# Averaging and Computing Normal Forms with Word Series Algorithms

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 267)

## Abstract

In the first part of the present work we consider periodically or quasiperiodically forced systems of the form $$(d/dt)x = \varepsilon f(x,t \omega )$$, where $$\varepsilon \ll 1$$, $$\omega \in \mathbb {R}^d$$ is a nonresonant vector of frequencies and $$f(x,\theta )$$ is $$2\pi$$-periodic in each of the d components of $$\theta$$ (i.e. $$\theta \in \mathbb {T}^d$$). We describe in detail a technique for explicitly finding a change of variables $$x = u(X,\theta ;\varepsilon )$$ and an (autonomous) averaged system $$(d/dt) X = \varepsilon F(X;\varepsilon )$$ 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 $$x(t) = u(X(t),t\omega ;\varepsilon )$$. 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 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’ $$(d/dt) x = \bar{g}(x)+\bar{f}(x)$$, where $$\bar{g}$$ and $$\bar{f}$$ are commuting vector fields and the flow of $$(d/dt) x = \bar{g}(x)$$ 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, $$\bar{g}$$, $$\bar{f}$$ and the invariants are written as series consisting of known vector-valued maps and universal scalar coefficients that may be found recursively.

## Keywords

Averaging High-order averaging Quasi-stroboscopic averaging Hamiltonian problems Near-integrable systems Normal forms

## Mathematics Subject Classification (2010)

34C20 34C29 70H05

## Notes

### Acknowledgements

A. Murua and J. M. Sanz-Serna have been supported by projects MTM2013-46553-C3-2-P and MTM2013-46553-C3-1-P from Ministerio de Economía y Comercio, Spain. Additionally A. Murua has been partially supported by the Basque Government (Consolidated Research Group IT649-13).

## References

1. 1.
Alamo, A., Sanz-Serna, J.M.: A technique for studying strong and weak local errors of splitting stochastic integrators. SIAM J. Numer. Anal. 54, 3239–3257 (2016)
2. 2.
Arnold, V.I.: Geometrical Methods in the Theory of Ordinary Differential Equations, 2nd edn. Springer, New York (1988)Google Scholar
3. 3.
Chartier, P., Murua, A., Sanz-Serna, J.M.: Higher-order averaging, formal series and numerical integration I: B-series. Found. Comput. Math. 10, 695–727 (2010)
4. 4.
Chartier, P., Murua, A., Sanz-Serna, J.M.: Higher-order averaging, formal series and numerical integration II: the quasi-periodic case. Found. Comput. Math. 12, 471–508 (2012)
5. 5.
Chartier, P., Murua, A., Sanz-Serna, J.M.: A formal series approach to averaging: exponentially small error estimates. DCDS A 32, 3009–3027 (2012)
6. 6.
Chartier, P., Murua, A., Sanz-Serna, J.M.: Higher-order averaging, formal series and numerical integration III: error bounds. Found. Comput. Math. 15, 591–612 (2015)
7. 7.
Fauvet, F., Menous, F.: Ecalle’s arborification-coarborification transforms and Connes-Kreimer Hopf algebra. Ann. Sci. Ec. Nom. Sup. 50, 39–83 (2017)
8. 8.
Jacobson, N.: Lie Algebras. Dover, New York (1979)
9. 9.
Kawski, M., Sussmann, H.J.: Nonommutative power series and formal Lie algebraic techniques in nonlinear control theory. In: Helmke, U., Pratzel-Wolters, D., Zerz, E. (eds.) Operators, Systems, and Linear Algebra, pp. 111–118. Teubner, Stuttgart (1997)
10. 10.
Hairer, E., Wanner, G.: On the Butcher group and general multi-value methods. Computing 13, 1–15 (1974)
11. 11.
Murua, A.: The Hopf algebra of rooted trees, free Lie algebras and Lie series. Found. Comput. Math. 6, 387–426 (2006)
12. 12.
Murua, A., Sanz-Serna, J.M.: Vibrational resonance: a study with high-order word-series averaging. Appl. Math. Nonlinear Sci. 1, 146–239 (2016)Google Scholar
13. 13.
Murua, A., Sanz-Serna, J.M.: Computing normal forms and formal invariants of dynamical systems by means of word series. Nonlinear Anal. 138, 326–345 (2016)
14. 14.
Murua, A., Sanz-Serna, J.M.: Word series for dynamical systems and their numerical integrators. Found. Comput. Math. 17, 675–712 (2017)
15. 15.
Reutenauer, C.: Free Lie Algebras. Clarendon Press, Oxford (1993)
16. 16.
Sanders, J.A., Verhulst, F., Murdock, J.: Averaging Methods in Nonlinear Dynamical Systems, 2nd edn. Springer, New York (2007)
17. 17.
Sanz-Serna, J.M., Calvo, M.P.: Numerical Hamiltonian Problems. Chapman and Hall, London (1994)
18. 18.
Sanz-Serna, J.M., Murua, A.: Formal series and numerical integrators: some history and some new techniques. In: Lei, G., Zhi, M. (eds.) Proceedings of the 8th International Congress on Industrial and Applied Mathematics (ICIAM 2015). Higher Education Press, Beijing, pp. 311–331 (2015)Google Scholar