228
R. dell’Erba
Fig. 14.1 Configuration of the lattice over different time (1, 10, 20 and 401) and PE1 contour plot
those that are close to the frame follow the neighbours with a little delay owing to the
different rule determining the displacement of the frame and of the followers. Thus,
they see a different situation with respect to, as an example, a central point. Note
as that the maximum PE1 area (red color in the plot) is not on the leaders line but
just one line on its left; this because, some points of the frame belong to the leader’s
neighbours and they always are close to them (they are subject to the frame rule)
lowering the PE1 value. We can avoid this convexity effect using a different frame
or mirroring the followers to obtain an infinite sample. Finally, it should be noted
how, at t = 401, the lattice is not back completely to the reference configuration
owing to the asymptotic process of relaxation. Consider now a central point j = 67
(sixth column, seventh row, points are numbered from left to right and from bottom
to up) of the lattice. The value of the PE1 increases notably when points are pulled,
after a delay owing to the propagation time as can be seen in Fig. 14.2; it decreases
when the leaders become followers subjected only to the rules leading to equilibrium
barycentre position. If we change point, the shape of the curve remains the same but
can be less o more flared. In Figs. 14.3 and 14.4, the evolution with time of the
coordinates of central point j = 67 and close to the leaders (j = 115) is shown.
Also, in this picture, we can recognize the coordinate x increases linearly (velocity
is constant), after a delay (less for j = 115), owing to the propagation time and later
decreases to the original position.
A light modification can generate instabilities and oscillations; as an example,
we can add to Eq. 1 a feedback term proportional to the difference between actual
and initial position to overshoot the old equilibrium position. The result is showed
in Figs. 14.5 and 14.6 (Same picture as 14.5 but different scale); note as also the
reaction time is changed.
R. dell’Erba
Fig. 14.1 Configuration of the lattice over different time (1, 10, 20 and 401) and PE1 contour plot
those that are close to the frame follow the neighbours with a little delay owing to the
different rule determining the displacement of the frame and of the followers. Thus,
they see a different situation with respect to, as an example, a central point. Note
as that the maximum PE1 area (red color in the plot) is not on the leaders line but
just one line on its left; this because, some points of the frame belong to the leader’s
neighbours and they always are close to them (they are subject to the frame rule)
lowering the PE1 value. We can avoid this convexity effect using a different frame
or mirroring the followers to obtain an infinite sample. Finally, it should be noted
how, at t = 401, the lattice is not back completely to the reference configuration
owing to the asymptotic process of relaxation. Consider now a central point j = 67
(sixth column, seventh row, points are numbered from left to right and from bottom
to up) of the lattice. The value of the PE1 increases notably when points are pulled,
after a delay owing to the propagation time as can be seen in Fig. 14.2; it decreases
when the leaders become followers subjected only to the rules leading to equilibrium
barycentre position. If we change point, the shape of the curve remains the same but
can be less o more flared. In Figs. 14.3 and 14.4, the evolution with time of the
coordinates of central point j = 67 and close to the leaders (j = 115) is shown.
Also, in this picture, we can recognize the coordinate x increases linearly (velocity
is constant), after a delay (less for j = 115), owing to the propagation time and later
decreases to the original position.
A light modification can generate instabilities and oscillations; as an example,
we can add to Eq. 1 a feedback term proportional to the difference between actual
and initial position to overshoot the old equilibrium position. The result is showed
in Figs. 14.5 and 14.6 (Same picture as 14.5 but different scale); note as also the
reaction time is changed.