2015/12/11 by A. Murua, Murua, A., J. M. Sanz‐Serna +1
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Numerical Analysis (math.NA) #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.1512.03601
openalex publication_date 2015/12/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
In the first part of the present work we consider periodically or quasiperiodically forced systems of the form (d/dt)x = εf(x,t ω), where ε≪ 1, ω∈ℝd is a nonresonant vector of frequencies and f(x,θ) is 2π-periodic in each of the d components of θ (i.e. θ∈\mathbbTd). We describe in detail a technique for explicitly finding a change of variables x = u(X,θ;ε) and an (autonomous) averaged system (d/dt) X = εF(X;ε) so that, formally, the solutions of the given system may be expressed in terms of the solutions of the averaged system by means of the relation x(t) = u(X(t),tω;ε). Here u and F are found as series whose terms consist of vector-valued maps weighted by suitable scalar coefficients. The maps are easily written down by combining the Fourier coefficients of f and the coefficients are found with the help of simple recursions. Furthermore these coefficients are \em universal in the sense that they do not depend on the particular f under consideration. In the second part of the contribution, we study problems of the form (d/dt) x = g(x)+f(x), where one knows how to integrate the "unperturbed" problem (d/dt)x = g(x) and f is a perturbation satisfying appropriate hypotheses. It is shown how to explicitly rewrite the system in the "normal form" (d/dt) x = g(x)+ f(x), where g and f are \em commuting vector fields and the flow of (d/dt) x = g(x) is conjugate to that of the unperturbed (d/dt)x = g(x). In Hamiltonian problems the normal form directly leads to the explicit construction of formal invariants of motion. Again, g, f and the invariants are written as series consisting of known vector-valued maps and universal scalar coefficients that may be found recursively.