Abstract
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.
Highlights
Determining the local stability of a dynamical system periodic solution is of primary interest in an engineering context since only stable solutions are experimentally encountered
Hill’s method often provides satisfactory results, the meaning of the computed eigenvalues is generally misunderstood and may lead to wrong results. This method is not of a trivial use since it requires to approximate the spectra of infinite-dimensional operators, i.e. to sort the most converged eigenvalues among all the numerical ones. In this Note, we propose a simple numerical strategy to sort these eigenvalues and properly determine the stability of periodic solutions of dynamical systems
We propose a simple criterion based on the energy distribution of the computed eigenvectors to extract the most converged eigenvalues (Floquet exponents) of the truncated problem and determine the local stability of the solution
Summary
Determining the local stability of a dynamical system periodic solution is of primary interest in an engineering context since only stable solutions are experimentally encountered. The monodromy matrix is a by-product of a shooting continuation method [2] The efficiency of those methods has been proved for years, but, inherently, the eigenvalues accuracy depends on the chosen time-steps size, leading to possible long time computations. Hill’s method often provides satisfactory results, the meaning of the computed eigenvalues is generally misunderstood and may lead to wrong results This method is not of a trivial use since it requires to approximate the spectra of infinite-dimensional operators, i.e. to sort the most converged eigenvalues among all the numerical ones. In this Note, we propose a simple numerical strategy to sort these eigenvalues and properly determine the stability of periodic solutions of dynamical systems. We confirm the convergence of the proposed Hill method and its considerable running time benefit as compared to the time-domain monodromy matrix computation
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