Complex \( \lambda \)-Plane (Linearised about the origin)4 multipliers. Local stability only; nonlinear saturation is shown in the time simulation.
\( \mathrm{Im}\,\lambda \)
\( \mathrm{Re}\,\lambda \)
4D Coupled Nonlinear System Dynamics
This page combines two different levels of description. The Floquet panel is computed from the linearisation about the origin, so it is a local statement about the zero solution. The time-series panel integrates the full nonlinear system, including weak cubic stiffness and amplitude-dependent damping, so it shows what a finite-amplitude perturbation does after the local instability has been triggered.
$$\ddot{x}_1 + (c_1 + \eta x_1^2)\dot{x}_1 + (\delta_1 + \epsilon \cos(2t))x_1 + \alpha x_1^3 + k(x_1 - x_2) + g\dot{x}_2 = 0$$
$$\ddot{x}_2 + (c_2 + \eta x_2^2)\dot{x}_2 + (\delta_2 + \epsilon \cos(2t))x_2 + \alpha x_2^3 + k(x_2 - x_1) - g\dot{x}_1 = 0$$
How to read the figure: the gyroscopic coupling \(g\) enters the velocity block as a skew-symmetric term. In the linearised periodic problem, that changes the monodromy matrix and can make the dominant Floquet multipliers leave the unit circle as a complex-conjugate pair rather than along the real axis. When that happens, the origin is linearly unstable through oscillatory growth. The nonlinear terms \(\alpha x^3\) and \(\eta x^2\dot{x}\) do not enter the Floquet calculation here; they only affect the separate time integration by limiting or reshaping the finite-amplitude response.
- Bounded nonlinear example: click Load Torus-like Example. This preset is intended to show a bounded, slowly modulated time trace after nonlinear saturation. It should be read as a qualitative nonlinear response example, not as a proof of a specific invariant torus from the diagnostics shown on this page alone.
- Complex-pair crossing example: click Load Complex-Pair Crossing, or set \(\delta_1=1.22\), \(\delta_2=0.37\), \(c_1=0.38\), \(c_2=0.11\), \(k=0.54\), \(g=0.65\), and increase \(\epsilon\) through about \(1.13\). Near that threshold the leading Floquet multipliers are a complex-conjugate pair that crosses the unit circle directly.
- Important caveat: the Floquet plane does not show the nonlinear attractor. It only shows the spectrum of the linearised return map at the origin. Any bounded oscillation, amplitude modulation, or quasiperiodic-looking response in the simulation belongs to the nonlinear system and must be interpreted separately from the local multiplier plot.