Spectral Submanifolds (SSMs): Nonlinear Model Reduction

• Spectral Submanifolds (SSMs) are uniquely smooth, invariant manifolds that extend linear modal subspaces to nonlinear systems under strict nonresonance conditions.
• They provide a mathematical foundation for reduced-order modeling by capturing the slow dynamics and backbone curves in dissipative oscillatory systems.
• SSMs leverage spectral quotients and damping characteristics to ensure uniqueness, robustness, and high differentiability in nonlinear modal analysis.

Spectral Submanifolds (SSMs) are uniquely smooth, invariant manifolds that extend the concept of modal subspaces from linear dynamical systems into the nonlinear regime. SSMs furnish the mathematical foundation for model reduction in dissipative nonlinear oscillatory systems by “continuing” selected spectral subspaces of the linearized system along recurrent sets such as fixed points, periodic orbits, or invariant tori. Originally developed to resolve ambiguities in the nonlinear extension of modal notions, SSM theory ensures existence, uniqueness, and robustness of these invariant manifolds under precise algebraic nonresonance conditions. Their differentiability and persistence are governed by spectral quotients derived from the real parts of the linear spectrum, leading to deep connections between damping, modal separation, manifold smoothness, and reduced-order modeling.

1. SSMs as the Nonlinear Continuation of Spectral Subspaces

Given a general nonlinear system linearized at an equilibrium, $\dot{x} = A x + f_{nl}(x)$, the eigenstructure of $A$ produces invariant spectral subspaces—collections of eigenvectors associated with subsets of the spectrum (e.g., slowest decaying modes). In the linear system, these subspaces are invariant, but in the nonlinear setting, there exists an infinite family of invariant manifolds tangent to any chosen spectral subspace at the origin. SSM theory singles out the unique invariant manifold that is the “smoothest” (i.e., with the highest differentiability or analyticity class), thereby resolving the ambiguity in the nonlinear extension. The SSM is defined as the invariant manifold tangent at the origin (or along an invariant set) to a spectral subspace $E$ and is the unique member in its smoothness class under nonresonance conditions. The nonresonance requirements prevent the occurrence of algebraic resonances up to a finite order determined by the so-called spectral quotient

$$\sigma(E) = \mathrm{Int}\left[\frac{\min_{\lambda \notin \mathrm{Spect}(A|_E)} \mathrm{Re}(\lambda)}{\max_{\lambda \in \mathrm{Spect}(A|_E)} \mathrm{Re}(\lambda)}\right],$$

thus controlling the maximal differentiability class in which uniqueness holds (Haller et al., 2016). For time-periodic or quasiperiodic systems, the corresponding quotient $\Sigma(E)$ incorporates the frequencies of external excitation.

2. Relationship to Nonlinear Normal Modes (NNMs)

In nonlinear modal analysis, NNMs are defined as recurrent motion sets (fixed points, periodic orbits, closures of quasiperiodic orbits) of the nonlinear system. The SSM is not a mode in itself but an invariant manifold attached asymptotically to an NNM, serving as the nonlinear counterpart to a modal subspace of the linearization along the NNM. In this hierarchy, the NNM forms the “backbone” while the SSM constitutes the “fiber”—the unique smoothest manifold along the NNM that is tangent to the selected modal subspace. This distinction generalizes and unifies classic modal analysis by rigorously incorporating nonlinearities and dissipation, extending the notion of modal structures beyond conservative or synchronous oscillations to include dissipative and forced systems (Haller et al., 2016).

3. Spectral Quotient, Damping, and Regularity

The spectral quotient (or absolute spectral quotient for non-autonomous systems) quantifies the separation in decay (or growth) rates between the modes within the chosen spectral subspace (master modes) and those outside (slave modes). It determines both the existence and the minimal regularity class in which the SSM is unique. Damping, even when weak, crucially influences the spectral quotient: small real parts in the master subspace and much larger (typically more negative) real parts outside provide a large quotient, necessitating a higher order of differentiability (or analytic continuation) for the SSM to be defined uniquely. This is of practical importance in real mechanical systems, where dissipation often makes the slow SSM highly attractive, capturing all but the shortest transients, and eligible for an exact reduced-order description (Haller et al., 2016).

