is jointly $C^{r}$ in x and a parameter $\mu$, the invariant manifold $\mathcal {M}_{1}$ is jointly $C^{r-L-1}$ in space and the parameter $\mu$. In particular, $C^{\infty }$ and analytic maps will have invariant manifolds that are $C^{\infty }$ and analytic, respectively, with respect to any parameters in the system.

1.2 Proof of Theorem 3

We now apply Theorem 5 to system (18). In this context, the space $\mathcal {P}$ is the finite-dimensional, real vector space $\mathcal {P}=\mathbb {R}^{N},$ and the mapping is the time-one map $\mathcal {F}=F^{1}:\mathcal {U\subset \mathcal {P}}\rightarrow \mathcal {P}$ of system (18). We further have

$$\begin{aligned} \mathcal {F}(0)=0,\quad \mathcal {A}=D\mathcal {F}(0)=DF^{1}(0)=e^{A}, \end{aligned}$$

(91)

and hence $\mathcal {A}$ is invertible. We have the spectra

$$\begin{aligned} \mathrm {spec}(\mathcal {A})= & {} \left\{ e^{\lambda _{1}},e^{\bar{\lambda }_{1}},\ldots ,e^{\lambda _{N}},e^{\bar{\lambda }_{N}}\right\} ,\nonumber \\ \mathrm {spec}(\mathcal {A}^{-1})= & {} \left\{ e^{-\lambda _{1}},e^{-\bar{\lambda }_{1}},\ldots ,e^{-\lambda _{N}},e^{-\bar{\lambda }_{N}}\right\} , \end{aligned}$$

(92)

where we have ordered the eigenvalues in an increasing order based on their real parts, i.e.,

$$\begin{aligned} \mathrm {Re}\lambda _{N}\le \ldots \le \mathrm {Re}\lambda _{1}<0, \end{aligned}$$

and listed purely real elements of the spectrum of $\mathcal {A}$ and $\mathcal {A}^{-1}$ twice to simplify our notation. Equation (92) implies that condition (0) of Theorem 5 is always satisfied.

For a given spectral subspace E, we let $\mathcal {P}_{1}=E,$ so that assumption (1) of Theorem 5 is satisfied. Because the real part of the spectrum of A is assumed to be strictly negative, the operator $\mathcal {A}$ defined in (91) satisfies assumptions (2–3) of Theorem 5.

Next we note that the smallest integer L satisfying

$$\begin{aligned} \left[ \mathrm {Spect}(\mathcal {A}_{1})\right] ^{L+1}\mathrm {Spect}(\mathcal {A}_{2}^{-1})\subset \left\{ z\in \mathbb {C}\,:\,\left| z\right| <1\right\} , \end{aligned}$$

is just the smallest integer that satisfies

$$\begin{aligned} \left[ e^{\max _{\lambda \in \mathrm {Spect}(A\vert _{E})}\mathrm {Re}\lambda }\right] ^{L+1}e^{\min _{\lambda \in \mathrm {Spect}(A)-\mathrm {Spect}(A\vert _{E})}\mathrm {Re}\lambda }<1. \end{aligned}$$

The solution of this inequality for a general real number L is

$$\begin{aligned} L>\frac{\min _{\lambda \in \mathrm {Spect}(A)-\mathrm {Spect}(A\vert _{E})}\mathrm {Re}\lambda }{\max _{\lambda \in \mathrm {Spect}(A\vert _{E})}\mathrm {Re}\lambda }-1, \end{aligned}$$

which, restricted to integer solutions, becomes

$$\begin{aligned} L\ge \sigma (E), \end{aligned}$$

with the relative spectral quotient $\sigma (E)$ defined in (15). The nonresonance condition (90) can then be written in our setting precisely in the form (20). Thus, under the assumptions of Theorem 3, the conditions of Theorem 5 are satisfied, and the statements of Theorem 3 are restatements of Theorem 5 in our present context.

1.3 Comparison with applicable results for normally hyperbolic invariant manifolds

Out of the three types of SSMs covered by Theorem 3, the existence of the slow SSMs (last column in Table 1) can also be deduced in a substantially weaker form from the classical theory of inflowing invariant normally hyperbolic invariant manifolds (Fenichel [15]). To show this, we first rescale variables via $x\rightarrow \delta x$ in system (18) to obtain the rescaled autonomous problem

$$\begin{aligned} \dot{x}=Ax+\delta \tilde{f}_{0}(x;\delta ),\quad \tilde{f}_{0}(x;\delta ):=\frac{1}{\delta ^{2}}f_{0}(\delta x). \end{aligned}$$

(93)

For $\delta =0$, this system coincides with the linearized system (6), while for $\delta >0$, it is equivalent to the full autonomous nonlinear system (18).

