Nonlinear Normal Modes in Dynamics

Updated 21 February 2026

Nonlinear normal modes (NNMs) are two-dimensional invariant manifolds in nonlinear systems that capture amplitude-dependent frequencies and modal interactions.
They are computed through methods such as shooting and continuation, invariant manifold formulations, Floquet theory, and modern deep learning techniques.
NNMs enable reduced-order modeling, energetically efficient control, and spectral analysis across mechanical, electrodynamic, and molecular systems.

Nonlinear normal modes (NNMs) constitute the fundamental building blocks of phase-space–invariant dynamics in nonlinear systems. Unlike linear normal modes, which are superpositions of harmonic oscillations with fixed shapes and frequencies, NNMs reflect the full nonlinear character of the governing equations, leading to amplitude-dependent frequencies, possible modal interactions, and geometrically intricate invariant structures. NNMs find application in analytical, computational, and experimental settings, spanning mechanical, electrodynamic, and molecular systems.

1. Mathematical Formulation and Geometric Structure

NNMs are rigorously defined as two-dimensional invariant manifolds embedded in the full phase space of a nonlinear dynamical system. Considering an $n$ –degree-of-freedom autonomous system

$\ddot{x}(t) + f(x(t),\dot{x}(t)) = 0, \qquad x\in\mathbb{R}^n,$

NNMs are characterized either as:

Rozenberg’s synchronous NNMs, where all coordinates exhibit proportional, time-periodic motion, i.e., $x_i(t)=a_i f(t),\ \forall i$ ; or,
Generalized (Shaw–Pierre) NNMs, in which each mode is a two-dimensional invariant manifold (through the fixed point) tangent to a modal eigenspace of the linearized system (Haller et al., 2016).

Each NNM may be further parameterized by action–phase variables $(r, \theta)$ , where the invariant manifold is locally described by

$x=W(r,\theta),\qquad \frac{\partial W}{\partial \theta} = \text{vector field on manifold},$

with

$\dot{\theta} = \Omega(r)$

yielding amplitude-dependent frequencies and nonlinear geometric distortions (Sachtler et al., 2024, Noël et al., 2016).

2. Computational Techniques for Nonlinear Normal Modes

Identification and computation of NNMs in finite-dimensional and distributed systems utilize several numerical and analytical strategies:

Shooting and Continuation: The periodic orbit formulation seeks states $x_0$ and periods $T$ such that $x(T;x_0)-x_0=0$ . A Newton–Raphson scheme iterates $F(x_0,T)=0$ with corrections computed by integrating the variational equations over one period, and branches are traced using pseudo-arclength continuation. This machinery underpins high-fidelity backbone curves $\Omega(E)$ and enables path-following through folds and internal resonances (Noël et al., 2016, Sachtler et al., 2024).
Invariant Manifold and PDE Methods: The invariance PDE for the NNM manifold $W(r,\theta)$ is solved—often by power series expansion—to arbitrary order by imposing that the manifold be flow-invariant. The main computational burden is the accurate construction of $W$ and resolution of the amplitude dependence (Simpson et al., 2020).
Floquet Theory and Variational Analysis: To assess the stability of a periodic orbit (NNM), one linearizes about the NNM and computes the monodromy (Floquet) matrix from the period $T$ , extracting multipliers. All transverse multipliers modulus $<\!1$ guarantee (local) invariance and modal isolation (Peng et al., 2020, Sachtler et al., 2024).
Group-Theoretical and Symmetry-Based Methods: For systems with discrete symmetry, especially lattices and molecular systems, group-theoretical selection rules and bush theory guarantee the existence of particular NNM families (e.g., bushes), grant closure of equations within these subspaces, and systematize mode coupling (Chechin et al., 2015, Chechin et al., 2021).

A representative table of computational approaches is below:

Methodology	Primary Use	Reference Examples
Shooting & continuation	Families of periodic NNMs	(Noël et al., 2016, Sachtler et al., 2024)
Invariant manifold PDEs	Global phase-space NNM structure	(Simpson et al., 2020)
Floquet multipliers	Stability/bifurcations	(Peng et al., 2020, Sachtler et al., 2024)
Group-theoretical	Symmetry-driven bushes	(Chechin et al., 2015, Chechin et al., 2021)

3. Key Properties: Energy Orthogonality, Backbone Curves, and Instabilities

NNMs support a variety of amplitude-dependent phenomena absent from linear modal analysis:

Amplitude-Dependent Frequencies ("Backbones"): The period $T$ (frequency $\Omega$ ) of an NNM generally varies strictly with energy, as captured by backbone curves $\Omega(E)$ . In mechanical and electrodynamic examples, the backbone's curvature (hardening/softening) encodes the nature of the nonlinearity (Kudrin et al., 2015, Kuether et al., 2016).
Energy Orthogonality: In certain analytically tractable nonlinear PDEs, such as exponential-nonlinearity Maxwell cavities, the total energy for a sum of NNMs is the sum of their energies—preserving orthogonality to all orders in nonlinearity (Kudrin et al., 2015).
Intermode Interactions and Modal Instability: NNMs persist as isolated invariant objects up to instabilities (e.g., modulational or Floquet instabilities), beyond which modal energy may rapidly delocalize—a mechanism underlying thermalization in FPU chains (Peng et al., 2020). The onset of instability typically follows universal scaling laws, e.g., inverse square-root divergence of the stability time near threshold.

