Variational superposed Gaussian approximation for time-dependent solutions of Langevin equations

Area of Science:

• Nonlinear dynamics
• Stochastic processes
• Computational physics

Background:

• Langevin equations model systems with noise.
• Chaotic signals pose challenges for traditional analysis methods like Fourier transforms.
• Understanding system responses to complex, noisy inputs is crucial.

Purpose of the Study:

• To introduce a novel variational superposed Gaussian approximation (VSGA) for solving Langevin equations with chaotic signals.
• To determine time-dependent parameters of Gaussian distributions using a variational principle.
• To analyze systems driven by chaotic signals where Fourier methods fail.

Main Methods:

• Applying the variational principle to derive time-dependent parameters for superposed Gaussian distributions.
• Utilizing the proposed VSGA to model systems driven by chaotic signals.
• Incorporating both white and colored Gaussian noise terms.
• Comparing VSGA results with Monte Carlo simulations for validation.

Main Results:

• VSGA accurately calculates time-dependent probability density functions (PDFs) and moments.
• Calculated PDFs from VSGA show excellent agreement with Monte Carlo simulations.
• The correlation between chaotic input and mean response was quantified.
• Aperiodic stochastic resonance was confirmed for both white and colored noise.

Conclusions:

• The proposed VSGA is a robust method for analyzing nonlinear systems driven by chaotic signals.
• VSGA provides accurate predictions of system dynamics and statistical properties.
• The study confirms the occurrence of aperiodic stochastic resonance in systems with chaotic forcing and noise.