# American Institute of Mathematical Sciences

October  2013, 33(10): 4411-4433. doi: 10.3934/dcds.2013.33.4411

## Construction of response functions in forced strongly dissipative systems

 1 Institute for Mathematics and its Applications, University of Minnesota, 307 Church St SE, Minneapolis, MN 55455, United States 2 Department of Mathematics, University of Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma, Italy 3 School of Mathematics, Georgia Institute of Technology, 686 Cherry St., Atlanta, GA 30332-0160, United States

Received  July 2012 Revised  December 2012 Published  April 2013

We study the existence of quasi--periodic solutions of the equation $ε \ddot x + \dot x + ε g(x) = ε f(\omega t)\ ,$ where $x: \mathbb{R} \rightarrow \mathbb{R}$ is the unknown and we are given $g:\mathbb{R} \rightarrow \mathbb{R}$, $f: \mathbb{T}^d \rightarrow \mathbb{R}$, $\omega \in \mathbb{R}^d$ (without loss of generality we can assume that $\omega\cdot k\not=0$ for any $k \in \mathbb{Z}^d\backslash\{0\}$). We assume that there is a $c_0\in \mathbb{R}$ such that $g(c_0) = \hat f_0$ (where $\hat f_0$ denotes the average of $f$) and $g'(c_0) \ne 0$. Special cases of this equation, for example when $g(x)=x^2$, are called the varactor problem'' in the literature.
We show that if $f$, $g$ are analytic, and $\omega$ satisfies some very mild irrationality conditions, there are families of quasi--periodic solutions with frequency $\omega$. These families depend analytically on $ε$, when $ε$ ranges over a complex domain that includes cones or parabolic domains based at the origin.
The irrationality conditions required in this paper are very weak. They allow that the small denominators $|\omega \cdot k|^{-1}$ grow exponentially with $k$. In the case that $f$ is a trigonometric polynomial, we do not need any condition on $|\omega \cdot k|$. This answers a delicate question raised in [8].
We also consider the periodic case, when $\omega$ is just a number ($d = 1$). We obtain that there are solutions that depend analytically in a domain which is a disk removing countably many disjoint disks. This shows that in this case there is no Stokes phenomenon (different resummations on different sectors) for the asymptotic series.
The approach we use is to reduce the problem to a fixed point theorem. This approach also yields results in the case that $g$ is a finitely differentiable function; it provides also very effective numerical algorithms and we discuss how they can be implemented.
Citation: Renato C. Calleja, Alessandra Celletti, Rafael de la Llave. Construction of response functions in forced strongly dissipative systems. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4411-4433. doi: 10.3934/dcds.2013.33.4411
##### References:
 [1] Jürgen Appell and Petr P. Zabrejko, "Nonlinear Superposition Operators," 95 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1990. doi: 10.1017/CBO9780511897450. [2] Roger Broucke and K. Garthwaite, A programming system for analytical series expansions on a computer, Celestial Mechanics, 1 (1969), 271-284. doi: 10.1007/BF01228844. [3] Renato Calleja, Alessandra Celletti and Rafael de la Llave, A KAM theory for conformally symplectic systems: efficient algorithms and their validation, MP_ARC # 11-188. Jour. Diff. Equ, to appear. [4] Renato Calleja and Rafael de la Llave, A numerically accessible criterion for the breakdown of quasi-periodic solutions and its rigorous justification, Nonlinearity, 23 (2010), 2029-2058. doi: 10.1088/0951-7715/23/9/001. [5] Alessandra Celletti and Luigi Chierchia, KAM stability and celestial mechanics, Mem. Amer. Math. Soc., 187 (2007), viii+134. [6] Corrado Falcolini and Rafael de la Llave, A rigorous partial justification of Greene's criterion, J. Statist. Phys., 67 (1992), 609-643. doi: 10.1007/BF01049722. [7] Mickaël Gastineau and Jacques Laskar, Development of trip: Fast sparse multivariate polynomial multiplication using burst tries, Computational Science - ICCS, Lecture Notes in Computer Science,(2006), 446-453. [8] Guido Gentile, Quasi-periodic motions in strongly dissipative forced systems, Ergodic Theory Dynam. Systems, 30 (2010), 1457-1469. doi: 10.1017/S0143385709000583. [9] Guido Gentile, Quasiperiodic motions in dynamical systems: Review of a renormalization group approach, J. Math. Phys., 51 (2010) pp. 34, 015207. doi: 10.1063/1.3271653. [10] Guido Gentile, Construction of quasi-periodic response solutions in forced strongly dissipative systems, Forum Mathematicum, 24 (2012), 791-808. doi: 10.1515/form.2011.084. [11] Guido Gentile, Michele V. Bartuccelli and Jonathan H. B. Deane, Summation of divergent series and Borel summability for strongly dissipative differential equations with periodic or quasiperiodic forcing terms, J. Math. Phys., 46 (2005) pp.20, 062704. doi: 10.1063/1.1926208. [12] Guido Gentile, Michele V. Bartuccelli and Jonathan H. B. Deane, Quasiperiodic attractors, Borel summability and the Bryuno condition for strongly dissipative systems, J. Math. Phys., 47 (2006) pp.10, 072702. doi: 10.1063/1.2213790. [13] Godfrey H. Hardy, "Divergent Series," Oxford, at the Clarendon Press, 1949. [14] Alex Haro, Automatic differentiation tools in computational dynamical systems, Univ. of Barcelona Preprint, (2008). [15] Alex Haro and Rafael de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: rigorous results, J. Differential Equations, 228 (2006), 530-579. doi: 10.1016/j.jde.2005.10.005. [16] Yu. S. Ilyashenko (editor), Nonlinear Stokes phenomena, Advances in Soviet mathematics, Am. Math. Soc., (1993). [17] Donald E. Knuth, "The Art of Computer Programming. Vol. 2: Seminumerical Algorithms," Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, third revised edition, 1997. [18] Thomas Runst and Winfried Sickel, "Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations," 3 of de Gruyter Series in Nonlinear Analysis and Applications. Walter de Gruyter & Co., Berlin, 1996. doi: 10.1515/9783110812411. [19] Alan D. Sokal, An improvement of Watson's theorem on Borel summability, J. Math. Phys., 21 (1980), 261-263. doi: 10.1063/1.524408. [20] Michael E. Taylor, "Partial Differential Equations. III," 117 of Applied Mathematical Sciences. Springer-Verlag, New York, 1997. [21] George N. Watson, A theory of asymptotic series, Lond. Phil. Trans. (A), 211 (1911), 279-313. doi: 10.1098/rsta.1912.0007.