4. Construction and Mathematical Characterization

The SSM is constructed as the solution to an invariance equation, typically represented as a local graph over the spectral subspace or parameterized via the so-called parameterization method. For the autonomous case, the invariance equation has the form

$A W(p) + f_{nl}(W(p)) = DW(p) R(p),$

where $W(p)$ parameterizes the SSM and $R(p)$ gives the induced reduced dynamics on the manifold. For nonautonomous (periodically or quasiperiodically forced) systems, the parameterization becomes time-dependent, and additional Fourier indices extend the expansion,

$$W_\epsilon(p, t) = W_0(p) + \epsilon W_1(p, t) + \cdots,$$

with time derivatives appearing in the invariance equation (Haller et al., 2016). Uniqueness and regularity are enforced by solving the invariance relations order-by-order up to the spectral quotient, under the nonresonance constraints. For each candidate spectral subspace, the nonresonance conditions

$$\langle m, \lambda \rangle_E \neq \lambda_l,\qquad 2 \le |m| \le \sigma(E),\qquad \lambda_l \notin \mathrm{Spect}(A|_E)$$

must hold to rule out resonant denominator smallness in the Taylor–Fourier expansion.

5. Model Reduction and Reduced-Order Dynamics

The most significant application of SSMs is in model reduction. The unique attracting SSM corresponding to the slowest decaying subspace contains asymptotically all long-term trajectories that are not captured by short-lived transients. By parameterizing the dynamics on the SSM,

$x = W(p),\qquad \dot{p} = R(p),$

one obtains a reduced-order model (ROM) that preserves the essential oscillatory, amplitude–frequency, damping, and resonance properties of the original system, but in a lower dimension. The reduced dynamics $R(p)$ may capture nonlinear modal interactions and, when written in polar form (amplitude and phase), yield backbone curves—amplitude–frequency relationships that reflect hardening or softening behavior and can be continued analytically or numerically (Haller et al., 2016). For systems with external periodic or quasiperiodic forcing, the reduced dynamics likewise include explicit time-dependence, and the existence of SSMs still guarantees the validity of reduced dynamics that capture phenomena such as forced responses, bifurcations, and modal energy transfer.

6. Computational Methods and Practical Considerations

SSMs are computed in practice either through direct series expansion (solving invariance equations via multi-index or Kronecker product algebra), solution of associated partial differential equations (for explicit parametrization), or using the parameterization method of Cabré, Fontich, and de la Llave. For high-dimensional or complex systems—especially those exhibiting strong mode separation due to damping—efficient computation requires sophisticated numerical schemes, and the order of expansion must match or exceed the spectral quotient to guarantee accuracy and uniqueness. The inclusion of damping, nonresonance verifications, and suitable selection of spectral subspaces are crucial steps in ensuring reliability and robustness of the SSM-based ROMs (Haller et al., 2016). For time-dependent systems, additional challenges arise from the necessity to handle Floquet multipliers (for periodic orbits) and nontrivial phase dependence.

A summary of the salient mathematical constructs is provided in the table below:

Key Concept	Mathematical Definition	Role in SSM Theory
Spectral Subspace $E$	Subspace of eigenvectors of $A$	Anchor for SSM construction
Spectral Quotient	$$\sigma(E)=\mathrm{Int}[\frac{\min_{\lambda\notin E}\mathrm{Re}(\lambda)}{\max_{\lambda\in E}\mathrm{Re}(\lambda)}]$$	Limits smoothness/regularity
Nonresonance Condition	$$\langle m,\lambda \rangle_E \neq \lambda_l, 2\le |m|\le \sigma(E)$$	Ensures uniqueness/existence
SSM Parameterization	$x=W(p)$ or $z=h^\epsilon(y,t)$	Manifold as graph/embedding
ROM Dynamics	$\dot{p}=R(p)$	Reduced-order model

7. Impact, Limitations, and Extensions

The SSM framework provides an exact, mathematically justified pathway for the reduction of dissipative nonlinear systems, bridging linear modal analysis and fully nonlinear phenomena—including backbone curve identification, bifurcation analysis, and robust model reduction. The uniqueness and robustness theorems anchor the use of SSMs as the “backbone” for data-driven modeling, control, and prediction in high-dimensional mechanical and structural systems.

Primary limitations arise in cases where the nonresonance conditions are violated or the spectral gap is small, in which case the smoothness and persistence of the SSM can be reduced, or multiple invariant structures may coexist. For highly nonlinear or degenerate cases, a deeper analysis may be necessary, possibly invoking extensions to fractional or mixed-mode SSMs.

In summary, spectral submanifolds constitute the mathematically rigorous backbone for nonlinear modal analysis and model reduction in dissipative oscillatory systems, furnishing unique, invariant, and maximally smooth manifolds attached to NNMs that enable extraction of dynamically faithful reduced-order models—both in theory and computation (Haller et al., 2016).