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

George Haller, Sten Ponsioen

arXiv:1602.00560v2math.DSnlin.PS

TL;DR

Nonlinear modal analysis lacks a unified, rigorous account of normal modes and invariant manifolds in dissipative and non-autonomous systems. The paper resolves this through unified terminology and mathematical results showing when spectral submanifolds exist uniquely, while highlighting damping and spectral gaps as important for model reduction.

  • Problem

    Nonlinear modal analysis lacks a unified relationship between normal modes and invariant manifolds, particularly in nonconservative and non-autonomous systems.

    Source: §1 · p. 1, §7 · p. 22

  • Method

    The paper defines nonlinear normal modes and spectral submanifolds within a unified framework and derives rigorous existence, uniqueness, smoothness, and robustness results.

    Source: §1 · p. 3, §9 · p. 36

  • Results

    NNMs are locally unique, while SSM uniqueness is governed by computable spectral quotients and nonresonance conditions; slow SSM uniqueness can be highly restricted in weakly damped systems.

    Source: §9 · p. 36

  • Takeaways & Limitations

    Rigorous SSM-based model reduction requires careful consideration of damping, spectral gaps, and the time scale targeted by the reduced model.

    Source: §8.3 · p. 33, §9 · p. 36

    The optimal SSM dimension depends on the time scale over which the reduced flow will approximate the full system dynamics.

    Source: §8.3 · p. 33

Abstract

from arXiv · show

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.

1 Introduction

The introduction identifies ambiguity and missing mathematical foundations in dissipative nonlinear normal-mode concepts, then proposes distinct NNMs and SSMs with rigorous existence, uniqueness, regularity, and robustness results for autonomous and forced systems. It emphasizes that these conditions depend strongly on spectral real parts, making damping essential for reliable model reduction.

Problem

  • Existing dissipative normal-mode concepts are ambiguous and lack firm foundations because periodic orbits are typically isolated while invariant manifolds may be absent or non-unique.The introduction also notes that proposed invariant surfaces may not persist nonlinearly, and truncated normal forms provide no a priori existence guarantee.Source: §1 · pp. 1–2
  • Formal Taylor expansions and numerical approximations do not ensure invariant-manifold existence or uniqueness, producing arbitrary reduced models when non-analytic manifold families differ globally.Euler’s example has a computable but divergent center-manifold series and infinitely many non-analytic center manifolds with different global shapes.Source: §1 · pp. 2–3

Proposed framework

  • The paper distinguishes NNMs as recurrent motions from SSMs as the smoothest invariant manifolds asymptotic to NNMs, covering autonomous and finitely forced systems.NNMs include fixed points, periodic orbits, and quasiperiodic-orbit closures; SSMs continue spectral subspaces along NNMs.Source: §1 · p. 3

Mathematical results

  • Classical and recent invariant-manifold theory yields computable existence, uniqueness, regularity, and robustness theorems for NNMs and SSMs from the linearized spectrum.The approach combines results by Fenichel, Cabré et al., and Haro and de la Llave.Source: §1 · p. 3

Implications

  • The relevant conditions depend more on the spectrum’s real parts than oscillation frequencies, so even weak damping must be included when constructing robust SSM-based reduced-order models.The paper illustrates these results with low-dimensional examples and discusses their model-reduction relevance.Source: §1 · p. 3

2 Set up

The study formulates weakly forced, damped mechanical systems and their first-order ODE representation to analyze nonlinear dynamics near an equilibrium. It seeks invariant solution sets that extend the relevance of linear oscillations beyond the small-amplitude, weak-forcing regime.

Mechanical formulation

  • The model describes n-degree-of-freedom mechanical systems with inertia, damping, gyroscopic, stiffness, follower-force, autonomous nonlinear, and weak external-forcing terms.The forcing has amplitude parameter 0 ≤ ϵ ≪ 1 and may be autonomous, periodic, or quasiperiodic depending on the frequency vector.Source: §2 · p. 4

Regularity assumptions

  • The nonlinearities and forcing are assumed to have class Cr regularity, where r may be finite, infinite, or analytic.The external forcing need not be nonlinear and can therefore remain large even when the state and velocity are small.Source: §2 · p. 4

Motivation

  • For ϵ = 0, the origin is an equilibrium, and linear oscillations are governed by the spectrum of the linearized system.A strict linear–nonlinear relationship is expected only near equilibrium and for small forcing, motivating nonlinear solution sets with broader phase-space influence.Source: §2 · p. 4

