Abstract
We consider a system of nonlocal equations driven by a perturbed periodic potential. We construct multibump solutions that connect one integer point to another one in a prescribed way. In particular, heteroclinic, homoclinic and chaotic trajectories are constructed. This is the first attempt to consider a nonlocal version of this type of dynamical systems in a variational setting and the first result regarding symbolic dynamics in a fractional framework.
Similar content being viewed by others
References
Alessio F., Bertotti M.L., Montecchiari P.: Multibump solutions to possibly degenerate equilibria for almost periodic Lagrangian systems. Z. Angew. Math. Phys. 50(6), 860–891 (1999). doi:10.1007/s000330050184 CODEN AZMPA8, ISSN 0044-2275
Berti M., Bolle P.: Variational construction of homoclinics and chaos in presence of a saddle-saddle equilibrium. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 9(3), 167–175 (1998) ISSN 1120-6330
Bessi U.: Global homoclinic bifurcation for damped systems. Math. Z. 218(3), 387–415 (1995). doi:10.1007/BF02571911 CODEN MAZEAX, ISSN 0025-5874
Bolotin S., MacKay R.: Multibump orbits near the anti-integrable limit for Lagrangian systems. Nonlinearity 10(5), 1015–1029 (1997). doi:10.1088/0951-7715/10/5/001 CODEN NONLE5, ISSN 0951-7715
Barrios B., Peral I., Soria F., Valdinoci E.: A Widder’s type theorem for the heat equation with nonlocal diffusion. Arch. Ration. Mech. Anal. 213(2), 629–650 (2014). doi:10.1007/s00205-014-0733-1 ISSN 0003-9527
Campanato S.: Proprietà di hölderianità di alcune classi di funzioni. Ann. Scuola Norm. Sup. Pisa (3) 17, 175–188 (1963)
Cozzi M., Passalacqua T.: One-dimensional solutions of non-local Allen–Cahn-type equations with rough kernels. J. Differ. Equ. 260(8), 6638–6696 (2016). doi:10.1016/j.jde.2016.01.006 ISSN 0022-0396
Caffarelli L., Silvestre L.: Regularity results for nonlocal equations by approximation. Arch. Ration. Mech. Anal. 200(1), 59–88 (2011). doi:10.1007/s00205-010-0336-4 ISSN 0003-9527
Cabré X., Sire Y.: Nonlinear equations for fractional Laplacians II: Existence, uniqueness, and qualitative properties of solutions. Trans. Am. Math. Soc. 367(2), 911–941 (2015). doi:10.1090/S0002-9947-2014-05906-0 ISSN 0002-9947
Coti Zelati V., Rabinowitz P.H.: Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials. J. Am. Math. Soc. 4(4), 693–727 (1991). doi:10.2307/2939286 ISSN 0894-0347
del Castillo-Negrete D.: Fractional diffusion models of nonlocal transport. Phys. Plasmas 13(8), 082308,16 (2006). doi:10.1063/1.2336114 CODEN PHPAEN, ISSN 1070-664X
Dipierro S., Figalli A., Valdinoci E.: Strongly nonlocal dislocation dynamics in crystals. Commun. Partial Differ. Equ. 39(12), 2351–2387 (2014). doi:10.1080/03605302.2014.914536 ISSN 0360-5302
Di Nezza E., Palatucci G., Valdinoci E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012). doi:10.1016/j.bulsci.2011.12.004 ISSN 0007-4497
Dipierro S., Palatucci G., Valdinoci E.: Dislocation dynamics in crystals: a macroscopic theory in a fractional Laplace setting. Commun. Math. Phys. 333(2), 1061–1105 (2015). doi:10.1007/s00220-014-2118-6 ISSN 0010-3616
Garroni A., Müller S.: A variational model for dislocations in the line tension limit. Arch. Ration. Mech. Anal. 181(3), 535–578 (2006). doi:10.1007/s00205-006-0432-7 ISSN 0003-9527
del Mar González M., Monneau R.: Slow motion of particle systems as a limit of a reaction-diffusion equation with half-Laplacian in dimension one. Discrete Contin. Dyn. Syst. 32(4), 1255–1286 (2012) ISSN 1078-0947
Maxwell T.O.: Heteroclinic chains for a reversible Hamiltonian system. Nonlinear Anal. 28(5), 871–887 (1997). doi:10.1016/0362-546X(95)00193-Y CODEN NOANDD, ISSN 0362-546X
McLean W.: Strongly elliptic systems and boundary integral equations, pp. xiv+357. Cambridge University Press, Cambridge. ISBN 0-521-66332-6; 0-521-66375-X.
