# American Institute of Mathematical Sciences

July  2022, 42(7): 3431-3463. doi: 10.3934/dcds.2022021

## Asymptotics of ODE's flow on the torus through a singleton condition and a perturbation result. Applications

 Univ Rennes, INSA Rennes, CNRS, IRMAR - UMR 6625, 35000 Rennes, France

*Corresponding author: Marc Briane

Received  June 2021 Revised  January 2022 Published  July 2022 Early access  March 2022

This paper deals with the long time asymptotics of the flow $X$ solution to the vector-valued ODE: $X'(t, x) = b(X(t, x))$ for $t\in \mathbb{R}$, with $X(0, x) = x$ a point of the torus $Y_d$. We assume that the vector field $b$ reads as $\rho\, \Phi$, where $\rho$ is a non negative regular function and $\Phi$ is a non vanishing regular vector field in $Y_d$. In this work, the singleton condition means that the Herman rotation set ${\mathsf{C}}_b$ composed of the average values of $b$ with respect to the invariant probability measures for the flow $X$ is a singleton $\{\zeta\}$. This first allows us to obtain the asymptotics of the flow $X$ when $b$ is a nonlinear current field. Then, we prove a general perturbation result assuming that $\rho$ is the uniform limit in $Y_d$ of a positive sequence $(\rho_n)_{n\in \mathbb{N}}$ satisfying $\rho\leq\rho_n$ and ${\mathsf{C}}_{\rho_n\Phi}$ is a singleton $\{\zeta_n\}$. It turns out that the limit set ${\mathsf{C}}_b$ either remains a singleton, or enlarges to the closed line segment $[0_{ \mathbb{R}^d}, \lim_n\zeta_n]$ of $\mathbb{R}^d$. We provide various corollaries of this result according to the positivity or not of some weighted harmonic means of $\rho$. These results are illustrated by different examples which highlight the alternative satisfied by ${\mathsf{C}}_b$. Finally, the singleton condition allows us to homogenize the linear transport equation induced by the oscillating velocity $b(x/{\varepsilon})$.

