# American Institute of Mathematical Sciences

March  2006, 5(1): 125-146. doi: 10.3934/cpaa.2006.5.125

## A Nekhoroshev theorem for some infinite--dimensional systems

 1 Dipartimento di Matematica, Università di Tor Vergata, via della ricerca scientifica, 00133, Roma, Italy

Received  May 2005 Revised  November 2005 Published  December 2005

We study the persistence for long times of the solutions of some infinite--dimensional discrete hamiltonian systems with formal hamiltonian $\sum_{i=1}^\infty h(A_i) + V(\varphi),$ $(A,\varphi)\in \mathbb R^{\mathbb N}\times \mathbb T^{\mathbb N}.$ $V(\varphi)$ is not needed small and the problem is perturbative being the kinetic energy unbounded. All the initial data $(A_i(0), \varphi_i(0)),$ $i\in \mathbb N$ in the phase--space $\mathbb R^{\mathbb N} \times \mathbb T^{\mathbb N},$ give rise to solutions with $|A_i(t) - A_i(0)|$ close to zero for exponentially--long times provided that $A_i(0)$ is large enough for $|i|$ large. We need $\frac{\partial h}{\partial A_i}(A_i(0))$ unbounded for $i\to+\infty$ making $\varphi_i$ a fast variable the greater is $i,$ the faster is the angle $\varphi_i$ (avoiding the resonances). The estimates are obtained in the spirit of the averaging theory reminding the analytic part of Nekhoroshev--theorem.
Citation: Paolo Perfetti. A Nekhoroshev theorem for some infinite--dimensional systems. Communications on Pure & Applied Analysis, 2006, 5 (1) : 125-146. doi: 10.3934/cpaa.2006.5.125
 [1] Björn Augner, Birgit Jacob. Stability and stabilization of infinite-dimensional linear port-Hamiltonian systems. Evolution Equations & Control Theory, 2014, 3 (2) : 207-229. doi: 10.3934/eect.2014.3.207 [2] Jinxin Xue. Continuous averaging proof of the Nekhoroshev theorem. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3827-3855. doi: 10.3934/dcds.2015.35.3827 [3] J. C. Robinson. A topological time-delay embedding theorem for infinite-dimensional cocycle dynamical systems. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 731-741. doi: 10.3934/dcdsb.2008.9.731 [4] Martin Schechter. Monotonicity methods for infinite dimensional sandwich systems. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 455-468. doi: 10.3934/dcds.2010.28.455 [5] Mohammed Elarbi Achhab. On observers and compensators for infinite dimensional semilinear systems. Evolution Equations & Control Theory, 2015, 4 (2) : 131-142. doi: 10.3934/eect.2015.4.131 [6] Paolo Perfetti. An infinite-dimensional extension of a Poincaré's result concerning the continuation of periodic orbits. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 401-418. doi: 10.3934/dcds.1997.3.401 [7] Massimiliano Guzzo, Giancarlo Benettin. A spectral formulation of the Nekhoroshev theorem and its relevance for numerical and experimental data analysis. Discrete & Continuous Dynamical Systems - B, 2001, 1 (1) : 1-28. doi: 10.3934/dcdsb.2001.1.1 [8] Stefano Pasquali. A Nekhoroshev type theorem for the nonlinear Klein-Gordon equation with potential. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3573-3594. doi: 10.3934/dcdsb.2017215 [9] Marta Lewicka, Mohammadreza Raoofi. A stability result for the Stokes-Boussinesq equations in infinite 3d channels. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2615-2625. doi: 10.3934/cpaa.2013.12.2615 [10] J. Gwinner. On differential variational inequalities and projected dynamical systems - equivalence and a stability result. Conference Publications, 2007, 2007 (Special) : 467-476. doi: 10.3934/proc.2007.2007.467 [11] María J. Garrido-Atienza, Oleksiy V. Kapustyan, José Valero. Preface to the special issue "Finite and infinite dimensional multivalued dynamical systems". Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : ⅰ-ⅳ. doi: 10.3934/dcdsb.201705i [12] H. O. Fattorini. The maximum principle for linear infinite dimensional control systems with state constraints. Discrete & Continuous Dynamical Systems - A, 1995, 1 (1) : 77-101. doi: 10.3934/dcds.1995.1.77 [13] Kening Lu, Alexandra Neamţu, Björn Schmalfuss. On the Oseledets-splitting for infinite-dimensional random dynamical systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1219-1242. doi: 10.3934/dcdsb.2018149 [14] Shin-Ichiro Ei, Hirofumi Izuhara, Masayasu Mimura. Infinite dimensional relaxation oscillation in aggregation-growth systems. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1859-1887. doi: 10.3934/dcdsb.2012.17.1859 [15] Chris Guiver, Mark R. Opmeer. Bounded real and positive real balanced truncation for infinite-dimensional systems. Mathematical Control & Related Fields, 2013, 3 (1) : 83-119. doi: 10.3934/mcrf.2013.3.83 [16] Xavier Cabré, Amadeu Delshams, Marian Gidea, Chongchun Zeng. Preface of Llavefest: A broad perspective on finite and infinite dimensional dynamical systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : ⅰ-ⅲ. doi: 10.3934/dcds.201812i [17] P. Adda, J. L. Dimi, A. Iggidir, J. C. Kamgang, G. Sallet, J. J. Tewa. General models of host-parasite systems. Global analysis. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 1-17. doi: 10.3934/dcdsb.2007.8.1 [18] Andrii Mironchenko, Hiroshi Ito. Characterizations of integral input-to-state stability for bilinear systems in infinite dimensions. Mathematical Control & Related Fields, 2016, 6 (3) : 447-466. doi: 10.3934/mcrf.2016011 [19] Helene Frankowska, Elsa M. Marchini, Marco Mazzola. A relaxation result for state constrained inclusions in infinite dimension. Mathematical Control & Related Fields, 2016, 6 (1) : 113-141. doi: 10.3934/mcrf.2016.6.113 [20] Xiaocai Wang, Junxiang Xu, Dongfeng Zhang. A KAM theorem for the elliptic lower dimensional tori with one normal frequency in reversible systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2141-2160. doi: 10.3934/dcds.2017092

2018 Impact Factor: 0.925