A harmonic-based method for computing the stability of periodic solutions of dynamical systems

By Arnaud Lazarus, Olivier Thomas. 2010

166 citations recorded by the source corpus.

In this Note, we present a harmonic-based numerical method to determine the local stability of periodic solutions of dynamical systems. Based on the Floquet theory and the Fourier series expansion (Hill method), we propose a simple strategy to sort the relevant physical eigenvalues among the expanded numerical spectrum of the linear periodic system governing the perturbed solution. By mixing the harmonic-balance method and asymptotic numerical method continuation technique with the developed Hill method, we obtain a purely-frequency based continuation tool able to compute the stability of the continued periodic solutions in a reduced computation time. To validate the general methodology, we investigate the dynamical behavior of the forced Duffing oscillator with the developed continuation technique.

Subfields: Biomedical Engineering, Civil and Structural Engineering, Computer Networks and Communications

View Paper: https://www.semanticscholar.org/paper/84fa0ae8891b2ab592a930ca774880ef75273466

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Biomedical Engineering

In Civil and Structural Engineering

In Computer Networks and Communications

Citations in the corpus

The harmonic balance method for bifurcation analysis of large-scale nonlinear mechanical systems (2015, background): “(47) However, only 2n eigenvalues among the complete set λ approximate the Floquet exponents λ̃ of the solution x∗ [63].”
Numerical computation of nonlinear normal modes in mechanical engineering (2016, methodology): “Stability can however be analyzed directly in the frequency domain using Hill’s method [53, 54].”
A high-order, purely frequency based harmonic balance formulation for continuation of periodic solutions: The case of non-polynomial nonlinearities (2008, methodology): “examples treated in this paper show a variety of such functions that might appear in nonlinear dynamical systems. Concerning future work, the companion frequency-based stability analysis presented in [8] could be adapted to general nonlinearities, following the same approach. Also, automatic diﬀeren”
The Harmonic Balance Method for Advanced Analysis and Design of Nonlinear Mechanical Systems (2014, unknown): “After sorting the obtained eigenvalues in order to eliminate the numerical ones, as proposed by Lazarus et al.[23], one can directly relates the remaining eigenvalues λ∗ to the Floquet exponents.”