Predicting attractor characteristics using Lyapunov exponents in a laser with injected signal

Area of Science:

• Nonlinear Dynamics
• Laser Physics
• Chaos Theory

Background:

• Coexisting attractors in laser systems exhibit complex dynamics.
• Understanding these dynamics is crucial for controlling laser behavior.

Purpose of the Study:

• To investigate the Lyapunov exponent (LE) patterns associated with coexisting attractors in a single-mode laser with an injected signal.
• To establish a predictive framework for attractor dynamics and bifurcations using LE spectra.

Main Methods:

• Analysis of Lyapunov exponent (LE) patterns for individual attractors.
• Characterization of Lyapunov spectra, including symmetric-like and asymmetric 'bubbles'.
• Comparison of attractor volume evolution with spectral formations under varying injected signal conditions.

Main Results:

• Each attractor possesses a unique LE pattern, bounded by a common spectral pattern starting and ending with two-zero LEs.
• Symmetric-like bubbles in LE spectra indicate proximity to dynamic changes; asymmetric bubbles signal bifurcations.
• Two-zero LEs are crucial for predicting period-doubling and torus dynamics, with convergence of three LEs to zero as a precursor.
• Changes in attractor volume correlate with spectral changes, indicating dynamic shifts.

Conclusions:

• The two-zero LE signature is a key, though not sufficient, predictor of attractor dynamics.
• Lyapunov spectra provide a robust method for forecasting dynamic transitions and bifurcations in laser systems.
• Attractor viability can be preliminarily assessed by introducing noise into the injected signal.