First-order formulation

  • Introducing x1 = q and x2 = ˙q yields the general first-order system ˙x = Ax + f0(x) + ϵf1(x, Ωt; ϵ), with f0(x) = O(|x|2).The state has dimension N = 2n for the mechanical system, while the general formulation is expressed using a constant matrix A and functions on U and the k-dimensional torus.Source: §2 · p. 4

3 Linear spectral geometry: Eigenspaces, normal modes, spectral subspaces, and invariant manifolds

The section builds the linear spectral geometry underlying modal analysis, from eigenvalues and real eigenspaces to spectral subspaces and their invariant manifolds. It shows that tangent invariant manifolds are generally nonunique, while smoothness and uniqueness are governed by spectral decay ratios.

3.1 Eigenvalues

  • Eigenvalues may be semisimple or possess Jordan blocks when algebraic multiplicity exceeds geometric multiplicity.Semisimplicity holds when algebraic and geometric multiplicities agree, including for distinct eigenvalues or symmetric matrices.Source: §3.1 · p. 5

3.2 Eigenspaces

  • Each distinct eigenvalue defines a real invariant eigenspace, with dimension alg(λj) for real eigenvalues and 2×alg(λj) for complex ones.Complex-conjugate eigenvalues generate the same real eigenspace, which is filled with trajectories of the linearized system.Source: §3.2 · p. 5

3.3 Linear normal modes

  • A linear normal mode is a periodic solution associated with a semisimple purely imaginary eigenvalue and fills its full eigenspace.For a typical simple mechanical mode, this eigenspace is a two-dimensional invariant plane; the origin is a singular, isolated normal mode.Source: §3.3 · p. 5

3.4 Spectral subspaces

  • Direct sums of distinct real eigenspaces are invariant spectral subspaces, including stable, unstable, and center subspaces classified by eigenvalue real parts.Damped mechanical systems commonly have only stable dynamics, making the full phase space stable and minimal spectral subspaces two-dimensional when eigenvalues are distinct and complex.Source: §3.4 · p. 6

3.5 Invariant manifolds in the linearized system

  • Invariant manifolds tangent to non-fast spectral subspaces are generally infinitely nonunique, whereas the fastest-decaying spectral subspace has a unique smooth invariant manifold.The flat graph is the unique analytic, and smoothest, invariant manifold over a non-fast spectral subspace; nonresonance conditions are required.Source: §3.5 · pp. 6–7, Figure 2 · p. 7

3.6 Spectral quotients

  • The spectral quotient measures maximal non-flat smoothness through the ratio of fastest external decay to slowest internal decay, motivating later existence and uniqueness results.Non-flat invariant graphs have only finitely many derivatives, while the spectral subspace itself is the only smoother member; relative and absolute quotients formalize these decay ratios.Source: §3.6 · pp. 8–9

4 Nonlinear spectral geometry: Nonlinear normal modes and spectral submanifolds

The paper distinguishes nonlinear normal modes (NNMs) as closures of multi-frequency recurrent motions from spectral submanifolds (SSMs) as smoothest invariant continuations of spectral subspaces along NNMs. This separates recurrent dynamical structures from geometrically defined invariant manifolds in dissipative systems.

Geometric distinction

  • NNMs and SSMs represent different dissipative-system structures: NNMs retain recurrent dynamics, whereas SSMs are nonlinear spectral-subspace continuations and need not be recurrent.The distinction generalizes Rosenberg’s normal modes for NNMs and Shaw–Pierre’s concept for SSMs.Source: §4 · pp. 9–11

Nonlinear normal modes

  • An NNM is the closure of a multi-frequency solution, including fixed-point, periodic, and quasiperiodic cases.Periodic NNMs are periodic orbits, while quasiperiodic NNMs form invariant tori when frequencies are rationally incommensurate.Source: §4 · p. 9

Spectral submanifolds

  • An SSM is an invariant subbundle along an NNM whose fibers smoothly continue a spectral subspace of the linearized system and have the highest available smoothness.For forced systems, the manifold is O(ϵ) C1-close to N × E; for zero forcing, it is tangent to E at the fixed point.Source: §4 · p. 10

Existence and uniqueness

  • SSM uniqueness is not automatic, because a smoothest invariant manifold may fail to exist or may not be unique among manifolds satisfying the defining tangency and perturbation conditions.The paper therefore derives conditions under which SSMs are unique and well-defined.Source: §4 · p. 10

5 Existence and uniqueness of NNMs

Classical invariant manifold theory closes the gap between formal NNM approximations and rigorous existence, uniqueness, persistence, and attraction results in dissipative systems. It establishes nearby trivial NNMs under autonomous forcing and unique attracting periodic or quasiperiodic NNMs under small quasiperiodic forcing.