Monneau R., Patrizi S.: Homogenization of the Peierls–Nabarro model for dislocation dynamics. J. Differ. Equ. 253(7), 2064–2105 (2012). doi:10.1016/j.jde.2012.06.019 CODEN JDEQAK, ISSN 0022-0396
Palatucci G., Savin O., Valdinoci E.: Local and global minimizers for a variational energy involving a fractional norm. Ann. Mat. Pura Appl. (4) 192(4), 673–718 (2013). doi:10.1007/s10231-011-0243-9 ISSN 0373-3114
Patrizi S., Valdinoci E.: Crystal dislocations with different orientations and collisions. Arch. Ration. Mech. Anal. 217(1), 231–261 (2015). doi:10.1007/s00205-014-0832-z ISSN 0003-9527
Patrizi S., Valdinoci E.: Homogenization and Orowan’s law for anisotropic fractional operators of any order. Nonlinear Anal. 119, 3–36 (2015). doi:10.1016/j.na.2014.07.010 ISSN 0362-546X
Rabinowitz, P.H.: Periodic and heteroclinic orbits for a periodic Hamiltonian system. Ann. Inst. H. Poincaré Anal. Non Linéaire 6(5), 331–346. http://www.numdam.org/item?id=AIHPC_1989__6_5_331_0. ISSN 0294-1449 (1989)
Rabinowitz P.H.: Heteroclinics for a reversible Hamiltonian system. Ergod. Theory Dyn. Syst. 14(4), 817–829 (1994). doi:10.1017/S0143385700008178 ISSN 0143-3857
Rabinowitz P.H.: A multibump construction in a degenerate setting. Calc. Var. Partial Differ. Equ. 5(2), 159–182 (1997). doi:10.1007/s005260050064 ISSN 0944-2669
Rabinowitz P.H.: Connecting orbits for a reversible Hamiltonian system. Ergod. Theory Dyn. Syst. 20(6), 1767–1784 (2000). doi:10.1017/S0143385700000985 ISSN 0143-3857
Rabinowitz, P.H., Coti Zelati, V.: Multichain-type solutions for Hamiltonian systems. In: Proceedings of the Conference on Nonlinear Differential Equations (Coral Gables, FL, 1999), vol. 5 of Electron. J. Differ. Equ. Conf., pp. 223–235 (electronic). Southwest Texas State Univ., San Marcos (2000)
Séré É.: Existence of infinitely many homoclinic orbits in Hamiltonian systems. Math. Z. 209(1), 27–42 (1992). doi:10.1007/BF02570817 CODEN MAZEAX, ISSN 0025-5874
Savin O., Valdinoci E.: \({\Gamma}\)-convergence for nonlocal phase transitions. Ann. Inst. H. Poincaré Anal. Non Linéaire 29(4), 479–500 (2012). doi:10.1016/j.anihpc.2012.01.006 ISSN 0294-1449
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by C. Liverani
Rights and permissions
About this article
Cite this article
Dipierro, S., Patrizi, S. & Valdinoci, E. Chaotic Orbits for Systems of Nonlocal Equations. Commun. Math. Phys. 349, 583–626 (2017). https://doi.org/10.1007/s00220-016-2713-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-016-2713-9