[Paper Review] Quasiperiodicity, bistability and chaos in the Landau-Lifshitz equation

Luis F. Alvarez, Oscar Pla+1 more|ArXiv.org|May 4, 1999

Quantum chaos and dynamical systems3 citations

TL;DR

This paper investigates the nonlinear dynamics of a single magnetic moment governed by the Landau-Lifshitz equation under transverse periodic driving. It reveals a rich sequence of behaviors—bistability, quasiperiodicity, and chaos—depending on the driving amplitude and applied field strength, with chaotic regimes emerging at higher perturbation amplitudes and quasiperiodic windows appearing periodically in the driving amplitude when the field exceeds the anisotropy field.

ABSTRACT

The dynamics of an individual magnetic moment is studied through the Landau-Lifshitz equation with a periodic driving in the direction perpendicular to the applied field. For fields lower than the anisotropy field and small values of the perturbation amplitude we have observed the magnetic moment bistability. At intermediate values we have found quasiperiodic bands alternating with periodic motion. At even larger values a chaotic regime is found. When the applied field is larger than the anisotropy one, the behavior is periodic with quasiperiodic regions. Those appear periodically in the amplitude of the oscillating field. Also, even for low values of the driving force, the moment is not parallel to the applied field.

Motivation & Objective

• To understand the complex nonlinear dynamics of a single magnetic moment under periodic external driving.
• To investigate how the interplay between anisotropy, applied field, and driving amplitude influences dynamic stability and periodicity.
• To identify the emergence of quasiperiodic and chaotic regimes in the system's response.
• To analyze the transition between bistable, periodic, and chaotic states as a function of control parameters.
• To explore the role of driving field amplitude and direction in inducing complex temporal behavior in magnetic systems.

Proposed method

• Modeling the magnetic moment dynamics using the Landau-Lifshitz equation with a periodic driving field perpendicular to the applied field.
• Applying numerical simulations to solve the Landau-Lifshitz equation under varying driving amplitudes and applied field strengths.
• Analyzing the system's response through time series, phase space trajectories, and power spectra to identify periodic, quasiperiodic, and chaotic behavior.
• Varying the applied field relative to the anisotropy field to explore transitions between different dynamical regimes.
• Using bifurcation analysis to map regions of bistability, quasiperiodicity, and chaos across parameter space.
• Focusing on the case where the driving field is perpendicular to the easy axis, with anisotropy and damping effects included.

Experimental results

Research questions

• RQ1 How does the driving amplitude influence the emergence of bistability in the magnetic moment dynamics?
• RQ2 What conditions lead to quasiperiodic motion, and how do these regions alternate with periodic windows?
• RQ3 At what driving amplitude does the system transition into a chaotic regime?
• RQ4 How does the applied field strength relative to the anisotropy field affect the occurrence of quasiperiodic behavior?
• RQ5 Why does the magnetic moment deviate from alignment with the applied field even at low driving amplitudes?

Key findings

• For applied fields below the anisotropy field and small driving amplitudes, the system exhibits bistability in the magnetic moment orientation.
• At intermediate driving amplitudes, quasiperiodic bands alternate with periodic motion, indicating complex frequency locking behavior.
• At higher driving amplitudes, the system enters a chaotic regime, as confirmed by irregular time series and broad power spectra.
• When the applied field exceeds the anisotropy field, periodic motion dominates, but quasiperiodic regions appear periodically as a function of the driving amplitude.
• Even for low driving amplitudes, the magnetic moment does not align with the applied field, indicating intrinsic non-equilibrium dynamics.
• The system displays a rich bifurcation structure with recurrent quasiperiodic windows, suggesting complex nonlinear dynamics in magnetic nanostructures.