Two-dimensional projections of several trajectories for Scheme 1 and Scheme 2. Calculations have been done with k 1 = 2, k 2 = 1, and k 3 = 0.6 for both schemes; these values are employed for all calculations in this work.
Pictorial representation of the attractiveness region in the N-dimensional phase-space. For points x belonging to one has that with characteristic rate ω(x) ⩾ 0: a trajectory which enters remains “entrapped” in it up to the EM.
Temporal profiles of frequencies ω n (see Eq. (29) ) for several orders n (on the right side) and only for n = 40 (on the left side), along a pair of trajectories coming “from above” and “from below” the perceived SM for Scheme 1 (insets). Open circles: expected profile (see Eq. (25) ) within the attractiveness region. Red triangles in the left panels: corresponding points on the phase-space and on the ω n profiles.
As in Figure 3 , here for Scheme 2.
Percentage of reached dyads for matrices V of several dimensions d as function of the maximum order n max employed in the algorithm. The used round-off parameters are ɛ eig = ɛ c = 0.05 (see the text for details).
Trend of best estimates of the limit percentages (%∞) of reached dyads as function of the matrix dimension d. Limit percentages are extrapolated by fits of the profiles in Figure 5 (see the insets). The blue circles are extra-points inserted for the trivial case d = 1. Upper panel: extrapolation from exponential fits; Lower panel: extrapolation from sigmoidal fits. Dashed lines are guides for the eye.
Explicit numbering of the cumulative index Q, characteristic rates h Q , and connectivity matrices M for the model cases of Scheme 1 and Scheme 2 (only the non-null elements are reported). The red borders delimit the reduced blocks sufficient to describe the autonomous evolution of the functions h 1, h 2, ⋅ ⋅ ⋅, h 6 for both schemes.
Percentage of recognized dyads from randomly generated matrices V of different dimensions d. Values are given for increasing maximum order, n max, which limits the length of the sequences considered to check the convergence to dyadic form. Results refer to i conv = 3 and to two choices of the cutoff parameters.
Article metrics loading...
Full text loading...