System Dynamics & Floquet Theory

We are visualizing a parametrically driven small-angle oscillator, which can be interpreted as the linearized form of a vertically forced pendulum near an equilibrium. For small angular displacements, $\sin x \approx x$, giving the damped Mathieu Equation:

The coefficients define the physics of the system:

• $x(t)$: The small displacement from the chosen equilibrium configuration.
• $c$: The linear viscous damping coefficient.
• $\delta$: The effective stiffness or natural frequency squared. If $\delta < 0$, the linear restoring term is reversed, so the origin behaves like an inverted equilibrium.
• $\epsilon$: The forcing amplitude, proportional to the vertical drive of the support.

Stability Analysis: Because the system is periodic with period $T = \pi$, we use Floquet Theory. We numerically integrate the fundamental matrix over one period to find the Monodromy Matrix, $\Phi(\pi)$. The eigenvalues of this matrix are the Floquet Multipliers, $\lambda_1$ and $\lambda_2$.

• If $|\lambda_i| < 1$ for both, perturbations decay geometrically. The resting state is asymptotically stable.
• If $|\lambda_i| > 1$ for either, the origin becomes unstable. The system enters Parametric Resonance, and small perturbations grow exponentially!