Assume now that the slow spectral subspace $E_{1,\ldots ,q}$ featured in Table 1 satisfies the strict inequality

$$\begin{aligned} \mathrm {Re}\lambda _{q+1}<\mathrm {Re}\lambda _{q}. \end{aligned}$$

This implies that $E_{1,\ldots ,q}$ is normally hyperbolic, i.e., all decay rates of the linearized system within $E_{1,\ldots ,q}$ are weaker than any decay rate transverse to $E_{1,\ldots ,q}$. Furthermore, a small compact manifold $\tilde{E}_{1,\ldots ,q}\subset E_{1,\ldots ,q}$ with boundary can be selected for the unperturbed limit ($\delta =0$) of system (93) such that $\dim \tilde{E}_{1,\ldots ,q}=\dim E_{1,\ldots ,q}$ and $\tilde{E}_{1,\ldots ,q}$ is inflowing invariant under the unperturbed limit of (93). This means that Ax points strictly outward on the boundary $\partial \tilde{E}_{1,\ldots ,q}$. Then, for $\delta >0$ small enough, the classic results of Fenichel [15] imply the existence of an invariant manifold $\tilde{W}(0)$ with boundary in system (93) that is $C^{1}$-close to $\tilde{E}_{1,\ldots ,q}$. Furthermore, $\dim \tilde{W}(0)=\dim E_{1,\ldots ,q}$ and the manifold $\tilde{W}(0)$ is of class $C^{\gamma }$, with

$$\begin{aligned} \gamma =\min \left( r,\mathrm {Int}\left[ \frac{\mathrm {Re}\lambda _{q+1}}{\mathrm {Re}\lambda _{q}}\right] \right) , \end{aligned}$$

(94)

which is the minimum of the degree of smoothness of (93) and the integer part of the ratio of the weakest decay rate normal to $\tilde{E}_{1,\ldots ,q}$ to the strongest decay rate inside $\tilde{E}_{1,\ldots ,q}$. Since $\delta >0$ has to be selected small in this result to keep the norm $\delta |\tilde{f}(0)|$ small enough, the above conclusion on the existence of $\tilde{W}(0)$ holds in a small enough neighborhood of $x=0$ in system (18).

This result might seem attractive at the first sight, as it requires no nonresonance conditions among the eigenvalues of the operator A. At the same time, the properties of $\tilde{W}(0)$ are substantially weaker than those obtained for $W_{1,\ldots ,q}(0)$ in Theorem 3. First, the degree $\gamma$ of differentiability for $\tilde{W}(0)$ (cf. formula (94)) is generally much lower than r, the degree of smoothness of system (18). In particular, even if (18) is analytic, the manifold $\tilde{W}(0)$ may well just be once continuously differentiable and hence cannot be sought in the form of a convergent Taylor expansion. Second, no uniqueness is guaranteed by the normal hyperbolicity results of Fenichel [15] for $\tilde{W}(0)$ within any class of invariant manifolds. Third, the whole argument is only applicable to slow SSMs, but not to intermediate and fast SSMs.

1.4 Comparison with results deducible from analytic linearization theorems

The analytic linearization theorem of Poincaré [34] concerns complex systems of differential equations of the form

$$\begin{aligned} \dot{y}=\Lambda y+g(y),\quad g(y)=\mathcal {O}\left( \left| y\right| ^{2}\right) , \end{aligned}$$

(95)

where $\Lambda \in \mathbb {C}^{N\times N}$ is diagonalizable and g(y) is analytic. If

• 1. all eigenvalues of $\Lambda$ lie in the same open half plane in the complex plane (e.g, $\mathrm {Re}\lambda _{j}<0$ for all j, as in our case), and

• 2. the nonresonance conditions $\left\langle m,\lambda \right\rangle \ne \lambda _{j}$ hold for all $l=1,\ldots ,N$ for all integer vectors $m=(m_{1},\ldots ,m_{N})$ with $m_{i}\ge 0,$ and $\sum _{i}m_{i}\ge 2$,

then there exists an analytic, invertible change of coordinates $z=h(y)$ in a neighborhood of the origin under which system (95) transforms to the linear system

$$\begin{aligned} \dot{z}=\Lambda z. \end{aligned}$$

(96)

The spectral subspaces of this linear system are all defined by analytic functions (trivially, flat graphs over themselves). As we discussed in Sect. 1, the spectral subspaces of nonresonant linear systems are in fact the only analytic invariant manifolds that are graphs over spectral subspaces.

