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. AmiratK. 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. AmiratK. 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. AmiratK. 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. BaesensJ. GuckenheimerS. 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. BrianeG. 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. BrianeG. 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.

show all references

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. AmiratK. 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. AmiratK. 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. AmiratK. 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. BaesensJ. GuckenheimerS. 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. BrianeG. 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. BrianeG. 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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