Citation: Marc Briane, Loïc Hervé. Asymptotics of ODE's flow on the torus through a singleton condition and a perturbation result. Applications. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3431-3463. doi: 10.3934/dcds.2022021
##### References:
 [1] G. Alessandrini and V. Nesi, Univalent $\sigma$-harmonic mappings, Arch. Rational Mech. Anal., 158 (2001), 155-171.  doi: 10.1007/PL00004242. [2] Y. Amirat, K. Hamdache and A. Ziani, Homogénéisation d'équations hyperboliques du premier ordre et application aux écoulements miscibles en milieu poreux, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 397-417.  doi: 10.1016/s0294-1449(16)30317-1. [3] Y. Amirat, K. Hamdache and A. Ziani, Homogénéisation par décomposition en fréquences d'une équation de transport dans $\mathbb{R}^n$, C. R. Acad. Sci. Paris Sér. I Math., 312 (1991), 37-40. [4] Y. Amirat, K. Hamdache and A. Ziani, Homogenisation of parametrised families of hyperbolic problems, Proc. Roy. Soc. Edinburgh Sect. A, 120 (1992), 199-221.  doi: 10.1017/S0308210500032091. [5] C. Baesens, J. Guckenheimer, S. Kim and R. S. MacKay, Three coupled oscillators: Mode-locking, global bifurcations and toroidal chaos, Phys. D, 49 (1991), 387-475.  doi: 10.1016/0167-2789(91)90155-3. [6] A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Studies in Mathematics and its Applications, 5, North-Holland Publishing Co., Amsterdam-New York, 1978. [7] Y. Brenier, Remarks on some linear hyperbolic equations with oscillatory coefficients, Proceedings of the Third International Conference on Hyperbolic Problems, (Uppsala 1990) Vol. I, II, Studentlitteratur, Lund, (1991), 119–130. [8] M. Briane, Isotropic realizability of a strain field for the two-dimensional incompressible elasticity system, Inverse Problems, 32 (2016), 065002, 22 pp. doi: 10.1088/0266-5611/32/6/065002. [9] M. Briane, Isotropic realizability of fields and reconstruction of invariant measures under positivity properties. Asymptotics of the flow by a nonergodic approach, SIAM J. App. Dyn. Sys., 18 (2019), 1846-1866.  doi: 10.1137/19M1240411. [10] M. Briane, Homogenization of linear transport equations. A new approach,, J. École Polytechnique - Mathématiques, 7 (2020), 479-495.  doi: 10.5802/jep.122. [11] M. Briane, G. W. Milton and V. Nesi, Change of sign of the corrector's determinant for homogenization in three-dimensional conductivity, Arch. Rat. Mech. Anal., 173 (2004), 133-150.  doi: 10.1007/s00205-004-0315-8. [12] M. Briane, G. W. Milton and A. Treibergs, Which electric fields are realizable in conducting materials?, ESAIM: Math. Model. Numer. Anal., 48 (2014), 307-323.  doi: 10.1051/m2an/2013109. [13] R. Caccioppoli, Sugli elementi uniti delle trasformazioni funzionali: Un teorema di esistenza e unicitá alcune sue applicazioni, Rend. Sem. Mat. Padova, 3 (1932), 1-15. [14] J. Carrand, Logarithmic bounds for ergodic averages of constant type rotation number flows on the torus: a short proof, arXiv: 2012.07481, 2020. [15] I. P. Cornfeld, S. V. Fomin and Y. G. Sinaĭ, Ergodic Theory, Translated from the Russian by A.B. Sosinskii, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 245, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4615-6927-5. [16] J. Franks, Recurrence and fixed points of surface homeomorphisms, Ergodic Theory Dynam. Systems, Charles Conley Memorial Issue, 8 (1988), 99–107. doi: 10.1017/S0143385700009366. [17] J. Franks and M. Misiurewicz, Rotation sets of toral flows, Proc. Amer. Math. Soc., 109 (1990), 243-249.  