Coupled Phase Oscillator

[MAL-Seeker Test Case] Coupled Phase Oscillator Model (6-Dimensional)

$$\omega_i - \frac{K}{N} \sum_{j=1}^{6} \sin(\theta_j - \theta_i) = 0 \quad (i = 1, 2, \dots, 6)$$

6-Variable Coupled Phase Oscillator Model
Valid Roots: 729 Points (within restricted bounds)

1. The Target Equation (Model of Synchronization)

The third test case for MAL-Seeker is the coupled oscillator model, famous as a mathematical model for "synchronization phenomena" in nonlinear dynamics. Here, we formulated it as a 6-dimensional system of equations where 6 oscillators mutually influence each other.

2. The "Infinite Illusions" Triggered by Trigonometric Functions

The most defining feature of this equation, and the biggest headache for search algorithms, is the presence of the $\sin$ function. Due to the periodicity of trigonometric functions, an infinite number of "false solutions" exist—solutions that merely shifted by $2\pi$ but carry the exact same physical meaning.

This is where MAL-Seeker's core concept, "the division of roles between human and Maru," shines. First, the human (developer) uses a "Scouter" to dynamically define a Bounding Box (valid phase space) to explore. Then, 100,000 "Marus" (search seeds) are released into that designated forest to gather only the valid solutions.

3. Execution Results

Here is the final terminal log of running MAL-Seeker in a local MacBook environment.

=========================================
  Total Valid Roots Found: 729  (Out of 2048 local minima)
  Execution Time: 53.942696 seconds
=========================================

Out of 2,048 local minima captured within the search domain, MAL-Seeker successfully filtered and extracted 729 Valid Roots that satisfied the boundary conditions. The computation time was approximately 53.9 seconds.

Dr.WataWata Insight:

For equations involving $\sin$ functions, solutions keep popping up endlessly no matter how deep you dig. The key is how to instruct the system on "when to stop" and "how to limit the range."
In this case, the human's preliminary assist (dynamically specifying the search range) paired perfectly with Marus' tenacious searching (the strike-based termination condition). Filtering down from 2,048 local minima to 729 true solutions in under a minute really highlights its flexible adaptability as a hybrid solver.

4. Analysis Environment

Analysis Engine: MAL-Seeker (Antares Ver 3.1.1 / C Language)
PC: MacBook
OS: macOS Monterey 12.7.6
CPU: 1.2GHz Dual-Core Intel Core m5
Memory: 8GB

1. The Target Equation (Model of Synchronization)

2. The "Infinite Illusions" Triggered by Trigonometric Functions

3. Execution Results

4. Analysis Environment

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