Nonlinear Mode-Crossing Dissipation

• Nonlinear mode-crossing dissipation is a phenomenon where coupled nonlinear modes create amplitude-dependent frequency shifts that trigger resonance crossings and energy exchange.
• The concept is modeled using Duffing-type oscillators, driven envelope equations, and nonlinear Schrödinger systems, revealing nonmonotonic and even negative effective damping.
• These insights enable advanced dissipation engineering and stabilization methods through phase locking, separatrix crossings, and controlled energy plateaus.

Nonlinear mode-crossing dissipation refers to dissipation phenomena that arise when coupled nonlinear modes in oscillatory, optical, or mechanical systems experience resonance crossings or internal resonances. These crossings, induced by amplitude-dependent frequency shifts or external drives, enable energy exchange between modes and can result in nontrivial, often nonmonotonic, dissipation dynamics fundamentally distinct from linear damping. The effect is manifest in mechanical resonators, photonic microcavities, levitated solids, and model nonlinear wave systems. Crucially, the interplay of nonlinearity, multimode coupling, and weak damping leads to rich dynamical structures: step-like energy loss, persistent phase locking with energy plateaus, separatrix crossings in phase space, and amplitude-dependent or even negative effective damping. Nonlinear mode-crossing dissipation not only modifies the standard understanding of dissipation in weakly damped systems but also underpins modern schemes for dissipation engineering and the stabilization of non-Hamiltonian attractors.

1. Fundamental Models of Nonlinear Mode-Crossing Dissipation

The canonical setting involves two or more coupled oscillatory modes with nonlinear interactions, each subject to weak intrinsic or engineered dissipation. Typical models comprise:

• Nonlinearly coupled oscillators: For mechanical systems, equations such as

$$m_1\ddot{q}_1 + m_1\gamma_1\dot{q}_1 + k_1 q_1 + \alpha_1 q_1^3 + g q_1^2 q_2 = 0,$$

$$m_2\ddot{q}_2 + m_2\gamma_2\dot{q}_2 + k_2 q_2 + \alpha_2 q_2^3 + g' q_1^3 = 0$$

capture Duffing-type nonlinearity and lowest-order cubic coupling, enabling internal $1$:$n$ resonant energy transfer (Wang et al., 2022, Chen et al., 2016, Shoshani et al., 2017).

• Driven envelope systems with dissipation: In autoresonant setups, slow envelope equations with quadratic nonlinear terms and explicit dissipative corrections,

$$A'(t) = -i(2tA + \frac{1}{2}A^*B + f) - \mu_1A, \quad B'(t) = -i(4tB + \frac{1}{4}A^2) - \mu_2B,$$

model chirped driving and nonlinear mode crossing under weak damping (Glebov et al., 2013).

• Optical/fiber systems: Nonlinear Schrödinger equations with small attenuation ($i\alpha\psi$) or coupled-mode models with amplitude-dependent detuning describe phase-space separatrix crossings and symmetry-broken recurrences in FPUT systems and photonic microcavities (Vanderhaegen et al., 2022, Bernard et al., 2017).
• Spatially periodic dissipative arrays: Linear and nonlinear Schrödinger equations with a spatially periodic imaginary potential exemplify selective long-lived (nondecaying) modes at Bragg-indices, showing dissipation “immunity” via mode structure and nonlinear soliton interactions (Fernández et al., 2014).

2. Resonant Mode Coupling, Internal Resonance, and Dissipation Enhancement

Nonlinear mode-crossing dissipation is fundamentally connected to amplitude-dependent frequency shifting—a hallmark of nonlinear systems—leading to spontaneous crossing of commensurate frequency conditions (internal resonance).

• Amplitude-Dependent Frequency Tuning: Each mode’s effective frequency depends on its amplitude via

$$\omega_i(A_i) = \omega_{i0} + \kappa_i A_i^2, \quad \kappa_i = \frac{3\alpha_i}{8 m_i \omega_{i0}},$$

such that as the amplitude of mode 1 decays, it can cross the resonance condition $n \omega_1(A_1) \approx \omega_2(A_2)$ (Wang et al., 2022).

• Energy Exchange and Dissipation Spikes: At resonance, nonlinear terms like $q_1^3 q_2$ (1:3 resonance) or $x(\theta) y(\theta)$ (frequency doubling) enable fast, resonantly-enhanced energy exchange.
  • For mechanical and optomechanical systems, this manifests as bursts of energy transfer that can generate “steps” in decay dynamics (e.g., discrete decreases in spin rate with each mode crossing) (Sourki et al., 4 Jan 2026).
  • In optical microcavities, nonlinear coupling produces dynamic detuning of Fano resonances, leading to dissipative reshaping or annihilation of resonance features (Bernard et al., 2017).

3. Non-Monotonic and Amplitude-Dependent Dissipation

A key signature of nonlinear mode-crossing dissipation is the strongly nonmonotonic, amplitude-dependent effective relaxation rate for the principal mode.

• Anomalous Effective Damping: The effective decay rate can be written as

$$\Gamma_{\rm eff}(A) = \Gamma_1 \left[ 1 + \zeta_{ad}\frac{A^4}{1 + (\Delta\Omega_{12} + A^2)^2} \right]$$

where $\Gamma_1$ is the background damping, $\zeta_{ad}$ encodes nonlinear coupling and damping asymmetry, and the term in $A$ is sharply peaked at resonance (Shoshani et al., 2017). This produces a “bump” in the decay rate as amplitude sweeps through resonance.