5 Existence and uniqueness of NNMs

  • Formal asymptotic expansions and numerical procedures do not by themselves establish that the targeted nonlinear normal modes exist.The analysis addresses this conceptual gap using classical invariant manifold theory.Source: §5 · p. 11, §5.2 · p. 12

5.1 Trivial NNM under autonomous external forcing (k = 0)

  • Under autonomous forcing and Reλ_j < 0, a unique nearby trivial NNM x_ϵ = τ(ϵ) exists for sufficiently small nonzero ϵ, attracts nearby trajectories, and depends on ϵ in a C^r fashion.The same dissipative assumption ensures the NNM and associated stable-manifold SSM definitions are meaningful.Source: §5.1 · p. 11, §5 · p. 11

5.2 Periodic and quasiperiodic NNM under non-autonomous external forcing (k ≥1)

  • For small nonzero ϵ and quasiperiodic forcing with k ≥ 1 frequencies, a unique attracting NNM x_ϵ(t) = ϵτ(Ω_1t, . . . , Ω_kt; ϵ) exists and depends on ϵ in a C^r fashion.The function τ is 2π-periodic in each forcing-phase argument.Source: §5.2 · p. 12
  • The non-autonomous existence and attraction results are independent of resonances between forcing frequencies and the imaginary parts of A’s eigenvalues, while the NNM’s periodic or quasiperiodic nature depends on ϵ.The theorem mathematically supports two-timing, harmonic balance, numerical continuation, and multi-mode Fourier expansions.Source: §5.2 · p. 12
  • The rigorous connection between the forced NNM and the x = 0 equilibrium holds for small |ϵ| but generally disappears for large |ϵ| as the local phase-space structure changes.This contrasts with intuitive forced-response connections discussed by Kuether et al..Source: §5.2 · p. 12

6 Spectral submanifolds in autonomous systems (k = 0)

This section examines spectral submanifolds in autonomous systems as smoothest invariant continuations of spectral subspaces, focusing on their existence, uniqueness, and classification under nonresonance conditions. It also constructs and assesses a high-order approximation of the slow spectral submanifold.

6.1 Main result

  • Theorem 3 establishes a class C^r SSM tangent to a spectral subspace, unique among C^{σ(E)+1} invariant manifolds under low-order nonresonance conditions.The SSM persists smoothly or analytically under small parameter perturbations when f0 has the corresponding regularity.Source: §6.1 · p. 13

6.2 Application to specific spectral subspaces

  • Fast SSMs require no nonresonance condition and are analytic for analytic f0, but they generally contain atypical fastest-decaying trajectories and have limited model-reduction relevance.The fast SSM is unique among the corresponding invariant manifolds and admits convergent Taylor expansions.Source: §6.2 · p. 15
  • Intermediate SSMs, including Shaw–Pierre surfaces and Peschek et al. surfaces, exist and are uniquely selected at class C^{σ(E)+1} when external low-order resonances are absent.Resonances within the selected spectral subspace are allowed, but Taylor expansions of order σ(E)+1 or higher identify the unique intermediate SSM.Source: §6.2 · p. 15
  • Slow SSMs require nonresonance between slow and faster modes, with uniqueness among C^{σ(E)+1} manifolds; their nonresonance order is independent of the number of slow modes.The section identifies slow SSMs as the most promising option for model reduction.Source: §6.2 · p. 15
  • Resonance violations can eliminate a C2 slow SSM or produce infinitely many smooth, even analytic, invariant manifolds tangent to the same spectral subspace.Thus, the nonresonance hypotheses are sharp for both existence and uniqueness.Source: §6.2 · pp. 17–18
  • In the damped mechanical example, the analytic fast SSM is unique among C1 manifolds, while the analytic slow SSM is unique among C6 manifolds because the required nonresonances through order 5 hold.The cubic expansion is nonunique for the slow SSM, whereas a sixth-order expansion uniquely captures it; the required order increases sharply with overdamping.Source: §6.2 · p. 19
  • The slow SSM is obtained by equating powers to determine its unknown coefficients, and odd-order terms vanish because of the example’s nonlinearity.Source: §6.2 · p. 20
  • Trajectories launched from the approximate slow SSM remain close to it and converge to the trivial NNM at the origin.Source: Figure 8 · p. 20
  • The sixth-order approximation, equivalent here to fifth order because fourth-order terms are absent, substantially reduces trajectory oscillations relative to the third-order approximation of Shaw and Pierre.The improvement reflects reduced effects from the approximate SSM’s lack of exact invariance.Source: §6.2 · p. 21

7 Spectral submanifolds in non-autonomous systems (k > 0)

This section develops the existence, uniqueness, and parameter dependence of SSMs attached to periodic or quasiperiodic NNMs in non-autonomous systems, while characterizing different SSM types and illustrating the theory with forced examples. It also constructs and validates a unique slow SSM for the periodically forced Shaw–Pierre example, where the manifold captures trajectories converging to a leading-order attracting periodic NNM.