doi: 10.1090/S0002-9939-1990-1021217-5. [18] B. Dacorogna, Direct Methods in the Calculus of Variations, Second Edition, Applied Mathematical Sciences, 78, Springer, New York, 2008. [19] D. Fried, The geometry of cross sections to flows, Topology, 21 (1982), 353-371.  doi: 10.1016/0040-9383(82)90017-9. [20] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001. [21] F. Golse, Moyennisation des champs de vecteurs et EDP, (French), [The averaging of vector fields and PDEs], Journées Équations aux Dérivées Partielles, (Saint Jean de Monts 1990), Exp. no. XVI, École Polytech. Palaiseau, 1990, 17 pp. [22] F. Golse, Perturbations de systèmes dynamiques et moyennisation en vitesse des EDP, C. R. Acad. Sci. Paris Sér. I Math., 314 (1992), 115-120. [23] M. R. Herman, Existence et non existence de tores invariants par des difféomorphismes symplectiques, (French), [Existence and nonexistence of tori invariant under symplectic diffeomorphisms], Séminaire sur les Équations aux Dérivées Partielles 1987-1988, XIV, École Polytech. Palaiseau, 1988, 24 pp. [24] M. W. Hirsch, S. Smale and R. L. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos, Second edition, Pure and Applied Mathematics, 60, Elsevier Academic Press, Amsterdam, 2004. [25] T. Y. Hou and X. Xin, Homogenization of linear transport equations with oscillatory vector fields, SIAM J. Appl. Math., 52 (1992), 34-45.  doi: 10.1137/0152003. [26] P.-E. Jabin and A.-E. Tzavaras, Kinetic decomposition for periodic homogenization problems, SIAM J. Math. Anal., 41 (2009), 360-390.  doi: 10.1137/080735837. [27] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187. [28] A. N. Kolmogorov, On dynamical systems with an integral invariant on the torus, Doklady Akad. Nauk SSSR (N.S.), 93 (1953), 763-766. [29] J. Llibre and R. S. MacKay, Rotation vectors and entropy for homeomorphisms of the torus isotopic to the identity, Ergodic Theory Dynam. Systems, 11 (1991), 115-128.  doi: 10.1017/S0143385700006040. [30] M. Misiurewicz and K. Ziemian, Rotation sets for maps of tori, J. London Math. Soc. (2), 40 (1989), 490-506.  doi: 10.1112/jlms/s2-40.3.490. [31] R. Peirone, Convergence of solutions of linear transport equations, Ergodic Theory Dynam. Systems, 23 (2003), 919-933.  doi: 10.1017/S014338570200144X. [32] C. L. Siegel, Note on differential equations on the torus, Annals of Mathematics, 46 (1945), 423-428.  doi: 10.2307/1969161. [33] E. Y. Sinaĭ, Dynamical Systems II, Ergodic Theory with Applications to Dynamical Systems and Statistical Mechanics, (Translated from the Russian), Encyclopaedia of Mathematical Sciences, 2, Springer-Verlag Berlin, 1989. [34] T. Tassa, Homogenization of two-dimensional linear flows with integral invariance, SIAM J. Appl. Math., 57 (1997), 1390-1405.  doi: 10.1137/S0036139996299820. [35] L. Tartar, Nonlocal effects induced by homogenization, Partial Differential Equations and the Calculus of Variations Vol. II, F. Colombini et al. (eds.), Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston, MA, 2 (1989), 925–938.

