(1a)–(1g) to reveal the presence of different timescales; determine how to group timescales that are present; implement geometric singular perturbation theory (GSPT) to set up reduced systems based on separation of timescales; and use the reduced systems to explain the mechanisms underlying the dynamics of the SB solutions.
This contradicts the assumption of using the reduced area as a robust order parameter, at least for the chick epiblast system.
In order to compare systems we will develop the necessary formulas for at least one multiple-A/D-based reduced STAP solution referred to as Beamspace STAP, where the number of adaptive channels is reduced to a manageable size, and then apply STAP on the reduced system using all available coherent pulses, giving us sufficient adaptive degrees of freedom.
Here, we aim to use the geometry of the reduced system in order to identify a manifold that separates the trajectories of the reduced problem into two distinct behaviors: those that return to an equilibrium or rest state and those that proceed to the fold curve where the fast dynamics again become important in establishing a spiking solution.
We can use this reduced system to explore how the system behaves as N increases.
While the reduced system used here has shortcomings, it also has important advantages in the context of the questions addressed here.
Thus, at present, the reduced system used here provides unique opportunities to examine these questions in an isolated, well-controlled fashion, with good access to network activity properties and the possibility to use advanced closed-loop techniques to expose and record adaptation kinetics.
Note that, in general, M ≠ M ^, i.e. solving the equation h v, m, n, μ, ε) = 0 for m = M v, n, μ, ε) does not yield the centre manifold for any ε, including ε = 0. Thus, the dynamics of the reduced system obtained using the quasi-steady-state reduction is, in general, different to the dynamics of the full system reduced to the centre manifold.
Methods to solve the reduced system of equations using iterative and direct solvers were developed and applied to determine the numerical benefit of using them through simple block upsetting and swash-plate forging simulations.
(P_{1}) and (P_{p}) deal with the reduced system of equations using the functions in lines 9-11 as well.