7 Spectral submanifolds in non-autonomous systems (k > 0)

  • Earlier non-autonomous SSM studies provided only formal calculations, leaving existence, uniqueness, convergence, and the connection to surviving NNMs unresolved.The unresolved NNM connection made the asymptotic behavior of trajectories in the proposed invariant manifolds unclear.Source: §7 · p. 22

7.1 Main result

  • Theorem 4 extends precise existence and uniqueness results from time-periodic to quasiperiodically forced systems, under asymptotic stability of the linearized dynamics and appropriate spectral nonresonance.The manifolds are unique among invariant surfaces at least CΣ(E)+1-times continuously differentiable in the x direction along the NNM.Source: §7.1 · p. 22
  • Under low-order nonresonance conditions, Theorem 4 guarantees a unique non-autonomous SSM attached to a periodic or quasiperiodic NNM, with dimension dim E + k and CΣ(E)+1 smoothness in x.Each time slice is O(ϵ) Cr-close to the spectral subspace E; if f0 and f1 are C∞ or analytic, the SSM depends on ϵ with the corresponding smoothness.Source: §7.1 · p. 22

7.2 Applications to specific spectral subspaces

  • Fast, intermediate, and slow non-autonomous SSMs correspond respectively to fastest-decaying, general, and slowest-decaying spectral subspaces, with trajectories asymptotic to the nontrivial NNM.Unlike the autonomous case, even fast SSMs require low-order nonresonance, while nonresonances involve only real parts of eigenvalues and therefore occur more readily.Source: §7.2 · p. 23
  • The periodic and quasiperiodic examples produce unique analytic slow SSMs, with periodic or quasiperiodic coefficient functions and trajectories converging to the corresponding unique NNM.The examples also show that generic invariant manifolds tangent to the slow subbundle may be only C4, whereas the distinguished SSM is analytic.Source: §7.2 · pp. 24–27
  • In the mechanical example, the slow SSM exists only for sufficiently small ϵ near a nonresonance threshold and is analytic and C6-unique, while the fast SSM requires no nonresonance condition but is only C2-unique.These conclusions follow from the relevant spectral quotients and nonresonance conditions in Table 2.Source: Table 2 · p. 23, §7.2 · p. 28
  • The leading-order approximation of the attracting periodic NNM is obtained after substituting the relevant parameter values into the governing expansion.Source: §7.2 · p. 29
  • The unique slow SSM W1(xϵ(t)), guaranteed by Theorem 4, is computed using a time-periodic Taylor expansion whose coefficient equations form a periodic boundary-value problem.The resulting ODEs are solved numerically, producing the slow SSM surface in extended phase space alongside the periodic NNM.Source: §7.2 · pp. 29–30
  • O(ϵ3) mean squared error separates the independently computed NNM from its projection onto the analytically computed SSM over one forcing period.For ϵ = 0.1, the instantaneous projection also shows a trajectory launched from the SSM converging to the analytically computed NNM.Source: Figure 14 · p. 30, §7.2 · p. 30

8 Relevance for model reduction

The section develops convergent expansions and reduced-order models on unique slow SSMs, while specifying how to choose their dimension and compute them locally or globally. It emphasizes that damping-dependent spectral gaps and quotient-based smoothness conditions determine approximation order, validity, and model fidelity.