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##### References:
 [1] G. Alessandrini and V. Nesi, Univalent $\sigma$-harmonic mappings, Arch. Rational Mech. Anal., 158 (2001), 155-171.  doi: 10.1007/PL00004242. [2] Y. Amirat, K. Hamdache and A. Ziani, Homogénéisation d'équations hyperboliques du premier ordre et application aux écoulements miscibles en milieu poreux, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 397-417.  doi: 10.1016/s0294-1449(16)30317-1. [3] Y. Amirat, K. Hamdache and A. Ziani, Homogénéisation par décomposition en fréquences d'une équation de transport dans $\mathbb{R}^n$, C. R. Acad. Sci. Paris Sér. I Math., 312 (1991), 37-40. [4] Y. Amirat, K. Hamdache and A. Ziani, Homogenisation of parametrised families of hyperbolic problems, Proc. Roy. Soc. Edinburgh Sect. A, 120 (1992), 199-221.  doi: 10.1017/S0308210500032091. [5] C. Baesens, J. Guckenheimer, S. Kim and R. S. MacKay, Three coupled oscillators: Mode-locking, global bifurcations and toroidal chaos, Phys. D, 49 (1991), 387-475.  doi: 10.1016/0167-2789(91)90155-3. [6] A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Studies in Mathematics and its Applications, 5, North-Holland Publishing Co., Amsterdam-New York, 1978. [7] Y. Brenier, Remarks on some linear hyperbolic equations with oscillatory coefficients, Proceedings of the Third International Conference on Hyperbolic Problems, (Uppsala 1990) Vol. I, II, Studentlitteratur, Lund, (1991), 119–130. [8] M. Briane, Isotropic realizability of a strain field for the two-dimensional incompressible elasticity system, Inverse Problems, 32 (2016), 065002, 22 pp. doi: 10.1088/0266-5611/32/6/065002. [9] M. Briane, Isotropic realizability of fields and reconstruction of invariant measures under positivity properties. Asymptotics of the flow by a nonergodic approach, SIAM J. App. Dyn. Sys., 18 (2019), 1846-1866.  doi: 10.1137/19M1240411. [10] M. Briane, Homogenization of linear transport equations. A new approach,, J. École Polytechnique - Mathématiques, 7 (2020), 479-495.  doi: 10.5802/jep.122. [11] M. Briane, G. W. Milton and V. Nesi, Change of sign of the corrector's determinant for homogenization in three-dimensional conductivity, Arch. Rat. Mech. Anal., 173 (2004), 133-150.  doi: 10.1007/s00205-004-0315-8. [12] M. Briane, G. W. Milton and A. Treibergs, Which electric fields are realizable in conducting materials?, ESAIM: Math. Model. Numer. Anal., 48 (2014), 307-323.  doi: 10.1051/m2an/2013109. [13] R. Caccioppoli, Sugli elementi uniti delle trasformazioni funzionali: Un teorema di esistenza e unicitá alcune sue applicazioni, Rend. Sem. Mat. Padova, 3 (1932), 1-15. [14] J. Carrand, Logarithmic bounds for ergodic averages of constant type rotation number flows on the torus: a short proof, arXiv: 2012.07481, 2020. [15] I. P. Cornfeld, S. V. Fomin and Y. G. Sinaĭ, Ergodic Theory, Translated from the Russian by A.B. Sosinskii, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 245, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4615-6927-5. [16] J. Franks, Recurrence and fixed points of surface homeomorphisms, Ergodic Theory Dynam. Systems, Charles Conley Memorial Issue, 8 (1988), 99–107. doi: 10.1017/S0143385700009366. [17] J. Franks and M. Misiurewicz, Rotation sets of toral flows, Proc. Amer. Math. Soc., 109 (1990), 243-249.  doi: 10.1090/S0002-9939-1990-1021217-5. [18] B. Dacorogna, Direct Methods in the Calculus of Variations, Second Edition, Applied Mathematical Sciences, 78, Springer, New York, 2008. [19] D. Fried, The geometry of cross sections to flows, Topology, 21 (1982), 353-371.  doi: 10.1016/0040-9383(82)90017-9. [20] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001. [21] F. Golse, Moyennisation des champs de vecteurs et EDP, (French), [The averaging of vector fields and PDEs], Journées Équations aux Dérivées Partielles, (Saint Jean de Monts 1990), Exp. no. XVI, École Polytech. Palaiseau, 1990, 17 pp. [22] F. Golse, Perturbations de systèmes dynamiques et moyennisation en vitesse des EDP, C. R. Acad. Sci. Paris Sér. I Math., 314 (1992), 115-120. [23] M. R. Herman, Existence et non existence de tores invariants par des difféomorphismes symplectiques, (French), [Existence and nonexistence of tori invariant under symplectic diffeomorphisms], Séminaire sur les Équations aux Dérivées Partielles 1987-1988, XIV, École Polytech. Palaiseau, 1988, 24 pp. [24] M. W. Hirsch, S. Smale and R. L. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos, Second edition, Pure and Applied Mathematics, 60, Elsevier Academic Press, Amsterdam, 2004. [25] T. Y. Hou and X. Xin, Homogenization of linear transport equations with oscillatory vector fields, SIAM J. Appl. Math., 52 (1992), 34-45.  doi: 10.1137/0152003. [26] P.-E. Jabin and A.-E. Tzavaras, Kinetic decomposition for periodic homogenization problems, SIAM J. Math. Anal., 41 (2009), 360-390.  doi: 10.1137/080735837. [27] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187. [28] A. N. Kolmogorov, On dynamical systems with an integral invariant on the torus, Doklady Akad. Nauk SSSR (N.S.), 93 (1953), 763-766. [29] J. Llibre and R. S. MacKay, Rotation vectors and entropy for homeomorphisms of the torus isotopic to the identity, Ergodic Theory Dynam. Systems, 11 (1991), 115-128.  doi: 10.1017/S0143385700006040. [30] M. Misiurewicz and K. Ziemian, Rotation sets for maps of tori, J. London Math. Soc. (2), 40 (1989), 490-506.  doi: 10.1112/jlms/s2-40.3.490. [31] R. Peirone, Convergence of solutions of linear transport equations, Ergodic Theory Dynam. Systems, 23 (2003), 919-933.  doi: 10.1017/S014338570200144X. [32] C. L. Siegel, Note on differential equations on the torus, Annals of Mathematics, 46 (1945), 423-428.  doi: 10.2307/1969161. [33] E. Y. Sinaĭ, Dynamical Systems II, Ergodic Theory with Applications to Dynamical Systems and Statistical Mechanics, (Translated from the Russian), Encyclopaedia of Mathematical Sciences, 2, Springer-Verlag Berlin, 1989. [34] T. Tassa, Homogenization of two-dimensional linear flows with integral invariance, SIAM J. Appl. Math., 57 (1997), 1390-1405.  doi: 10.1137/S0036139996299820. [35] L. Tartar, Nonlocal effects induced by homogenization, Partial Differential Equations and the Calculus of Variations Vol. II, F. Colombini et al. (eds.), Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston, MA, 2 (1989), 925–938.
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