Recall that the composition of two analytic functions is analytic and the inverse of an invertible analytic function is also analytic. We can, therefore, transform back the spectral subspaces of (96) under the analytic inverse mapping $h^{-1}(z)$ to conclude that (95) also has unique analytic SSMs tangent at the origin to any selected spectral subspace of the operator $\Lambda$. (Indeed, if (95) had more than one such analytic SSMs, then those would have to transform to nontrivial analytic SSMs of (96) under h(y), but no such nontrivial analytic SSMs exist in (96).) The unique analytic SSMs over spectral subspaces of (95) can in turn be extended to smooth global invariant manifolds under the reverse flow map of (95) up to the maximum time of definition of backward solutions.

Cirillo et al. [11] touch on parts of this argument for the existence of two-dimensional SSMs in autonomous nonlinear systems, without establishing uniqueness and analyticity in detail. These authors involve the Koopman operator (cf. Mezić [27]) in their arguments, but all spectral subspaces of a linear mapping are well defined without the need to view them as zero sets of Koopman eigenfunctions. (These subspaces are in fact the only invariant manifolds of the linearized system (96) out of the infinitely many that are expressible as zero sets of Koopman eigenfunctions under the nonresonance conditions given above.) Furthermore, as shown by the argument above, the restriction to two-dimensional SSMs is not necessary either.

The line of reasoning we gave above for the existence of autonomous SSMs is complete but applicable only under assumptions that limit its applicability in practice. Specifically, SSMs obtained from the analytic linearization are applicable only when the linear operator A in system (6) has no resonances, not even inside any of the spectral subspaces. This latter assumption is a limitation, as the main motivation in the nonlinear normal mode literature for multi-mode Pierre–Shaw-type invariant surfaces is precisely to deal with internal resonances inside a spectral subspace $E_{j_{1},\ldots ,j_{q}}.$ Furthermore, unlike Theorem 3, Poincaré’s result guarantees uniqueness only for analytic dynamical systems and only within the class of analytic SSMs. This is again a limitation in practice, as no finite order can be deduced over which a Taylor expansion will only approximate the unique SSM. A relaxation of Poincaré’s analytic setting to the case of finite differentiability is available (Sternberg [43]). In that setting, however, the uniqueness of SSMs can no longer be concluded within any function class, given that the local linearizing transformation h(y) is no longer unique.

Existence, uniqueness and persistence for non-autonomous SSMs ($k>0$)

First, we recall a more abstract result of Haro and de la Llave [18] on quasiperiodic mappings and their subwhiskers, which we subsequently apply to our setting.

1.1 Invariant tori and their spectral subwhiskers in quasiperiodic maps

We fix the finite-dimensional phase space $\mathcal {P=\mathbb {R}}^{N}\times \mathbb {T}^{k}$. On an open subset $\mathcal {U}=U\times \mathbb {T}^{k}\subset \mathcal {P}$ of this phase space, we consider a map $\mathcal {F}_{1}:\mathcal {U}\rightarrow \mathbb {R}^{N}$. For some $r\in \mathbb {N}\cup \{\infty ,a\}$ and $s\ge 2,$ we will say that the map $\mathcal {F}_{1}$ is of class ${C^{r,s},} \hbox {if}\,\mathcal {F}_{1}(x,\phi )$ is $C^{r}$ in its second argument $\phi \mathbb {\in T}^{k}$, and jointly $C^{r+s}$ in both of its arguments $(x,\phi )\in U\times \mathbb {T}^{k}$. In other words, if $\mathcal {F}_{1}\in C^{r,s}$ then $\partial _{\phi }^{i}\partial _{x}^{j}\mathcal {F}_{1}$ exists and is continuous for all indices $(i,j)\in \mathbb {N}^{2}$ satisfying $i\le r$ and $i+j\le r+s.$

Next we assume that for any $\phi \in \mathbb {T}^{k}$, the map $\mathcal {F}_{1}(\,\cdot \,,\phi$) is a local diffeomorphism. For a constant phase shift vector $\Delta \in \mathbb {R}^{k}$, we define the quasiperiodic mapping $\mathcal {F}=\left( \mathcal {F}_{1},\mathcal {F}_{2}\right) :\mathcal {U}\times \mathbb {T}^{k}\rightarrow \mathcal {P}$ as

$$\begin{aligned} \mathcal {F}(x,\phi )=\left( \mathcal {F}_{1}(x,\phi ),\mathcal {F}_{2}(\phi )\right) :=\left( \mathcal {F}_{1}(x,\phi ),\phi +\Delta \right) . \end{aligned}$$

Assume that $\mathcal {F}_{1}(0,\phi )=0$, i.e., $\mathcal {K}=\left\{ 0\right\} \times \mathbb {T}^{k}$ is an invariant torus for the map $\mathcal {F}$. Let $K:\mathbb {T}^{k}\rightarrow \mathbb {R}^{n}$ be a parametrization of the torus $\mathcal {K}$.

