4D Coupled Floquet Theory

4D Coupled Floquet

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$ \delta_1 $ (Stiffness 1): 1.00

$ \delta_2 $ (Stiffness 2): 2.50

$ c_1 $ (Damping 1): 0.00

$ c_2 $ (Damping 2): 0.50

$ k $ (Coupling): 1.00

$ \epsilon $ (Forcing): 0.00

STABLE

Coupled Mechanics

Time Series: $x_{1,2}(t)$

Scale: $\pm 1.00$

Pendulum 1 Pendulum 2

Complex $ \lambda $-Plane (4D) 4 Multipliers. Click to perturb.

$ \mathrm{Im}\,\lambda $

$ \mathrm{Re}\,\lambda $

4D Coupled System Dynamics

We expanded the base damped Mathieu equation by introducing a second parametrically driven oscillator and coupling them with a linear spring $k$.

\ddot{x}_1 + c_1 \dot{x}_1 + (\delta_1 + \epsilon \cos(2t))x_1 + k(x_1 - x_2) = 0$$ $$\ddot{x}_2 + c_2 \dot{x}_2 + (\delta_2 + \epsilon \cos(2t))x_2 + k(x_2 - x_1) = 0

Exotic Instabilities: Expanding to 4 dimensions (position and velocity for both pendulums) unlocks exotic behaviors that are mathematically forbidden in the 2D system.

Co-existing Instabilities: You can observe multiple distinct modes of instability simultaneously. For example, by using negative stiffnesses, both pendulums can undergo extreme real-axis resonances concurrently.