• Persistent Negative Dissipation (Coherent Plateaus): At exact resonance and with appropriate initial phase, mode 1 can be phase-locked, causing energy anti-dissipation: $\Gamma_{\rm eff}<0$ for finite intervals. Mode 1’s amplitude remains nearly constant, drawing energy from mode 2 (“reservoir”) via nonlinear coupling (Chen et al., 2016, Wang et al., 2022).
• Relaxation Pathway Bifurcation: Depending on initial phases, the system may enter the phase-locked state with energy plateau or bypass it, leading to standard monotonic decay. This bifurcation is determined solely by the initial relative phase and not the total energy input (Wang et al., 2022).

4. Phase Space Structures, Separatrix Crossing, and Symmetry Breaking

Nonlinear mode-crossing dissipation is underpinned by rich phase space structures and separatrix crossing processes.

• Hamiltonian Geometry and Separatrix Crossing: In conservative models, phase space is separated by heteroclinic structures (e.g., in the FPUT recurrence, by the Akhmediev-breather separatrix). Weak damping causes slow drift of the system’s invariants, leading to critical thresholds where the system crosses from one class of orbits (unshifted to shifted recurrences) to another (Vanderhaegen et al., 2022).
• Quantized Criticality: The attenuation at which the system switches recurrence symmetry (or crossing pathway) is quantized; explicit perturbative formulas yield the sequence of critical attenuation values at which separatrix crossing occurs, matching experimental observation in Raman-controlled fiber links (Vanderhaegen et al., 2022).
• Persistent Phase-Locked States: In nonlinear resonators, internal resonance domains appear as basins of attraction in phase space corresponding to multi-stable “locked” states, with transition boundaries sharply controlled by initial conditions (Wang et al., 2022).

5. Manifestations Across Physical Platforms

Nonlinear mode-crossing dissipation has been experimentally detected and theoretically characterized in diverse systems:

Platform	Key Nonlinear Coupling	Dissipation Dynamics
Nanomechanical resonators (Shoshani et al., 2017, Chen et al., 2016)	$q_1^3 q_2$ or $q_1^2 q_2$ (1:3, 1:2) cubic nonlinearities	Nonmonotonic $\Gamma_{\rm eff}(A)$, phase-locked energy plateaus, multistability
Levitated rotors (Sourki et al., 4 Jan 2026)	Spin–mode coupling $f(\dot\alpha)x_j$	Discrete step-like angular velocity loss, quadratic decay envelope, injection-locked stabilization
Fiber optics (Vanderhaegen et al., 2022)	Nonlinear Schrödinger formation of FPUT recurrences	Damping-induced separatrix crossing, symmetry-broken recurrences
Photonic microcavities (Bernard et al., 2017)	Thermo-optic nonlinear coupled-mode interaction	Complete mode extinction, bistable lineshapes driven by dissipative nonlinearity
Periodic dissipative lattices (Fernández et al., 2014)	Intrinsic periodic $\delta$-dissipation, NLS nonlinearity	Long-lived or nondecaying linear/nonlinear states at resonant Bloch indices

These systems demonstrate that the core features—resonance-enhanced nonlinear mode coupling, amplitude-dependent dissipation, and intermittent or persistent energy plateaus—are universal, not restricted to a specific frequency range or platform.

6. Implications for Dissipation Engineering and Control

Nonlinear mode-crossing dissipation enables and, in some cases, limits novel strategies for system stabilization and dissipation control:

• Dissipation Engineering: By exploiting internal resonance (via appropriate selection of nonlinearity strength, detuning, and mode structure), systems can be engineered to support extended energy retention intervals, “autonomous” oscillation plateaus, or even negative effective damping for predefined timescales without continuous external input (Chen et al., 2016, Wang et al., 2022).
• Stabilization via Injection Locking: In magnetic levitated rotors, weak counter-acting torques produce stabilization at fixed rotation rates for days (mechanical injection locking), governed by the slope of $\Gamma_{\rm eff}(\Omega)$ at the locking point; locking bandwidth and stability are analytically predicted and confirmed experimentally (Sourki et al., 4 Jan 2026).
• Non-Hamiltonian Attractors & Noise Sensitivity: The emergence of nonlinear attractors and phase-locked states highlights a regime intermediate between pure, low-dimensional nonlinear dynamics and high-dimensional chaos, sensitive to noise and drive parameters. This is significant for quantum regime investigations and ultra-sensitive force or acceleration sensors (Sourki et al., 4 Jan 2026).

7. Extensions, Special Cases, and Future Directions

• Mode Selectivity in Dissipative Lattices: In spatially periodic dissipative arrays, only Bloch modes at the Brillouin-zone center or edge are immune to absorption, leading to robust nonlinear mode formation even in the presence of strong background dissipation (Fernández et al., 2014).
• Nonlinearity Class Generality: The mechanisms do not rely on a particular nonlinearity type (Duffing, cubic–linear, Kerr/thermo-optic), provided the system supports amplitude-dependent frequency shifts and resonance crossings.
• Quantized Thresholds and Critical Phenomena: The quantized, staircase-like sequence of dissipation-induced transitions—separatrix crossings, symmetry breaking—point toward broader universality classes, motivating exploration in other nonlinear dynamical and quantum optical systems (Vanderhaegen et al., 2022).
• Prospects: The engineering of dissipation landscapes, selective “loss minimization” for specific nonlinear modes, and control of high-Q macroscopic or quantum mechanical objects are anticipated directions, with applications across metrology, frequency standards, and fundamental studies of nonequilibrium statistical mechanics.

Nonlinear mode-crossing dissipation thus occupies a central role at the interface of nonlinear dynamics, dissipation engineering, and resonantly coupled multimode systems, offering both fundamental insight and practical potential for the design and control of advanced oscillatory and wave-based devices.