All simulations in this figure used a 3-dimensional analytical system in which the fast noise in the perceived stimulus *S* _{p} was absorbed into the 2D model, which was then driven in Task 2 through the space spanned by the behavior *B*, the memory *M*, and the position *P* by the true stimulus *S* _{t} (see , Section 4). To reveal the canonical dynamics, all variability of the true environment was also eliminated in this figure, so that the environment consisted simply of repeats of the single canonical sequence of three free and one attached seaweed strips, all exactly of length τ. All simulations started from the initial conditions *B*(0) = 0, *M*(0) = 0, *P*(0) = 0. *A*: Representative steady-state limit cycle trajectories reached by the system with different values of τ, all with *f* = 0.1. The time required to complete one such cycle is the “limit cycle period” in this figure (see also *C*, bottom). *B*: Values of the limit cycle period, color-coded according to the scales shown on the right, over values of τ ranging from 1 to 250 and *f* from 0 to 1. In the red region, the limit cycle reached from the initial conditions *B*(0) = 0, *M*(0) = 0, *P*(0) = 0 consists of three successful ingestions and a successful egestion, as in simulations 1–3 of *A*; in the blue region, it consists of a failed egestion, as in simulation 4 of *A*. The locations of the four simulations in *A* are marked. Below is shown a representative section through the plot, again at *f* = 0.1, and under it the corresponding performance, given by 3τ/(limit cycle period) for a “successful” limit cycle and 0 for a “failed” limit cycle. With no variability, the smallest possible period of the successful limit cycle is exactly 9τ and the highest performance therefore exactly 1/3 (, Section 3.3). *C*: Over an intermediate range of τ, the two kinds of limit cycle coexist. Here, with τ = 150 and *f* = 0.1, a small perturbation moved the system from the successful to the failed limit cycle (see further and , Section 6.6). The corresponding time series of the behavior *B* is shown below.

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