Next, we define the torus-transverse Jacobian

$$\begin{aligned} M(\phi )=D_{x}\mathcal {F}_{1}(0,\phi ) \end{aligned}$$

(97)

of the mapping component $\mathcal {F}_{1}$, and let $\nu :\mathbb {T}^{k}\rightarrow \mathbb {R}^{N}$ be any bounded mapping from the k-dimensional torus into $\mathbb {R}^{N}$. We then define the transfer operator $\mathcal {T}_{\Delta }:\nu \mapsto \mathcal {T}_{\Delta }\nu$ as a functional that maps the function $\nu$ into the function

$$\begin{aligned} \left[ \mathcal {T}_{\Delta }\nu \right] (\phi )=D_{x}\mathcal {F}_{1}(0,\phi -\Delta )\nu (\phi -\Delta ). \end{aligned}$$

(98)

Note that $\mathcal {T}_{\Delta }$ is just the torus-transverse component of the mapping $(\phi -\Delta ,\nu (\phi -\Delta ))\mapsto (\phi ,\left[ \mathcal {T}_{\Delta }\nu \right] (\phi ))$ which maps the vector $\nu (\phi -\Delta )$, an element of the normal space of the torus $\mathcal {K}$ at the base point $(0,\phi -\Delta )$, under the linearized map $D\mathcal {F}$ into a vector in the normal space of $\mathcal {K}$ at the base point $(0,\phi )$.

As long as $\nu$ is taken from the class of bounded functions, the spectrum of the operator $\mathcal {T}_{\Delta }$ does not depend on the smoothness properties of $\nu$ (see Theorem 2.12, Haro and de la Llave [18]). We will need the annular hull of the spectrum of $\mathcal {T}_{\Delta }$, defined as

$$\begin{aligned} \mathcal {A}=\left\{ ze^{i\alpha }\,:\,\,z\in \mathrm {Spect}\mathcal {T}_{\Delta },\,\,\alpha \in \mathbb {R}\right\} . \end{aligned}$$

(99)

This set is a union of circles in the complex plane, with each circle obtained by rotating an element of the spectrum of $\mathcal {T}_{\Delta }$.

We make the following assumptions:

• (0) The spectrum of the operator $\mathcal {T}_{\Delta }$ does not intersect the complex unit circle, i.e.,

  $$\begin{aligned} \mathrm {Spect}\mathcal {T}_{\Delta }\cap \left\{ z\in \mathbb {C}\,:\,\left| z\right| =1\right\} =\emptyset . \end{aligned}$$

• (1) There exists a decomposition of $N \mathcal {K}$, the normal bundle of $\mathcal {K}$, into a direct sum

  $$\begin{aligned} N\mathcal {K}=P_{1}\oplus P_{2} \end{aligned}$$

  of two $C^{r}$ subbundles, $P_{1},P_{2}\subset N\mathcal {K}$, such that $P_{1}$ is invariant under $M(\phi )$. As a consequence, a representation of $M(\phi )$ with respect to this decomposition is given by

  $$\begin{aligned} M=\left( \begin{array}{cc} M_{1}(\phi ) &{} B(\phi )\\ 0 &{} M_{2}(\phi ) \end{array}\right) . \end{aligned}$$

  The corresponding restrictions of the transfer operator $\mathcal {T}_{\Delta }$ onto functions mapping into $P_{1}$ and $P_{2}$ will be denoted as $\mathcal {T}_{1,\Delta }$ and $\mathcal {T}_{2,\Delta }$. The annular hulls $\mathcal {A}_{j}$ of the spectra of these restricted operators can be defined similarly to $\mathcal {A}$:

  $$\begin{aligned}&\mathcal {A}_{j}=\left\{ ze^{i\alpha }\,:\,\,z\in \mathrm {Spect}\mathcal {T}_{j,\Delta },\,\,\alpha \in \mathbb {R}\right\} ,\quad j=1,2,\nonumber \\&\quad \mathcal {A}_{1}\cup \mathcal {A}_{2}=\mathcal {A}. \end{aligned}$$

