[Paper Review] Forced symmetry breaking as a mechanism for rogue bursts in a dissipative nonlinear dynamical lattice

Priya Subramanian, Edgar Knobloch+1 more|arXiv (Cornell University)|Mar 12, 2022

Nonlinear Photonic Systems51 references3 citations

TL;DR

This paper proposes forced symmetry breaking in a dissipative nonlinear lattice as a mechanism for rogue wave bursts, demonstrating that localized, extreme-amplitude excursions emerge via spatiotemporal dynamics in a ring of diffusively coupled oscillators. The key result is the emergence of bimodal amplitude distributions with peaks at vanishing and very large amplitudes, indicating rogue-like events without requiring integrability or Hamiltonian structure.

ABSTRACT

We propose an alternative to the standard mechanisms for the formation of rogue waves in a non-conservative, nonlinear lattice dynamical system. We consider an ODE system that features regular periodic bursting arising from forced symmetry breaking. We then connect such potentially exploding units via a diffusive lattice coupling and investigate the resulting spatio-temporal dynamics for different types of initial conditions (localized or extended). We find that in both cases, particular oscillators undergo extremely fast and large amplitude excursions, resembling a rogue wave burst. Furthermore, the probability distribution of different amplitudes exhibits bimodality, with peaks at both vanishing and very large amplitude. While this phenomenology arises over a range of coupling strengths, for large values thereof the system eventually synchronizes and the above phenomenology is suppressed. We use both distributed (such as a synchronization order parameter) and individual oscillator diagnostics to monitor the dynamics and identify potential precursors to large amplitude excursions. We also examine similar behavior with amplitude-dependent diffusive coupling.

Motivation & Objective

• To explore alternative mechanisms for rogue wave formation in non-conservative, nonlinear systems beyond integrable or Hamiltonian frameworks.
• To investigate how forced symmetry breaking in a lattice of coupled oscillators leads to extreme spatiotemporal events resembling rogue waves.
• To identify precursors and dynamical signatures of large-amplitude excursions in such systems.
• To examine the role of coupling strength and coupling type (including amplitude-dependent) in suppressing or enabling rogue burst phenomena.
• To extend the understanding of rogue wave emergence in dissipative systems by leveraging a model with well-understood single-oscillator dynamics.

Proposed method

• The study employs a ring lattice of identical oscillators governed by a normal form ODE system derived from a Hopf bifurcation with broken D4 symmetry.
• Each oscillator exhibits intermittent bursts due to forced symmetry breaking, with trajectories escaping to and returning from infinity in finite time.
• The oscillators are coupled via diffusive coupling, enabling spatial coherence and propagation of extreme events across the lattice.
• Numerical simulations are performed for both localized and extended initial conditions to probe spatiotemporal dynamics.
• Diagnostics include synchronization order parameters, individual oscillator amplitude tracking, and probability distribution analysis of amplitudes.
• Amplitude-dependent diffusive coupling is also examined to assess its impact on burst formation and suppression.

Experimental results

Research questions

• RQ1 Can forced symmetry breaking in a dissipative, nonlinear lattice generate rogue wave-like bursts without relying on integrability or Hamiltonian structure?
• RQ2 How do localized and extended initial conditions affect the emergence of extreme-amplitude events in the coupled lattice?
• RQ3 What dynamical indicators or precursors can be used to predict large-amplitude excursions in such systems?
• RQ4 How does the strength of diffusive coupling influence the occurrence and suppression of rogue bursts?
• RQ5 Does amplitude-dependent coupling alter the statistical properties of the amplitude distribution or the burst dynamics?

Key findings

• For weak to moderate coupling, the lattice system exhibits rogue bursts with amplitudes reaching up to 10^9, consistent with extreme event statistics.
• The amplitude probability distribution is bimodal, with distinct peaks at vanishing amplitudes and very large amplitudes, indicating rare but extreme events.
• For large coupling strengths, the system synchronizes and rogue burst phenomena are suppressed, indicating a critical coupling threshold.
• Both localized and extended initial conditions lead to the emergence of spatially and temporally localized extreme events resembling rogue waves.
• Distributed diagnostics such as the synchronization order parameter and individual oscillator diagnostics successfully identify precursors to large-amplitude excursions.
• Amplitude-dependent coupling modifies burst dynamics, suggesting tunability of extreme event occurrence through coupling design.