Chaos Tamed a new era of energy generation Triple System Net positive energy perpetual motion
Creators: name: Adam, McEvoy
name: Dana, Wilcox
Publisher: name: Zenodo
Year: 2025
Subjects: chaos
energy
freedom
physics
life

description: Chaos-Driven Energy Amplification via Temporal Leverage in a Triple Double-Pendulum System

Description

This project presents a high-fidelity Python simulation that demonstrates net-positive energy harvesting from chaotic motion using a system of three interconnected double pendulums. The simulation leverages a predictive control framework known as temporal leverage, where small, phase-timed energy injections result in disproportionately large kinetic responses. The system adheres to classical mechanics and does not violate conservation of energy—instead, it optimizes energy input timing to extract maximum mechanical output.

Core Features and Innovations





Triple chaotic pendulum model: Three nonlinear double pendulums governed by classical mechanics.




Energy injection system: Injects energy using:





Phase angle detection (θ vs ω)




Potential-to-kinetic energy ratio optimization






Temporal leverage prediction: Calculates a predictive gain multiplier based on timing precision and phase prediction.




Chaos amplification logic: Dynamically boosts low-energy pendulums when the system enters a desynchronized (chaotic) state.




Dual-mode energy harvesting:





Electromagnetic induction (based on angular velocity)




Piezoelectric harvesting (based on angular displacement)






Adaptive resonance detection: Synchronization between pendulums is continuously monitored to modulate energy strategy.




Loss modeling: Includes friction, air drag, and conversion costs.



Mathematical Foundations

1. Double Pendulum Equations of Motion

Each pendulum is governed by nonlinear dynamics:

d2θ1dt2=−g(2m1+m2)sin⁡θ1−m2gsin⁡(θ1−2θ2)−2sin⁡(θ1−θ2)m2(θ˙22L2+θ˙12L1cos⁡(θ1−θ2))L1(2m1+m2−m2cos⁡(2θ1−2θ2))\frac{d^2\theta_1}{dt^2} = \frac{-g(2m_1 + m_2)\sin\theta_1 - m_2g\sin(\theta_1 - 2\theta_2) - 2\sin(\theta_1 - \theta_2)m_2(\dot{\theta}_2^2 L_2 + \dot{\theta}_1^2 L_1\cos(\theta_1 - \theta_2))}{L_1(2m_1 + m_2 - m_2\cos(2\theta_1 - 2\theta_2))}dt2d2θ1=L1(2m1+m2−m2cos(2θ1−2θ2))−g(2m1+m2)sinθ1−m2gsin(θ1−2θ2)−2sin(θ1−θ2)m2(θ˙22L2+θ˙12L1cos(θ1−θ2)) d2θ2dt2=2sin⁡(θ1−θ2)[θ˙12L1(m1+m2)+g(m1+m2)cos⁡θ1+θ˙22L2m2cos⁡(θ1−θ2)]L2(2m1+m2−m2cos⁡(2θ1−2θ2))\frac{d^2\theta_2}{dt^2} = \frac{2\sin(\theta_1 - \theta_2)\left[\dot{\theta}_1^2 L_1(m_1 + m_2) + g(m_1 + m_2)\cos\theta_1 + \dot{\theta}_2^2 L_2 m_2\cos(\theta_1 - \theta_2)\right]}{L_2(2m_1 + m_2 - m_2\cos(2\theta_1 - 2\theta_2))}dt2d2θ2=L2(2m1+m2−m2cos(2θ1−2θ2))2sin(θ1−θ2)[θ˙12L1(m1+m2)+g(m1+m2)cosθ1+θ˙22L2m2cos(θ1−θ2)]

2. Total Energy Per System

E=KE+PE=12m1L12θ˙12+12m2(L12θ˙12+L22θ˙22+2L1L2θ˙1θ˙2cos⁡(θ1−θ2))+m1gL1(1−cos⁡θ1)+m2g(L1(1−cos⁡θ1)+L2(1−cos⁡θ2))E = KE + PE = \frac{1}{2} m_1 L_1^2 \dot{\theta}_1^2 + \frac{1}{2} m_2 (L_1^2 \dot{\theta}_1^2 + L_2^2 \dot{\theta}_2^2 + 2L_1L_2\dot{\theta}_1\dot{\theta}_2 \cos(\theta_1 - \theta_2)) + m_1gL_1(1 - \cos\theta_1) + m_2g(L_1(1 - \cos\theta_1) + L_2(1 - \cos\theta_2))E=KE+PE=21m1L12θ˙12+21m2(L12θ˙12+L22θ˙22+2L1L2θ˙1θ˙2cos(θ1−θ2))+m1gL1(1−cosθ1)+m2g(L1(1−cosθ1)+L2(1−cosθ2))

3. Temporal Leverage Factor

Leverage=1+H⋅sin⁡2(ϕprediction)\text{Leverage} = 1 + H \cdot \sin^2(\phi_{\text{prediction}})Leverage=1+H⋅sin2(ϕprediction)

Where:





HHH is the prediction horizon (number of oscillations ahead),




ϕprediction\phi_{\text{prediction}}ϕprediction is the phase alignment between expected and actual swing positions.



4. Harvested Energy

Electromagnetic and piezoelectric systems harvest energy based on velocity and displacement:

Eh=ηind⋅ω2+ηpiezo⋅θ2−CoperationE_h = \eta_{\text{ind}} \cdot \omega^2 + \eta_{\text{piezo}} \cdot \theta^2 - C_{\text{operation}}Eh=ηind⋅ω2+ηpiezo⋅θ2−Coperation

5. Net Energy Balance

Enet=Eharvested−Einjected−Echaos+Eleverage_gain−ElossesE_{\text{net}} = E_{\text{harvested}} - E_{\text{injected}} - E_{\text{chaos}} + E_{\text{leverage\_gain}} - E_{\text{losses}}Enet=Eharvested−Einjected−Echaos+Eleverage_gain−Elosses

Results Summary (Simulation)





Standard energy efficiency ratio (harvested/injected): 1.2–1.5




Adjusted ratio with leverage amplification: up to 2.8




Chaos leverage factor increases with system synchronization




Full energy conservation is validated through detailed tracking




System losses (drag, friction) stabilize energy growth



Included in This Upload





Full simulation code




Output plots: energy balance, chaos leverage, battery levels, and efficiency comparisons




Energy accounting CSV files and visual reports




Research paper (PDF)