  (100)

• (2) The annular hull of $\mathrm {Spect}(\mathcal {T}_{1,\Delta })$ lies strictly inside the complex unit circle, i.e., $\mathcal {A}_{1}\subset \left\{ z\in \mathbb {C}\,:\,\left| z\right| <1\right\}$

• (3) For the smallest integer $L\ge 1$ satisfying

  $$\begin{aligned} \mathcal {A}_{1}^{L+1}\mathcal {A}^{-1}\subset \left\{ z\in \mathbb {C}\,:\,\left| z\right| <1\right\} , \end{aligned}$$

  (101)

  we have

  $$\begin{aligned} \mathcal {A}_{1}^{i}\cap \mathcal {A}_{2}=\emptyset \end{aligned}$$

  (102)

  for every integer $i\in [2,L]$ (in case $L\ge 2)$

• (5) $L+1\le s$

We then have the following result:

Theorem 6

(Haro and de la Llave [18]) Under assumptions (0–5):

• (i) There exists an invariant manifold $\mathcal {M}_{1}\subset \mathcal {P}$ that is a $C^{r,s}$ embedding of the subbundle $P_{1}$ into $\mathcal {P}$, and is tangent to $P_{1}$ along the torus $\mathcal {K}$.

• (ii) The invariant manifold $\mathcal {M}_{1}$ is unique among all $C^{r,L+1}$ invariant manifolds of $\mathcal {F}$ that are tangent to the subbundle $P_{1}$ along the torus $\mathcal {K}$. That is, every two $C^{r,L+1}$ invariant manifolds with this tangency property will coincide in a neighborhood of $\mathcal {K}$.

• (iii) There exists a map $R:P_{1}\rightarrow P_{1}$ that is a polynomial of degree not larger than L in the variable $\Delta$, of class $C^{r}$ in x and $C^{\infty }$ in $\phi$, and there exists a $C^{r,s}$ map $W:U_{1}\subset P_{1}\rightarrow \mathcal {P}$, defined over an open tubular neighborhood $U_{1}$ of the zero section of $P_{1}$, satisfying

  $$\begin{aligned}&R(0,\phi )=0,\quad D_{1}R(0,\phi )=M_{1},\\&\quad W(0,\phi )=K(\phi ),\quad \pi _{P_{1}}D_{1}W(0,\phi )=I{}_{P_{1}},\\&\quad \pi _{E_{2}}D_{2}W(0,\phi )=0 \end{aligned}$$

  for all $\phi \mathbb {\in \mathbb {T}}^{k}$, such that W serves as an embedding of $\mathcal {M}_{1}$ from $P_{1}$ to $\mathcal {P}$, and R represents the pull-back of the dynamics on $\mathcal {M}_{1}$ to $U_{1}$ under this embedding. Specifically, we have

  $$\begin{aligned} \mathcal {F}_{1}(W(\eta ,\phi ),\phi )=W(R(\eta ,\phi ),\phi +\Delta ) \end{aligned}$$

  in the tubular neighborhood $U_{1}$.

• (iv) If we further assume that for some integer $L_{-}\ge 2,$ we have $\mathcal {A}_{1}^{i}\cap \mathcal {A}_{1}=\emptyset$ for every integer $i\in [L_{-},L],$ then R can be chosen to be a polynomial of degree not larger than $L_{-}-1.$

• (v) If $\mathcal {A}_{2}\cap \left\{ z\in \mathbb {C}\,:\,\left| z\right| =1=\emptyset \right\}$ (i.e., the torus $\mathcal {K}$ is normally hyperbolic), then statements (i)–(iv) remain valid under small enough $C^{r,s}$ perturbations of the map $\mathcal {F}_{1}$. In particular, the invariant manifold $\mathcal {M}_{1}$ and its parametrization persist smoothly under small enough changes in parameters $\mu \in \mathbb {R}^{p}$ as long as for the new variable $\tilde{\phi }=\left( \phi ,\mu \right)$, the function $\mathcal {F}_{1}(x,\tilde{\phi })$ is of class $C^{r,s}$.

These results have been collected, with minor notational changes, from Theorem 4.1 and Remark 4.7 of Haro and de la Llave [18].