Modern NNM identification extends beyond analytical models to experimental and data-driven frameworks:

Phase-Quadrature and Force Appropriation: Experimental protocols aim to isolate NNMs by imposing harmonic excitation tuned so that response and forcing are in phase quadrature. However, with underactuation (fewer inputs than modal degrees of freedom), this technique may substantially deviate from the true NNM locus except in narrow amplitude/frequency bands identifiable via phase error analysis (Renson et al., 2018).
Broadband Forcing and System Identification: Nonlinear subspace approaches extract state-space models from broadband (multisine) input–output data, enabling numerical continuation in modal coordinates to recover the NNM backbone and associated amplitude–frequency relations (Noël et al., 2016).
Physics-Constrained Deep Learning and Normalizing Flows: Recent work employs invertible deep architectures (autoencoders, normalizing flows) trained with physics-motivated loss functions (consistency with dynamics, modal independence) to extract nonlinear modal coordinates and NNMs directly from high-dimensional measured data. Modal decomposition in latent spaces enables backbone extraction, mode reconstruction, and long-range system prediction—even in experimental regimes where analytical models are unavailable (Rostamijavanani et al., 11 Mar 2025, Rostamijavanani et al., 23 Jan 2025).

The structure of NNMs underpins advanced strategies for control and model reduction:

Energetically Efficient Control: By stabilizing trajectories on NNM manifolds and gently ramping modal amplitudes, closed-loop controllers can accomplish complex maneuvers (e.g., swing-up of weakly actuated double pendulums) with minimal energy input, as NNMs represent passive system oscillations (Sachtler et al., 2024).
Reduced-Order and Surrogate Modeling: NNMs serve as nonlinear reduction bases, supporting the construction of low-dimensional models that preserve modal invariance and capture essential amplitude-dependent dynamics, outperforming linear projections and proper orthogonal decomposition (POD) in highly nonlinear settings. Neural manifold learning and LSTM regression in NNM coordinates accelerate simulation and enable efficient system analysis under varying external inputs (Simpson et al., 2020).

6. Symmetry, Selection Rules, and Coupled-Mode Theory

Systems with spatial or discrete symmetry (molecular, lattice, electrical) admit further specialization through the theory of bushes:

Bushes of Nonlinear Normal Modes: A bush comprises a finite-dimensional invariant subspace of NNMs, closed under the full nonlinear dynamics by symmetry selection rules. Excitation of a root mode within a bush ensures that energy remains trapped, with dynamics governed by coupled amplitude equations. Density functional theory investigations confirm that these symmetry-determined NNMs and their bushes define isolated dynamical structures in realistic molecular and condensed-matter systems (Chechin et al., 2015, Chechin et al., 2021, Chechin et al., 2022).
Selection Rules and Conservation Laws: Only modes belonging to certain symmetry-adapted subspaces may interact nonlinearly—preventing or allowing targeted energy transfer. Selection rules precisely identify root modes and bushes immune to cross-excitation, with experimental protocols exploiting these properties for controlled mode excitation and energy routing (Chechin et al., 2022).
Topological and Nonlinear Extensions: In strongly nonlinear regimes (e.g., topological interface lattices), NNM backbones and associated geometric phases (e.g., Zak phase) predict nonlinear generalizations of topologically protected states, with stability and excitability thresholds determined by topological invariants and backbone merging points (Tempelman et al., 2021).

7. Advanced Spectral Perspectives and Koopman Operator Methods

Spectral operator theory provides a global framework for the characterization of NNMs:

Koopman Operator and Spectral Characterization: NNMs correspond to zero-level sets of certain Koopman eigenfunctions. The full state flow can be expressed as a spectral sum over products of these eigenfunctions, with the dynamics on the NNM mapped onto a linear oscillator in the modal coordinates. This enables global parametrization of the invariant manifold, removing the obstructions of local “folding” that compromise classical master–slave coordinate expansions (Cirillo et al., 2015, Das et al., 2024).
Extensions and Limitations: Koopman-based NNM identification is global and constructive under non-resonant, hyperbolic equilibria. Proximity to internal resonances or non-hyperbolic points can affect convergence and accuracy. As dissipation decreases, the spectral quotient increases and formal existence and uniqueness of smooth spectral submanifolds (SSMs) must be carefully reassessed (Haller et al., 2016, Das et al., 2024).

Overall, the rigorous construction, computation, and exploitation of nonlinear normal modes unify large classes of nonlinear vibrational behaviors, enabling sophisticated analysis, control, and data-driven modal decomposition across physical domains from mechanics to electrodynamics and fluids. Recent advances leverage deep learning, symmetry theory, and spectral operator analysis to extend the reach of NNMs into complex, high-dimensional, and experimentally challenging systems.