Universal Chaos Harvesting and Stabilization Across Nonlinear Dynamical Systems
Creators: name: Adam, McEvoy
name: Dana, Wilcox
Publisher: name: Zenodo
Year: 2025
Subjects: Three body Problem
chaos
energy
lorenz
railgun
triple pendulum
Duffing Oscillator
Double Pendulum

description: Title

Universal Chaos Harvesting and Stabilization Across Nonlinear Dynamical Systems



Description

This submission presents a unified simulation framework capable of stabilizing and harvesting energy from a wide range of chaotic systems under physically realistic constraints. The system includes predictive damping, energy accounting, and constraint enforcement for ensuring structural safety and energy realism. All models were evaluated over long timeframes and confirmed to maintain physical fidelity across motion, energy, and chaos control.



Tested Models and Mathematical Descriptions

1. Double Pendulum





System: Two masses linked by rods under gravity.




Purpose: Classical testbed for chaotic motion.




Equations:





bash

θ₁'' = f(θ₁, θ₂, θ₁', θ₂')  
θ₂'' = f(θ₁, θ₂, θ₁', θ₂')





2. Triple Pendulum





System: Three coupled pendulum arms.




Purpose: Extreme chaos and sensitivity.




Equations:





bash

dθ₁/dt = ω₁  
dω₁/dt = f(θ₁, θ₂, θ₃, ω₁, ω₂, ω₃)  
...





3. Duffing Oscillator





System: Nonlinear oscillator with cubic stiffness.




Purpose: Demonstrates bistability and period doubling.




Equation:





arduino

x'' + δx' + αx + βx³ = γcos(ωt)





4. Lorenz System





System: Atmospheric convection model.




Purpose: Canonical chaotic attractor.




Equations:





bash

dx/dt = σ(y - x)  
dy/dt = x(ρ - z) - y  
dz/dt = xy - βz





5. Magnetic Pendulum





System: Pendulum influenced by magnetic field torque.




Purpose: Compact demonstration of chaotic bifurcation.




Equation:





bash

θ'' + γθ' = (qB sin(θ)) / m





6. Railgun Model





System: High-current rail acceleration simulation.




Purpose: Models nonlinear high-force motion.




Equation:





java

x'' = (I² * L) / m - Dv  
I = V / (R + L)





Simulation Enhancements





Realistic damping: Proportional and velocity-based drag for Earth-like air resistance




Material constraints: Limits on angular velocity and displacement based on maraging steel




Energy harvesting logic: Acceleration-phase energy collection with efficiency scaling




Full energy validation: Tracks kinetic + potential energy for each model




Model comparison suite: Visual plots compare energy, acceleration, and harvested yield




Per-model output: Each system saves its own acceleration and energy trace charts





Chaos Prediction and Control Logic

Purpose

To prevent uncontrolled growth and ensure valid results, a predictive chaos control engine actively monitors each system for signs of structural or numerical instability.



How It Works

At every time step:





Threshold MonitoringChecks whether angular velocity or displacement is nearing a hard limit (e.g., 90% of max allowed).




Predictive BrakingApplies forward-looking damping if limits are approached:



cpp

if abs(omega) > 0.9 * MAX_VELOCITY:
    domega -= 2.0 * DAMPING_COEFFICIENT * omega






Energy-Constrained HarvestingOnly positive acceleration is harvested:



ini

harvested = sum(max(0, a)) * efficiency






Validation ChecksIf harvested energy exceeds physical system energy:



pgsql

if harvested > max(E_total): raise warning






Continuous LoggingEvery model logs total energy over time for visual and numerical verification.





What This Enables





Stability in long chaotic runs




Real-time response to chaotic buildup




Physical conservation of energy




Safe operation under extreme nonlinearity





Results Summary

All models:





Completed full 300s test windows




Maintained stability under chaotic motion




Produced energy outputs validated against physical limits



Top-performing systems include the Railgun, Lorenz Attractor, and Double Pendulum, each demonstrating high harvested energy while remaining bounded.



Conclusion

This project demonstrates a functional, general-purpose numerical solution framework for chaotic systems. It effectively predicts, stabilizes, and harnesses chaotic motion under real-world constraints, providing practical solutions across multi-body, electromechanical, and nonlinear physical domains. While not symbolic, this work offers a working, simulation-based answer to the control and utilization of chaos, including in domains associated with the three-body problem.