1.2 Proof of Theorem 4

We consider eq. (41) but will work with its equivalent autonomous form

$$\begin{aligned} \dot{x}= & {} Ax+f_{0}(x)+\epsilon f_{1}(x,\phi ,\epsilon ),\nonumber \\ \dot{\phi }= & {} \Omega . \end{aligned}$$

(103)

We will state the smoothness assumptions on $f_{0}$ and $f_{1}$ in more detail later. By (v) of Theorem 6, we can first establish the existence of various spectral submanifolds attached to the invariant torus $\mathcal {K}_{0}=\left\{ 0\right\} \times \Pi ^{k}$ of the $\epsilon =0$ limit of (103). We then conclude the existence of similar submanifolds attached to the quasiperiodic normal mode $x_{\epsilon }(t),$ represented by a perturbed invariant torus $\mathcal {K}_{\epsilon }$ for $\epsilon >0$ in the full perturbed system (103).

In the context of the above theorem, we are working on the phase space $\mathcal {P}\mathbb {=R}^{N}\times \mathbb {T}^{k}$ and an open neighborhood $\mathcal {U}=U\times \mathbb {T}^{k}$, where $U\subset \mathbb {R}^{N}$ is an open neighborhood of the fixed point $x=0$ of (41). We define the mapping $\mathcal {F}$ as the time-one map of the autonomous system (103) for $\epsilon =0$. , i.e.,

$$\begin{aligned} \mathcal {F}(x,\phi )= & {} \left( F_{0}^{1}(x),\phi +\Omega \right) :\mathcal {U}\rightarrow \mathcal {P},\nonumber \\ \mathcal {F}_{1}(x)= & {} F_{0}^{1}(x),\nonumber \\ \mathcal {F}_{2}(\phi )= & {} \phi +\Omega , \end{aligned}$$

(104)

with the map $F_{0}^{1}$ denoting the time-one map of $\dot{x}=Ax+f_{0}(x).$ By our assumptions, we have $\mathcal {F}_{1}(0)=0$, and hence, the torus $\mathcal {K}_{0}=$ is an invariant torus for the map $\mathcal {F}$ for $\epsilon =0$.

The Jacobian of the x-dynamics at $x=0$, as defined in (97), is

$$\begin{aligned} M(\phi )=D_{x}F_{0}^{1}(0)=e^{A}, \end{aligned}$$

and the transfer operator defined in (98) takes the form

$$\begin{aligned} \left[ \mathcal {T}_{\Omega }\nu \right] (\phi )=e^{A}\nu (\phi -\Omega ). \end{aligned}$$

We now Fourier expand the general function $\nu :\mathbb {T}^{k}\rightarrow \mathbb {R}^{N}$ as

$$\begin{aligned} \nu (\phi )=\sum _{\left| m\right| =1}^{\infty }\nu _{m}e^{i\left\langle m,\phi \right\rangle },\quad m\in \mathbb {Z}^{n}. \end{aligned}$$

Be definition, $\lambda \in \mathbb {C}$ is in the spectrum of the operator $\mathcal {T}_{\Omega }$ if $\left[ \lambda I-\mathcal {T}_{\Omega }\right] ^{-1}$ does not exist. After Fourier-expanding $\mathcal {T}_{\Omega }\nu$, we see that the non-invertibility of $\lambda I-\mathcal {T}_{\Omega }$ is equivalent to the non-solvability of

$$\begin{aligned} \sum _{\left| m\right| =1}^{\infty }\left( \lambda I-e^{-i\left\langle m,\Omega \right\rangle }e^{A}\right) \nu _{m}e^{i\left\langle m,\phi \right\rangle }=\sum _{\left| m\right| =1}^{\infty }\tilde{\nu }_{m}e^{i\left\langle m,\phi \right\rangle } \end{aligned}$$

for the coefficients $\nu _{m}$, where $\tilde{\nu }_{m}$ is arbitrary but fixed. This non-solvability arises precisely when

$$\begin{aligned} \det \left[ e^{A}-\lambda e^{i\left\langle m,\Omega \right\rangle }I\right] =0, \end{aligned}$$

i.e., when $\lambda e^{i\left\langle m,\Omega \right\rangle }$ is contained in the spectrum $e^{A}$. We conclude that the spectrum of $\mathcal {T}_{\Omega }$ is given by

$$\begin{aligned} \mathrm {Spect}\left( \mathcal {T}_{\Omega }\right) =\left\{ e^{\lambda _{j}-i\left\langle m,\Omega \right\rangle }:\,\,j=1,\ldots ,d;\,\,\,m\in \mathbb {N}^{k}\right\} , \end{aligned}$$

