Construct a MizerSim of the linearised limit cycle

Using the leading complex eigenvector from getStability(), constructs a MizerSim object covering one period of the limit cycle in the linear approximation. The result can be inspected with all standard mizer plotting functions (e.g. plotBiomass(), plotSpectra()).

Usage

getLimitCycleSim(x, amplitude = 0.1, t_save = 0.1, ...)

Arguments

x: A MizerParams object at a steady state, typically the output of steadyNewton(), or the list returned by getStability().
amplitude: Maximum relative perturbation $\max_w |\delta N(t,w)|/N^*(w)$ across the limit cycle. Default 0.1.
t_save: The time interval between saved time steps in the returned MizerSim. Defaults to 0.1.
...: Additional arguments forwarded to getStability() when x is a MizerParams object.

Value

A MizerSim object whose time axis spans one period $[0, T]$ of the linearised limit cycle.

Details

Mathematical background

Near a Hopf bifurcation the dominant eigenvalue of the linearised one-step map is a complex conjugate pair $\lambda = r e^{i\theta}$ with $r \approx 1$ and angular frequency $\theta = 2\pi/T$ per time step. The linearised perturbation is $$\delta N(t) = A\,\operatorname{Re}[e^{i\theta t}\,\mathbf{v}],$$ where $\mathbf{v}$ is the leading complex eigenvector (normalised so $\max_w |\mathbf{v}(w)| = 1$) and $A$ is chosen so that the maximum relative perturbation $\max_{t,w}|\delta N(t,w)|/N^*(w) = $ amplitude. The full state at each time step is $$N(t) = \max(N^* + \delta N(t),\; 0)$$ (clipping prevents negative abundances at large amplitudes).

The resource is placed at its quasi-static semichemostat equilibrium $n_{pp}^*(N(t))$ at each step, consistent with the reduced Jacobian used by getStability().

The returned MizerSim has times running from 0 to $T$ (the period in time steps, typically years).