Biology Reference
In-Depth Information
where x
¼
cv, and P
¼
cp in Eq. (15.25), or
dx
=
dt
¼
f(cv
;
cp
;
fl)
;
or
(15.27)
“(rate of change in x) is a function of x, control parameter cp and fluctuation fl” , or most
abstractly
(15.28)
Þ
0
¼
ð
f(
;
;
Þ
(15.29)
where (_____ )' indicates a time derivative of whatever is inside the parenthesis,
f is a mathematical function, and the underlines represent “place holders” which
can be filled with appropriate variables, numbers, or words. That is, although
Eqs. 15.25-15.29 all look different, their meaning is the same, and this is because
Equation (15.25) can be integrated with respect to time t, resulting in:
cv
¼
F(t
;
cp
;
IC
;
fl)
(15.30)
where F is a new function different from f, t is time, and IC is the integration
constant whose numerical value is determined by initial conditions. According to
Eq. (15.30), the so-called trajectories (see 1 below) in
t-cv plots
depend on
initial
conditions.
Some of the basic concepts and principles embodied in the coordination law,
Eq. (15.25), can be visualized using the skateboarder as an analogy. The skate-
boarder moving up and down the walls of the empty swimming pool is a convenient
metaphor to illustrate a set of important concepts in nonlinear dynamics:
1.
Coordination variable,
cv: The position of the skateboarder on the
x
-axis which
varies with time, increasing (movement from left to right) or decreasing (move-
ment from right to left) as the skateboarder moves up and down the pool surface
acted upon by gravity. The plot of cv against time, t, is known as
trajectories
.
The shapes of the trajectories differ (i.e., the trajectories evolve in time in
different ways) depending on
initial condition
(i.e., the numerical value of cv
at t
0) and the
control parameter
cp, which is in the present case the curvature
of the pool surface.
2.
Stable fixed point
also called
attractor
: The skateboarder always returns to the
bottom of the pool to minimize its gravitational potential energy.
3.
Unstable fixed point
also called
repeller
: The skateboarder resting on the top of
the hill is
unstable
because he/she can be easily pushed off the peak position. If
the skateboarder is unperturbed (e.g., by randomly fluctuating directions of
wind), he/she can remain at the precarious position forever.
4.
Potential landscape
often designated as the
cv-V plot
: The relation between the
gravitational potential energy of skateboarder's body, V, and its position on the
x
-axis which fixes its
z
-axis due to the constraint imposed by the pool surface.
¼