(105)

where $\lambda _{j}$ are the eigenvalues of A, listed in (7). By the definition (99), the annular hull of $\mathrm {Spect}\mathcal {T}_{\Omega }$ is therefore

$$\begin{aligned} \mathcal {A}=\left\{ z\in \mathbb {C}\,:\,\left| z\right| =e^{\mathrm {Re}\lambda _{j}}:\,\,j=1,\ldots ,d\right\} . \end{aligned}$$

(106)

For later reference, the analogous annular hull defined for the inverse of A is then

$$\begin{aligned} \mathcal {A}^{-1}=\left\{ z\in \mathbb {C}\,:\,\left| z\right| =e^{\mathrm {-Re}\lambda _{j}}:\,\,j=1,\ldots ,d\right\} . \end{aligned}$$

By assumption (42), Eq. (105) implies that hypotheses (0–2) of Theorem 6 are satisfied. To verify the remaining assumptions of the theorem, we note that the smallest integer L satisfying

$$\begin{aligned} \mathcal {A}_{1}^{L+1}\mathcal {A}^{-1}\subset \left\{ z\in \mathbb {C}\,:\,\left| z\right| <1\right\} \end{aligned}$$

is just the smallest integer that satisfies

$$\begin{aligned} \left[ e^{\max _{\lambda \in \mathrm {Spect}(A\vert _{E})}\mathrm {Re}\lambda }\right] ^{L+1}e^{\min _{\lambda \in \mathrm {Spect}(A)}\mathrm {Re}\lambda }<1. \end{aligned}$$

The solution of this inequality for a general real L is given by

$$\begin{aligned} L>\frac{\min _{\lambda \in \mathrm {Spect}(A)}\mathrm {Re}\lambda }{\max _{\lambda \in \mathrm {Spect}(A\vert _{E})}\mathrm {Re}\lambda }-1. \end{aligned}$$

The integer solutions of this inequality therefore satisfy

$$\begin{aligned} L\ge \Sigma (E), \end{aligned}$$

with the absolute spectral quotient $\sigma (E)$ defined in (16). The nonresonance condition (102) can be written in our setting precisely in the form (43). Thus, under the assumptions of Theorem 4, the conditions of Theorem 6 are satisfied. The statements of Theorem 4 are then just restatements of Theorem 6 in our present context.

1.3 Comparison with applicable results for normally hyperbolic invariant manifolds

As in the autonomous case, the existence of slow non-autonomous SSMs (last column of Table 2) could also be deduced in a substantially weaker form from the classic theory of inflowing invariant normally hyperbolic invariant manifolds (Fenichel [15]).

Following the approach taken in Appendix “Comparison with applicable results for normally hyperbolic invariant manifolds” for the autonomous case, we let $\delta =\sqrt{\epsilon }$ and use the rescaling $x\rightarrow \delta x$ in system (103) to obtain the equivalent dynamical system

$$\begin{aligned} \dot{x}= & {} Ax+\delta \left[ \tilde{f}_{0}(x;\delta )+f_{1}(\delta x,\phi )\right] ,\nonumber \\ \dot{\phi }= & {} \Omega . \end{aligned}$$

(107)

Assume that the slow spectral subspace $E_{1,\ldots ,q}$ featured in row (1) Table 2 satisfies the strict inequality

$$\begin{aligned} \mathrm {Re}\lambda _{q+1}<\mathrm {Re}\lambda _{q}. \end{aligned}$$

This implies that in the $\delta =0$ limit of system (107), the torus bundle $\mathcal {K}_{0}\times E_{1,\ldots ,q}$ is a normally hyperbolic invariant manifold, i.e., all decay rates of the linearized system within $\mathcal {K}_{0}\times E_{1,\ldots ,q}$ are weaker than any decay rate transverse to $E_{1,\ldots ,q}$. Furthermore, a small compact manifold $\mathcal {K}_{0}\times \tilde{E}_{1,\ldots ,q}\subset \mathcal {K}_{0}\times E_{1,\ldots ,q}$ with boundary can be selected such that $\dim \tilde{E}_{1,\ldots ,q}=\dim E_{1,\ldots ,q}$ and $\mathcal {K}_{0}\times \tilde{E}_{1,\ldots ,q}$ is inflowing invariant under the flow of (107) for $\delta =0$. This specifically means that the vector field $(Ax,\Omega )$ points strictly outward on the boundary $\partial (\mathcal {K}_{0}\times \tilde{E}_{1,\ldots ,q})=\mathcal {K}_{0}\times \partial \tilde{E}_{1,\ldots ,q}$ of $\mathcal {K}_{0}\times \tilde{E}_{1,\ldots ,q}$. Then, for $\delta >0$ small enough, the results of Fenichel [15] imply the existence of an invariant manifold $\tilde{W}$ with boundary in system (107) that is $\mathcal {O}(\delta )\, C^{1}$-close to $\mathcal {K}_{0}\times \tilde{E}_{1,\ldots ,q}$ within a small neighborhood of $\mathcal {K}_{0}$. Furthermore, $\dim \tilde{W}=\dim \tilde{E}_{1,\ldots ,q}+k$ and the manifold $\tilde{W}$ is of class $C^{\gamma }$ with the integer $\gamma$ defined in (94).

