Polynomial Stochastic Dynamical Indicators

TL;DR

This paper introduces three polynomial stochastic dynamical indicators to analyze the impact of uncertainty on dynamical systems, providing tools for phase space mapping and stability assessment under parametric uncertainty.

Contribution

It presents novel polynomial-based indicators derived from Lyapunov exponents and polynomial expansions, enhancing uncertainty quantification in dynamical systems analysis.

Findings

Indicators effectively depict phase space under uncertainty

They identify robust initial conditions and stability regions

Numerical experiments validate the indicators' usefulness

Abstract

This paper introduces three types of dynamical indicators that capture the effect of uncertainty on the time evolution of dynamical systems. Two indicators are derived from the definition of Finite Time Lyapunov Exponents while a third indicator directly exploits the property of the polynomial expansion of the dynamics with respect to the uncertain quantities. The paper presents the derivation of the indicators and a number of numerical experiments that illustrates the use of these indicators to depict a cartography of the phase space under parametric uncertainty and to identify robust initial conditions and regions of practical stability in the restricted three-body problem.