False Asymptotic Instability Behavior at Iterated Functions with Lyapunov Stability in Nonlinear Time Series

TL;DR

This paper investigates false asymptotic instability in iterated nonlinear functions, revealing how empirical observations can misrepresent true Lyapunov stability due to finite time effects.

Contribution

It introduces a framework to identify false instability behaviors in iterated functions caused by empirical observation constraints.

Findings

Discontinuous g(x) can mimic instability in finite observations

Lyapunov stability may be falsely inferred from empirical data

Asymptotic stability can be hidden in short-term dynamics

Abstract

Empirically defining some constant probabilistic orbits of f(x) and g(x) iterated high-order functions, the stability of these functions in possible entangled interaction dynamics of the environment through its orbit's connectivity (open sets) provides the formation of an exponential dynamic fixed point b = S(n+1) as a metric space (topological property) between both iterated functions for short time lengths. However, the presence of a dynamic fixed point f(g(x)) (x+1) can identify a convergence at iterations for larger time lengths of b (asymptotic stability in Lyapunov sense). Qualitative (QDE) results show that the average distance between the discontinuous function g(x) to the fixed point of the continuous function f(x) (for all possible solutions), might express fluctuations of g(x) on time lengths (instability effect). This feature can reveal the false empirical asymptotic instability behavior between the given domains f and g due to time lengths observation and empirical constraints within a well-defined Lyapunov stability.