Parametric resonance induced chaos in magnetic damped driven pendulum

Abstract

A damped driven pendulum with a magnetic driving force, appearing from a solenoid, where ac current flows is considered. The solenoid acts on the magnet, which is located at the free end of the pendulum. In this system, the existence and interrelation of chaos and parametric resonance is theoretically examined. Derived analytical results are supported by numerical simulations and conducted experiments.

Extracted equations

• m*ℓ*α'' = F_x*cos(α) + F_y*sin(α) - m*g*sin(α) - q*α'
• α'' = -α*(ω₀² + h*cos(2ωt)) - q*α'
• where F_x = -∂U/∂x, F_y = -∂U/∂y, and U is the dipole-dipole interaction energy

Paper claims vs. our run

• Parametric resonance condition: h ≥ 2|ω² - ω₀² + 2qω|
  not-testable
  no fidelity score recorded
• Parametric instability growth rate: s = (ω₀² + h/2 - ω² - qω)/(2ω)
  not-testable
  no fidelity score recorded
• Chaos occurs when parametric resonance conditions are satisfied
  not-testable
  no fidelity score recorded
• Lyapunov exponent for small angles matches theoretical growth rate s
  not-testable
  no fidelity score recorded
• System exhibits large-amplitude oscillations from small initial perturbations when parametrically unstable
  not-testable
  no fidelity score recorded

Parameters

Run notes

equations tried: ["m*ℓ*α'' = F_x*cos(α) + F_y*sin(α) - m*g*sin(α) - q*α'", "α'' = -α*(ω₀² + h*cos(2ωt)) - q*α'", 'where F_x = -∂U/∂x, F_y = -∂U/∂y, and U is the dipole-dipole interaction energy']