8.1 Expansions for NNMs

  • Theorems 1 and 2 yield unique NNMs depending Cr-smoothly on amplitude, enabling Taylor expansions in autonomous systems and Taylor–Fourier expansions in non-autonomous systems.Autonomous NNMs are expanded in powers of ϵ, whereas non-autonomous NNMs include multi-frequency Fourier coefficients.Source: §8.1 · p. 31

8.2 Expansions for slow SSMs

  • Theorems 3 and 4 underpin reduced-order models on slow SSMs, whose trajectories remain active longest and require expansion order at least σ(E)+1 or Σ(E)+1 for uniqueness.These orders depend on fastest- and slowest-decaying damping rates and may be large even when the full spectrum has real parts near zero.Source: §8.2 · p. 31

8.3 The optimal dimension of the slow SSM

  • The optimal slow SSM dimension is selected from spectral gaps to balance dimensional reduction against the time scale over which the reduced flow must represent the full dynamics.The minimal SSM has dimension equal to the algebraic multiplicity of λ1; it is one-dimensional for a simple real eigenvalue and two-dimensional for a simple complex one.Source: §8.3 · pp. 32–33

8.4 Implications for the computation of NNMs and slow SSMs

  • Theorems 3 and 4 provide the mathematical foundation for systematically computing unique slow SSMs anchored to fixed-point, periodic, or quasiperiodic NNMs.The computational constructions use local expansions, PDEs, or global parametrizations while enforcing the relevant smoothness and nonresonance conditions.Source: §8.4 · p. 33, §8.4.1 · p. 33

8.4.1 Local Taylor–Fourier expansion for slow SSMs

  • Under analyticity and low-order nonresonance conditions, slow SSMs admit convergent Taylor or Fourier–Taylor expansions whose truncations approximate the unique manifolds.Lower-order truncations can also approximate other, less smooth invariant manifolds with similar tangent spaces.Source: §8.4.1 · pp. 33–34

8.4.2 Local PDEs for slow SSMs

  • PDE-based SSM approximations should penalize derivatives of order σ(E1,...,q)+1 or Σ(E1,...,q)+1 to exclude infinitely many less smooth invariant manifolds.This prescription addresses non-uniqueness arising when PDEs are solved with ill-posed or undetermined boundary conditions.Source: §8.4.2 · p. 34

8.4.3 Global parametrization of slow SSMs

  • Global parametrization avoids the fold-related breakdown of graph methods by embedding the spectral subspace into phase space and recursively solving for the embedding and reduced-flow coefficients.The reduced flow is exactly conjugate to a polynomial function whose order is no larger than K = Σ(E1,...,q).Source: §8.4.3 · pp. 34–36

9 Conclusions · A Existence, uniqueness and analyticity issues for invariant manifolds tangent to eigenspaces · A.1 Modified Euler example of a non-analytic but C∞center manifold

The paper unifies dissipative nonlinear modal analysis through rigorously characterized NNMs and SSMs, while showing that spectral quotients and damping govern SSM uniqueness and model-reduction relevance. A modified Euler example demonstrates that a center manifold can be C∞ but non-analytic.

9 Conclusions

  • The unified terminology provides rigorous existence, uniqueness, smoothness, and robustness results for NNMs and SSMs in dissipative systems.Source: §9 · p. 36
  • NNMs describe recurrent fixed-point, periodic, and quasiperiodic motions, whereas SSMs are the smoothest invariant manifolds asymptotic to them along spectral subbundles.Source: §9 · p. 36
  • NNMs are locally unique, and SSMs are unique over spectral subspaces without low-order resonances, with autonomous and non-autonomous conditions governed by σ(E) and Σ(E), respectively.Source: §9 · p. 36
  • Slow SSMs are most relevant for model reduction but have restricted uniqueness, requiring a distinguishing Taylor order exceeding the strongest-to-weakest decay-rate ratio.This ratio may be large even under weak damping, making damping essential for rigorous SSM-based reduction.Source: §9 · p. 36
  • SSMs remove the ambiguity of invariant manifolds tangent to modal subspaces, whose differences grow substantially away from fixed points and hinder defensible extension.Source: §9 · pp. 36–37

A.1 Modified Euler example of a non-analytic but C∞center manifold

  • For the modified Euler system, eigenvalues λ1 = 0 and λ2 = −1 yield a center manifold W c(0) tangent to the x axis at the origin.The classic center manifold theorem guarantees its existence.Source: §A.1 · p. 37
  • The center-manifold coefficients satisfy aj = (j − 1)!, so the formal series diverges for every x ≠ 0 and the manifold is C∞ but not analytic near the origin.Source: §A.1 · p. 37

A.2 Uniqueness and analyticity issues for invariant manifolds in linear systems … C Existence, uniqueness and persistence for autonomous SSMs

The paper distinguishes broad non-uniqueness of smooth invariant manifolds from analytically selected solutions, while identifying boundary, spectral, and hyperbolicity conditions governing existence and persistence. It also shows how numerical procedures can create perceived uniqueness without resolving the underlying ambiguity.

