The
Case of
Nonlinear
Dynamical System
“Attractors” –
Teleological?
Part 1. of the
Series on
Nonlinear
Dynamical
Systems.
Dear Reader,
Consider the “limit-cycle
attractor”, a closed curve orbit in a state-space, approached by the
state-space trajectory – i.e., by the solution-geometry of a nonlinear
dynamical system’s system-describing total-differential equation – but it is only
approached asymptotically, never reached or
even touched by that state-space trajectory for all finite [realistic] values
of the time-variable, t
< ¥.
The limit-cycle thus constitutes
a ‘‘‘transcendental’’’ state-space structure/process/closed loop of
states, ‘‘‘located at’’’ “t = ¥”, and of a kind not found among
the solutions of linear total-differential equations, but
prominent among the solutions of nonlinear, total-differential,
dynamical, and strictly-causal, deterministic equations.
¿Does such a “limit-cycle attractor” actually “attract” the state-space trajectory of the solution to
the nonlinear “ordinary” differential equation, that harbors that limit-cycle
trajectory as its “essence”; as its final, “t = ¥” solution, irresistibly “pulling” upon that, winding,
trajectory from the “transcendent”, “eternal”, “t =
¥”
future, and “reeling it in” to, like Sisyphus, forever orbit that limit-cycle
loop of states, closer and closer to that loop, but never finally merging with
it, into it, for all, finite, futurity?
¿Are nonlinear total-differential equations, with “limit-cycle”
solutions, or even those with “strange attractor” solutions – “aperiodic
deterministic flows” in state-space; so-called “chaos” – just
a mathematical implementation of Aristotle’s doctrine of teleology, of “final causes”?
No.
A nonlinear differential
equation is “nonlinear”, or “of higher degree” [higher than, or just different
than, the 1st degree] because, in
it, the, e.g., x(t), unknown-“function-of-time”-to-be-solved-for, operates upon itself;
multiplies itself, or at least one of that function-unknown’s
derivatives operates upon itself; multiplies itself –
x(t)1 ´ x(t)1 = x(t)2;
x(t)1 ´ x(t)1 ´ x(t)1 = x(t)3,
etc.;
(dx(t)/dt)2,
etc.
This can lead
to an “unbounded”, unlimited “explosion-to-infinity”
of the equation’s solution state-vector, due to a division-by-zero “singularity”
for a critical time-moment value in the modeled-history of that system.
However, in
limit-cycle-solution nonlinear dynamical total-differential equations, this ‘self-interaction’
– of the state-function with itself – leads to a cumulative self-constraint
upon the values of that state-vector, x(t), as its t-value increases.
Thus, starting from the initial
state/state-vector,
x(0), of such a system, with
each advance of each of its time-values,
t, to its “next” value, t +
Dt, the system’s state-space path or trajectory
constrains itself, bending itself a little more toward the limit-cycle loop,
until the ‘cumulation’ of these ‘self-bendings’ brings it, for high-enough values
of t, into a shrinking orbit that increasingly converges
toward that cyclically-repeating loop of states, without ever
finally entering that loop.
This is not,
mystical, “teleology”.
This is self-consistent
self-determination.
For more
information regarding these
Seldonian insights, and to read and/or download, free
of charge, PDFs and/or JPGs of Foundation books, other texts, and images, please see:
and
https://independent.academia.edu/KarlSeldon
For partially pictographical, ‘poster-ized’ visualizations of many of these Seldonian insights -- specimens of ‘dialectical art’ – as well as dialectically-illustrated books
published by
the F.E.D. Press, see –
https://www.etsy.com/shop/DialecticsMATH
¡ENJOY!
Regards,
Miguel
Detonacciones,
Voting Member, Foundation Encyclopedia Dialectica [F.E.D.];
Elected Member, F.E.D. General Council;
Participant, F.E.D. Special Council for Public Liaison;
Officer, F.E.D. Office of Public Liaison.
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