The limitations of this approach are identical to those discussed in Appendix “Comparison with applicable results for normally hyperbolic invariant manifolds.”

1.4 Comparison with results deducible from analytic linearization theorems

A time-quasiperiodic extension of the linearization theorem of Poincaré [34] (cf. Appendix “Comparison with results deducible from analytic linearization theorems”) is given by Belaga [5] (cf. Arnold [1]), covering differential equations of the form

$$\begin{aligned} \dot{y}= & {} \Lambda y+g(y,\phi ),\quad g(y,\phi )=\mathcal {O}\left( \left| y\right| ^{2}\right) ,\end{aligned}$$

(108)

$$\begin{aligned} \dot{\phi }= & {} \Omega , \end{aligned}$$

(109)

where $\Lambda \in \mathbb {C}^{N\times N}$is diagonalizable, $\phi \in \mathbb {T}^{k}$ and $g(y,\phi )$ is analytic. If

• 1. all eigenvalues of $\Lambda$ lie in the same open half plane in the complex plane (e.g, $\mathrm {Re}\lambda _{j}<0$ for all j in our setting), and

• 2. the nonresonance conditions $\lambda _{l}\ne \left\langle m,\lambda \right\rangle +i\left\langle p,\Omega \right\rangle$ hold for all integer vectors $m\in (m_{1},\ldots ,m_{N})$, with $m_{i}\ge 0,$ and $\sum _{i}m_{i}\ge 2$, and for all $p\in \mathbb {Z}^{k}$ ,

then there exists an analytic, invertible change of coordinates $z=h(y)$ in a neighborhood of the origin under which system (108) transforms to

$$\begin{aligned} \dot{z}= & {} \Lambda z,\nonumber \\ \dot{\phi }= & {} \Omega . \end{aligned}$$

(110)

The spectral subbundles of the trivial normal mode $\left\{ z=0\right\} \times \mathbb {T}^{k}$ in this system are all defined by analytic functions, given as direct products of flat graphs over any spectral subspace of $\Lambda$ with the torus $\mathbb {T}^{k}$. It follows from our discussion in Sect. 1 that these flat subbundles are the only analytic spectral subbundles of (110). Then, following the argument in Appendix section “Comparison with results deducible from analytic linearization theorems,” we conclude that (108) also has unique analytic, quasiperiodic SSMs, tangent at the origin to any selected spectral subspace of the operator $\Lambda$. These unique analytic SSMs over spectral subspaces of (108) can in turn be extended to smooth global invariant manifolds under the reverse flow map of (108) up to the maximum time of definition of backward solutions.

This construct has all the practical limitations already discussed Appendix “Comparison with results deducible from analytic linearization theorems,” plus two more. First, resonances with the external forcing are also excluded by the above nonresonance assumptions. Second, the term representing external, time-dependent forcing must be fully nonlinear in the phase space variables. The latter is rarely the case in mechanical models.

We close by noting that in the case of $k=1$ (single-frequency forcing), the above results of Belaga can be extended to cover time-periodic dependence in the linear operator $\Lambda$ as well (see Arnold [1]). This is the mechanical setting for the formal manifold calculations of Sinha et al. [42] and Redkar et al. [35]. The limitations of the linearization approach discussed above remain valid for this extension as well. In contrast, a direct application of Theorem 5 to the Poincaré map of (108) with $k=1$ gives sharp existence, persistence and uniqueness results for SSMs, assuming that the Floquet multipliers associated with the time-dependent linearization are known.

Similarly, if $\Lambda$ has quasiperiodic $(k>1)$ dependence on $\phi$, Theorem 6 formally applies to the quasiperiodic map associated with the linearized system, giving sharp existence, persistence and uniqueness results for SSMs in the nonlinear system. In this general case, however, the spectrum of the transfer operator $\left[ \mathcal {T}_{\Delta }\Omega \right] (\phi )$ defined in (98) is not known and requires a case-by-case analysis. For this reason, we have assumed throughout this paper the common mechanical setting in which the operator A of the linearized system is time-independent.