A.2 Uniqueness and analyticity issues for invariant manifolds in linear systems

  • For a fixed spectral subspace, infinitely many invariant surfaces can exist, but under Theorem 3’s nonresonance conditions only the flat analytic solution is unique; other solutions are not C∞.Non-characteristic boundary data generate unique local PDE solutions, and different boundary surfaces or values produce infinitely many invariant surfaces tangent to the same subspace.Source: §A.2 · pp. 38–39

A.3 Uniqueness issues for invariant manifolds obtained from numerical solutions of PDEs

  • In nonlinear systems, quasilinear invariance PDEs likewise require non-characteristic boundary data for local uniqueness, yet generally admit infinitely many smooth invariant manifolds extending to the fixed point.The exception is manifolds over the q fastest modes, for which the strong stable manifold theorem guarantees uniqueness at the origin.Source: §A.3 · p. 39
  • Numerical studies commonly overlook this non-uniqueness, even for two-dimensional manifolds tangent to the slowest-decaying spectral subspace, where it already applies.The issue therefore affects the simplest q = 1, dim E1 = 2 computations used in nonlinear normal-mode studies.Source: §A.3 · pp. 39–40
  • Peschek et al. impose conditions on the characteristic a = 0 boundary, so their procedure is unique only for fast modes, not slow or intermediate modes.Infinitely many solutions satisfy the same quadratic tangency conditions in polar coordinates because the selected boundary is characteristic.Source: §A.3 · p. 40
  • Renson et al. select one solution by initializing the nested-domain computation with the spectral subspace, making perceived uniqueness an artifact of the numerical procedure.Their later domain boundaries are non-characteristic, but the initial condition singles out one manifold from arbitrarily many candidates.Source: §A.3 · p. 40

B Existence, uniqueness and persistence of non-autonomous NNMs

  • For non-autonomous systems, the trivial normal mode becomes an invariant torus, and Reλi ≠ 0 makes it normally hyperbolic because transverse rates dominate tangent rates.Fenichel’s result alone does not establish persistence in the required C0, C∞, or Ca classes; Haro and de la Llave provide the relevant torus-persistence result.Source: B Existence, uniqueness and… · p. 41

C Existence, uniqueness and persistence for autonomous SSMs

  • The autonomous SSM analysis begins by recalling Cabré, Fontich and de la Llave’s abstract results for mappings in Banach spaces before applying them to the system.This establishes the abstract framework used for the subsequent existence, uniqueness, and persistence theory.Source: C Existence, uniqueness and… · p. 41

C.1 Spectral submanifolds for mappings on complex Banach spaces · C.2 Proof of Theorem 3

The appendix establishes existence, uniqueness, parametrization, and parameter regularity of invariant manifolds for Banach-space mappings, then applies these results to prove Theorem 3 using the time-one map and relative spectral quotient.

C.1 Spectral submanifolds for mappings on complex Banach spaces

  • Under assumptions (0)–(5), an invariant C^r manifold M1 exists for F and is tangent to the invariant subspace P1 at the fixed point.The setting assumes a fixed point of a C^r map on a real or complex Banach space and a bounded-projection decomposition P = P1 ⊕ P2.Source: §C.1 · pp. 41–42
  • M1 is unique locally among C^{L+1} invariant manifolds tangent to P1 at the fixed point.Any two such manifolds coincide in a neighborhood of 0.Source: §C.1 · p. 42
  • The manifold dynamics admit a polynomial reduced map R of degree at most L and an embedding K satisfying F ◦ K = K ◦ R.K embeds M1 from P1 into P, while R pulls the dynamics back to the parameter domain and has linearization A1 at 0.Source: §C.1 · p. 42
  • If F is jointly C^r in state and parameter, M1 is jointly C^{r−L−1} in both, retaining C∞ or analytic parameter dependence when F has that regularity.This gives parameter regularity for the invariant manifold under the stated smoothness assumptions.Source: §C.1 · p. 42

C.2 Proof of Theorem 3

  • For system (18), Theorem 5 is applied to its finite-dimensional time-one map F = F1 on P = R^N, with P1 chosen as a spectral subspace E.The proof uses the linearized operator’s invertibility and the assumed negative real parts of its spectrum to satisfy the relevant spectral assumptions.Source: §C.2 · p. 43
  • The smallest admissible integer L equals the integer relative spectral quotient σ(E), and the nonresonance condition (90) becomes condition (20).Consequently, under Theorem 3’s assumptions, all hypotheses of Theorem 5 hold and Theorem 3 follows as its specialization.Source: §C.2 · p. 43

C.3 Comparison with applicable results for normally hyperbolic invariant manifolds

Classical Fenichel theory can recover existence of slow SSMs under a normal-hyperbolicity condition, but with substantially weaker smoothness, uniqueness, and scope than Theorem 3.

