The Case of Nonlinear Dynamical System “Attractors” – Teleological? Part 1. of the Series on Nonlinear Dynamical Systems.

 

 

 

 

 

 

 

 

 

 

 

 

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:

 

www.dialectics.info

 

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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