Data Analysis: Calculating and Plotting the Relationship Between Signal-to-Noise Ratio and Noise Intensity of the Double-Well Duffing Oscillator Based on Matlab

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🔥 Content Introduction

The Duffing oscillator, as a classic nonlinear dynamical system, has attracted significant attention due to its rich dynamical behaviors. In particular, the double-well Duffing oscillator exhibits unique stochastic resonance phenomena under external periodic driving and noise disturbances. Studying the relationship between the signal-to-noise ratio (SNR) and noise intensity of this oscillator not only aids in understanding the noise behavior of nonlinear systems but also provides a theoretical basis for applications such as signal detection and energy harvesting. This article will explain how to calculate and plot the relationship between the SNR and noise intensity of the double-well Duffing oscillator, involving model establishment, numerical simulation, data processing, and result analysis.

1. Double-Well Duffing Oscillator Model

The double-well Duffing oscillator can be described by the following second-order nonlinear ordinary differential equation:

x'' + δx' - αx + βx^3 = Acos(ωt) + ξ(t)

Where:

x represents the displacement of the oscillator;
x' and x'' represent velocity and acceleration, respectively;
δ is the damping coefficient, describing the energy dissipation of the system;
α and β are parameters related to the shape of the potential wells, determining the depth and width of the double wells. Typically, α > 0 and β > 0 ensure the existence of two stable equilibrium points, forming a double-well structure;
A and ω are the amplitude and frequency of the external periodic driving, respectively;
ξ(t) represents Gaussian white noise, with statistical properties given by:

<ξ(t)> = 0 (mean is zero)
<ξ(t)ξ(t')> = 2Dδ(t - t') (correlation function is the Dirac function), where D represents the noise intensity.

The above equation describes an oscillator affected by damping, nonlinear restoring force, periodic driving, and random noise. The double-well structure is key to the complex behaviors exhibited by this system, while noise introduces randomness, allowing the oscillator to transition between the two potential wells.

2. Numerical Simulation Method

Since the equation of the Duffing oscillator is nonlinear, it is usually difficult to obtain an analytical solution. Therefore, numerical simulation methods are required to solve this equation. Common numerical methods include:

Runge-Kutta Method: The Runge-Kutta method is a high-order single-step integration method characterized by high accuracy and good stability. The fourth-order Runge-Kutta method (RK4) is the most commonly used choice. This method approximates the true solution by estimating derivatives over multiple time steps.
Euler-Maruyama Method: A numerical method specifically designed for solving stochastic differential equations. It is the stochastic version of the Euler method, suitable for systems with additive noise.

To more accurately simulate the double-well Duffing oscillator, it is recommended to use the Runge-Kutta method in conjunction with methods for handling random noise. The specific implementation steps are as follows:

*   **Discretize Time:** Divide the time axis into discrete time steps `Δt`.  
*   **Convert to First-Order System:** Transform the second-order equation into two first-order equations:  
    ```  
    x' = v  
    v' = αx - βx^3 - δv + Acos(ωt) + ξ(t)  
    ```  
*   **Iterative Calculation:** Use the Runge-Kutta method to iteratively calculate the above first-order equations, obtaining the displacement `x(t)` and velocity `v(t)` at each time step.  
*   **Add Noise:** At each time step, generate a random number following a normal distribution with a mean of 0 and a standard deviation of `√(2DΔt)`, then add it as `ξ(t)` to the velocity update equation.

3. Calculation of Signal-to-Noise Ratio (SNR)

The signal-to-noise ratio is an indicator that measures the strength of the signal relative to the noise intensity. For the Duffing oscillator, it reflects the clarity of the periodic driving signal against the noise background. The steps to calculate SNR are as follows:

Time Domain Data Collection: Obtain the time series data of the oscillator’s displacement x(t) through numerical simulation. Ensure that the data is sufficiently long for subsequent spectral analysis. Typically, multiple cycles of the driving signal need to be simulated, discarding the initial transient process.
Spectral Analysis: Perform a Fourier transform (FFT) on the time domain data to obtain the frequency spectrum X(f). The frequency spectrum shows the energy distribution of the signal at different frequencies.
Determining Signal Power and Noise Power: In the frequency spectrum, find the peak corresponding to the driving frequency ω, which represents the signal power P_signal. Select a frequency range near the peak, excluding the influence of the signal peak, and calculate the average power within that frequency range as the noise power P_noise.
Calculate SNR: The signal-to-noise ratio can be calculated using the following formula:
```
SNR = 10log10(P_signal / P_noise)
```
The unit of SNR is decibels (dB).

4. Plotting the Relationship Between SNR and Noise Intensity

To plot the relationship between SNR and noise intensity, the following steps need to be performed:

Loop Through Noise Intensities: Set a series of different noise intensity D values.
Numerical Simulation: For each noise intensity D value, perform numerical simulation to obtain the time series data of displacement x(t).
Calculate SNR: Calculate the SNR value corresponding to each noise intensity using the aforementioned method.
Plot the Relationship Graph: Plot a scatter plot or curve graph with noise intensity D on the x-axis and SNR on the y-axis.

5. Result Analysis and Discussion

After plotting the relationship graph between SNR and noise intensity, the results can be analyzed and discussed. Typically, the following phenomena are observed:

Stochastic Resonance: At lower noise intensities, SNR increases with the increase of noise intensity, reaching a peak. This indicates that moderate noise can enhance the strength of the signal, which is the phenomenon of stochastic resonance. Noise facilitates the transition of the oscillator between the two potential wells, thereby responding better to the periodic driving signal.
Over-Noise Suppression: When the noise intensity exceeds a certain threshold, SNR decreases with the increase of noise intensity. This is because excessive noise can drown out the signal, making it difficult to detect effectively.
Parameter Influence: The parameters of the Duffing oscillator (e.g., damping coefficient δ, double-well shape parameters α and β, and the amplitude A and frequency ω of the driving signal) significantly affect the relationship between SNR and noise intensity. For example, a larger damping will weaken the system’s response to noise, thereby affecting the strength of stochastic resonance.

⛳️ Running Results

🔗 References

[1] Zhang Mei, Luo Shiyu, Shao Mingzhu. Channel Effect and Optical Bistability Effect of Superlattice Quantum Wells [J]. Semiconductor Optoelectronics, 2008, 29(3):4. DOI:CNKI:SUN:BDTG.0.2008-03-012.

[2] Cao Baofeng, Li Peng, Li Xiaoqiang, et al. Weak Pulse Signal Detection and Parameter Estimation Based on Strongly Coupled Duffing Oscillator [J]. Acta Physica Sinica, 2019(8):11. DOI:10.7498/aps.68.20181856.

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