C.3 Comparison with applicable results for normally hyperbolic invariant manifolds

  • Fenichel theory [17] yields a slow invariant manifold after rescaling the system, provided the selected slow spectral subspace satisfies the stated strict normal-hyperbolicity inequality.The rescaled problem reduces to the linearized system at δ = 0 and is equivalent to the nonlinear system for δ > 0.Source: §C.3 · pp. 43–44
  • For sufficiently small δ, the resulting manifold has the slow subspace’s dimension, is C1-close to a compact inflowing manifold, and exists only near x = 0.Its smoothness is Cγ, where γ is controlled by the system’s smoothness and a ratio of transverse to internal decay rates.Source: §C.3 · p. 44
  • Unlike Theorem 3, Fenichel’s manifold may have much lower differentiability, lack a convergent Taylor expansion even for analytic systems, and lack uniqueness within any invariant-manifold class.These limitations arise despite the absence of eigenvalue nonresonance conditions.Source: §C.3 · p. 44
  • The normal-hyperbolicity argument applies only to slow SSMs, not to intermediate or fast SSMs.Its existence conclusion is also restricted to a sufficiently small neighborhood of the origin.Source: §C.3 · p. 44

C.4 Comparison with results deducible from analytic linearization theorems · D Existence, uniqueness and persistence for non-autonomous SSMs (k > 0)

Under Poincaré’s nonresonance and half-plane assumptions, analytic linearization yields unique analytic SSMs that can extend globally, but only within a restrictive analytic setting. The non-autonomous case begins by invoking an abstract result on quasiperiodic mappings and sub-whiskers.

C.4 Comparison with results deducible from analytic linearization theorems

  • Poincaré’s theorem requires all eigenvalues to lie in one open half-plane and satisfy the stated integer nonresonance conditions.Source: §C.4 · p. 44
  • These assumptions provide an analytic invertible coordinate change that transforms the nonlinear system into a linear one near the origin.Source: §C.4 · p. 44
  • Because nonresonant linear systems have only flat analytic invariant graphs over spectral subspaces, the transformed system has no nontrivial analytic SSMs.Source: §C.4 · p. 44
  • Pulling spectral subspaces back through the analytic inverse establishes unique analytic SSMs tangent to selected spectral subspaces, extendable to smooth global manifolds under the reverse flow.Source: §C.4 · p. 45
  • The argument exceeds earlier two-dimensional Koopman-based treatments by establishing uniqueness and analyticity without restricting SSMs to two dimensions.Source: §C.4 · p. 45
  • Analytic linearization is limited by the absence of all resonances, including internal resonances, and guarantees uniqueness only for analytic systems within the analytic class.Source: §C.4 · p. 45
  • Finite-differentiability extensions remove analytic assumptions but lose uniqueness because the local linearizing transformation is no longer unique.Source: §C.4 · p. 45

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

  • For non-autonomous SSMs with k > 0, the development first recalls Haro and de la Llave’s abstract result on quasiperiodic mappings and sub-whiskers.Source: D Existence, uniqueness and… · p. 45

D.1 Invariant tori and their spectral sub-whiskers in quasiperiodic maps

For quasiperiodic maps with an invariant torus, a spectral sub-whisker exists as an invariant manifold tangent to a selected normally contracting subbundle. It is locally unique, admits reduced polynomial dynamics, and persists under suitable spectral separation and small perturbations.

D.1 Invariant tori and their spectral sub-whiskers

  • Theorem 6 establishes an invariant manifold M1 tangent to the subbundle P1 along the invariant torus K.M1 is a Cr,s embedding of P1 into P.Source: §D.1 · pp. 46–47
  • M1 is unique among Cr,L+1 invariant manifolds tangent to P1 along K, with any two such manifolds coinciding near K.Source: §D.1 · p. 47
  • The manifold dynamics pull back through an embedding W to a reduced map R that is polynomial of degree at most L in ∆, Cr in x, and C∞ in φ.W is defined over an open tubular neighborhood of the zero section of P1 and represents M1 in P.Source: §D.1 · p. 47
  • If Ai ∩ A1 = ∅ for every integer i ∈ [L−, L], the reduced map R can instead be chosen polynomial of degree at most L−−1.Source: §D.1 · p. 47
  • When A2 avoids the unit circle, the invariant manifold, reduced dynamics, and parametrization persist under sufficiently small Cr,s perturbations of F1 and parameter changes.This condition is identified with normal hyperbolicity of the torus K.Source: §D.1 · p. 47

D.2 Proofs of Theorem (4) · D.3 Comparison with applicable results for normally hyperbolic invariant manifolds

The proof recasts the non-autonomous system autonomously, derives the transfer-operator spectrum and verifies Theorem 6 conditions through the absolute spectral quotient. Comparison with Fenichel theory shows weaker existence and regularity results for slow non-autonomous spectral submanifolds under normal-hyperbolicity assumptions.

