Dynamical integrity assessment of stable equilibria: a new rapid iterative procedure

Abstract

A new algorithm for the estimation of the robustness of a dynamical system’s equilibrium is presented. Unlike standard approaches, the algorithm does not aim to identify the entire basin of attraction of the solution. Instead, it iteratively estimates the so-called local integrity measure, that is, the radius of the largest hypersphere entirely included in the basin of attraction of a solution and centred in the solution. The procedure completely overlooks intermingled and fractal regions of the basin of attraction, enabling it to provide a significant engineering quantity in a very short time. The algorithm is tested on four different mechanical systems of increasing dimension, from 2 to 8. For each system, the variation of the integrity measure with respect to a system parameter is evaluated, proving the engineering relevance of the results provided. Despite some limitations, the algorithm proved to be a viable alternative to more complex and computationally demanding methods, making it a potentially appealing tool for industrial applications.

Stability of a nonideally excited Duffing oscillator

“…uncertainties on the values of the system parameters) [1]. In addition, some effort should be devoted to determining or estimating the basins of attraction of the different attractors present in the system [20,40]. This would help to quantify the probability of resonant capture and passage through resonance.…”

“…Result 4.1 Assuming d < 0, and for initial conditions sufficiently close to the fixed point corresponding to z = −1, solutions of the averaged system (24) and solutions of the exact system (20) satisfy…”

“…If this trajectory is tracked for the time length on which the averaged system is valid (O(1/ √ ε)), it is found to be repelled out of the equilibrium. Therefore, it can be stated that this fixed point, which is unstable in the averaged system (24), is also unstable in the exact system (20).…”

“…Note that an overbar has been used to emphasize the difference between the original variables {a, β, σ, φ} (solutions of system (20)) and the averaged variables { ā, β, σ , φ} (solutions of system ( 22)), whose relation is given by…”

“…In order to avoid making the main body of the paper too long and cumbersome, the details of the proof can be found in Appendix I. In summary, Result 4.1 allows stating that, if the fixed point corresponding to z = −1 is asymptotically stable in the averaged system (24), the stability is preserved in the exact system (20).…”