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##### References:
 [1] Jürgen Appell and Petr P. Zabrejko, "Nonlinear Superposition Operators," 95 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1990. doi: 10.1017/CBO9780511897450. [2] Roger Broucke and K. Garthwaite, A programming system for analytical series expansions on a computer, Celestial Mechanics, 1 (1969), 271-284. doi: 10.1007/BF01228844. [3] Renato Calleja, Alessandra Celletti and Rafael de la Llave, A KAM theory for conformally symplectic systems: efficient algorithms and their validation, MP_ARC # 11-188. Jour. Diff. Equ, to appear. [4] Renato Calleja and Rafael de la Llave, A numerically accessible criterion for the breakdown of quasi-periodic solutions and its rigorous justification, Nonlinearity, 23 (2010), 2029-2058. doi: 10.1088/0951-7715/23/9/001. [5] Alessandra Celletti and Luigi Chierchia, KAM stability and celestial mechanics, Mem. Amer. Math. Soc., 187 (2007), viii+134. [6] Corrado Falcolini and Rafael de la Llave, A rigorous partial justification of Greene's criterion, J. Statist. Phys., 67 (1992), 609-643. doi: 10.1007/BF01049722. [7] Mickaël Gastineau and Jacques Laskar, Development of trip: Fast sparse multivariate polynomial multiplication using burst tries, Computational Science - ICCS, Lecture Notes in Computer Science,(2006), 446-453. [8] Guido Gentile, Quasi-periodic motions in strongly dissipative forced systems, Ergodic Theory Dynam. Systems, 30 (2010), 1457-1469. doi: 10.1017/S0143385709000583. [9] Guido Gentile, Quasiperiodic motions in dynamical systems: Review of a renormalization group approach, J. Math. Phys., 51 (2010) pp. 34, 015207. doi: 10.1063/1.3271653. [10] Guido Gentile, Construction of quasi-periodic response solutions in forced strongly dissipative systems, Forum Mathematicum, 24 (2012), 791-808. doi: 10.1515/form.2011.084. [11] Guido Gentile, Michele V. Bartuccelli and Jonathan H. B. Deane, Summation of divergent series and Borel summability for strongly dissipative differential equations with periodic or quasiperiodic forcing terms, J. Math. Phys., 46 (2005) pp.20, 062704. doi: 10.1063/1.1926208. [12] Guido Gentile, Michele V. Bartuccelli and Jonathan H. B. Deane, Quasiperiodic attractors, Borel summability and the Bryuno condition for strongly dissipative systems, J. Math. Phys., 47 (2006) pp.10, 072702. doi: 10.1063/1.2213790. [13] Godfrey H. Hardy, "Divergent Series," Oxford, at the Clarendon Press, 1949. [14] Alex Haro, Automatic differentiation tools in computational dynamical systems, Univ. of Barcelona Preprint, (2008). [15] Alex Haro and Rafael de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: rigorous results, J. Differential Equations, 228 (2006), 530-579. doi: 10.1016/j.jde.2005.10.005. [16] Yu. S. Ilyashenko (editor), Nonlinear Stokes phenomena, Advances in Soviet mathematics, Am. Math. Soc., (1993). [17] Donald E. Knuth, "The Art of Computer Programming. Vol. 2: Seminumerical Algorithms," Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, third revised edition, 1997. [18] Thomas Runst and Winfried Sickel, "Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations," 3 of de Gruyter Series in Nonlinear Analysis and Applications. Walter de Gruyter & Co., Berlin, 1996. doi: 10.1515/9783110812411. [19] Alan D. Sokal, An improvement of Watson's theorem on Borel summability, J. Math. Phys., 21 (1980), 261-263. doi: 10.1063/1.524408. [20] Michael E. Taylor, "Partial Differential Equations. III," 117 of Applied Mathematical Sciences. Springer-Verlag, New York, 1997. [21] George N. Watson, A theory of asymptotic series, Lond. Phil. Trans. (A), 211 (1911), 279-313. doi: 10.1098/rsta.1912.0007.
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