D.2 Proofs of Theorem (4)

  • The proof formulates the dynamics on P = R^N × T^k and uses the time-one map at ϵ = 0, for which K_0 is invariant.This establishes the invariant-torus setting needed for the transfer-operator analysis.Source: §D.2 · p. 47
  • The perturbed quasiperiodic normal mode is represented by an invariant torus K_ϵ, to which spectral submanifolds continue from those attached to K_0.The construction begins with the ϵ = 0 invariant torus and transfers the resulting submanifolds to the perturbed system.Source: §D.2 · p. 47
  • The transfer operator has spectrum determined by the eigenvalues of A shifted by torus frequencies, yielding annular radii e^Reλj.The annular hull is therefore the set of complex numbers with moduli e^Reλj.Source: §D.2 · p. 48
  • The proof verifies Theorem 6 by reducing its smoothness requirement to an inequality governed by the absolute spectral quotient σ(E).The nonresonance condition in the present setting is exactly condition (43), so Theorem 4 follows as a restatement of Theorem 6.Source: §D.2 · p. 48

D.3 Comparison with applicable results for normally hyperbolic invariant manifolds

  • Fenichel theory can also yield slow non-autonomous invariant manifolds, but only in a substantially weaker form than the stated spectral-submanifold results.This comparison concerns the slow non-autonomous SSMs listed in Table 2.Source: §D.3 · p. 49
  • After δ = √ϵ rescaling, a strict spectral inequality makes K_0 × E_1,...,q normally hyperbolic, with internal decay weaker than transverse decay.A compact inflowing invariant subset with boundary can then be selected inside this bundle.Source: §D.3 · p. 49
  • For sufficiently small δ > 0, Fenichel theory gives an invariant manifold Ẇ that is O(δ) C1-close, has dimension dim E_1,...,q + k, and class C^γ.The manifold is close to the selected compact inflowing invariant subset near K_0.Source: §D.3 · p. 49
  • The normal-hyperbolicity comparison retains the same limitations identified for the corresponding autonomous analysis.These limitations constrain the scope of the Fenichel-based alternative.Source: §D.3 · p. 49

D.4 Comparison with results deducible from analytic linearization theorems

Analytic linearization yields unique analytic quasiperiodic SSMs under same-half-plane and nonresonance assumptions, but excludes forcing resonances and generally requires fully nonlinear forcing. Direct application of the paper’s theorems instead provides sharper SSM results for periodic and quasiperiodic settings, subject to spectral information.

Analytic linearization

  • If all eigenvalues share an open half-plane and satisfy the stated nonresonance conditions, an analytic coordinate change transforms the system into ˙z = Λz, ˙φ = Ω.The nonresonance conditions exclude both internal and external-forcing resonances.Source: §D.4 · p. 49
  • The linearized system’s flat spectral subbundles imply unique analytic quasiperiodic SSMs tangent to any selected spectral subspace, extendable to smooth global invariant manifolds under reverse flow.These extensions remain valid up to the maximum time for which backward solutions are defined.Source: §D.4 · pp. 49–50

Limitations

  • The linearization approach excludes forcing resonances and assumes forcing is fully nonlinear in phase-space variables, a condition rarely met in mechanical models.These limitations apply to the time-periodic extension involving time-dependent linear operators as well.Source: §D.4 · p. 50

Direct theorem application

  • For single-frequency forcing, applying Theorem 5 to the Poincaré map gives sharp SSM existence, persistence, and uniqueness results when the Floquet multipliers are known.This contrasts with the limitations of the analytic linearization approach used for the mechanical setting of Sinha et al. and Redkar et al..Source: §D.4 · p. 50
  • For quasiperiodic linear dependence, Theorem 6 formally gives analogous sharp results, but the transfer-operator spectrum is unknown and requires case-by-case analysis.The paper therefore assumes the common mechanical setting of a time-independent linearized operator.Source: §D.4 · p. 50

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