On symmetry breaking bifurcations in reversible systems

L. M. Lerman ; D. V. Turaev

Russian journal of nonlinear dynamics, Tome 8 (2012) no. 2, pp. 323-343.

Résumé

We review results on local bifurcations in reversible systems (flows and diffeomorphisms) which lead to the creation of pairs attractor–repellor at bifurcations from symmetric equilibria (for flows) and fixed points (for diffeomorphisms). We consider bifurcations of co-dimension 1 in systems of small dimensions (2,3, and 4).

Détail

Keywords: reversible system, reversible diffeomorphism, symmetric equilibrium, symmetric fixed point, loss of symmetry.
Mots-clés : bifurcation

Bibliographie

[1] Borisov A. V., Mamaev I. S., “Strannye attraktory v dinamike keltskikh kamnei”, Negolonomnye dinamicheskie sistemy: Integriruemost, khaos, strannye attraktory, Sb. st., eds. A. V. Borisov, I. S. Mamaev, Inst. kompyutern. issled., M.–Izhevsk, 2002, 296–326 | MR

[2] Bochner S., Montgomery D., “Locally compact groups of differentiable transformations”, Ann. of Math. (2), 47:4 (1946), 639–653 | DOI | MR | Zbl

[3] Bibikov Yu. N., Local theory of nonlinear analytic ordinary differential equations, Lecture Notes in Math., 702, Springer, Berlin–Heidelberg, 1979, 147 pp. | MR | Zbl

[4] Broer H. W., “Quasiperiodic flow near a codimension one singularity of a divergence free vector field in dimension three”, Dynamical systems and turbulence (Warwick, 1980), Lecture Notes in Math., 898, Springer, Berlin–New York, 1981, 75–89 | DOI | MR

[5] Broer H. W., Huitema G. B., Sevryuk M. B., Quasi-periodic motions in families of dynamical systems: Order amidst chaos, Lecture Notes in Math., 1645, Springer, Berlin, 1996, 196 pp. | MR

[6] Delshams A., Gonchenko S. V., Gonchenko V. S., Lázaro J. T., Sten'kin O., Abundance of attracting, repelling and elliptic periodic orbits in two-dimensional reversible maps, [math.DS], 25 Jan 2012, arXiv: 1201.5357

[7] Devaney R. L., “Reversible diffeomorphisms and flows”, Trans. Amer. Math. Soc., 218 (1976), 89–113 | DOI | MR | Zbl

[8] Fiedler B., Liebscher S., Alexander J. C., “Generic Hopf bifurcation from lines of equilibria without parameters: 1. Theory”, J. Differential Equations, 167:1 (2000), 16–35 | DOI | MR | Zbl

[9] Gavrilov N. K., “O nekotorykh bifurkatsiyakh sostoyaniya ravnovesiya s odnim nulevym i paroi chisto mnimykh kornei”, Metody kachestvennoi teorii differentsialnykh uravnenii, Mezhvuz. temat. sb. nauchn. trudov, ed. E. A. Leontovich-Andronova, GGU, Gorkii, 1978, 33–40

[10] Gavrilov N. K., Mezhvuz. temat. sb. nauchn. trudov, Metody kachestvennoi teorii differentsialnykh uravnenii, ed. E. A. Leontovich-Andronova, Gorkii, GGU, 1985

[11] Fenichel N., “Persistence and smoothness of invariant manifolds for flows”, Indiana Univ. Math. J., 21:3 (1971), 193–226 | DOI | MR | Zbl

[12] Glebsky L. Yu., Lerman L. M., “On the small stationary self-localized solutions for generalized $1$D Swift–Hohenberg equation”, Chaos, 5:3 (1995), 424–431 | DOI | MR | Zbl

[13] Gonchenko S. V., Lamb J., Rios I., Turaev D. V., Attractors and repellers near generic reversible elliptic points, Preprint

[14] Gonchenko A. S., Gonchenko S. V., Kazakov A. O., “Novye rezultaty o khaoticheskoi dinamike «keltskogo kamnya»”, Nelineinaya dinamika (to appear)

[15] Holmes Ph., Guckenheimer J., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Appl. Math. Sci., 42, 2nd ed., Springer, New York, 1983, 480 pp. ; Gukenkheimer Dzh., Kholms P., Nelineinye kolebaniya, dinamicheskie sistemy i bifurkatsii vektornykh polei, Inst. kompyutern. issled., M.–Izhevsk, 2002, 560 pp. | MR | Zbl

[16] Haragus M., Iooss G., Local bifurcations, center manifolds, and normal forms in infinite-dimensional dynamical systems, Universitext, Springer, London; EDP Sciences, Les Ulis, 2011, 329 pp. | MR | Zbl

[17] Iooss G., Pérouème M.-C., “Perturbed homoclinic solutions in reversible $1 : 1$ resonance vector fields”, J. Differential Equations, 102:1 (1993), 62–88 | DOI | MR | Zbl

[18] Kuznetsov Yu. A., Elements of applied bifurcation theory, Appl. Math. Sci., 112, 3rd ed., Springer, New York, 2004, 631 pp. | MR | Zbl

[19] Lamb J. S. W., Roberts J. A. G., “Time-reversal symmetry in dynamical systems: A survey”, Phys. D, 112:1-2 (1998), 1–39 | DOI | MR | Zbl

[20] Lamb J. S. W., Capel H. W., “Local bifurcations on the plane with reversing point group symmetry”, Chaos Solitons Fractals, 5:2 (1995), 271–293 | DOI | MR | Zbl

[21] Lamb J.S.W., Stenkin O. V., “Newhouse regions for reversible systems with infinitely many stable, unstable and elliptic periodic orbits”, Nonlinearity, 17:4 (2004), 1217–1244 | DOI | MR | Zbl

[22] Lerman L., “Breaking hyperbolicity for smooth symplectic toral diffeomorphisms”, Regul. Chaotic Dyn., 16:2-3 (2010), 196–211 | MR

[23] Lerman L., “Isoenergetical structure of integrable Hamiltonian systems in extended neighborhoods of simple singular points: Three degrees of freedom”, Methods of qualitative theory of differential equations and related topics: Supplement, Amer. Math. Soc. Transl. Ser. 2, 200, eds. L. Lerman, G. Polotovsky, L. Shilnikov, AMS, Providence, RI, 2000, 219–242 | MR

[24] Pérouème M.-C., “Bifurcation from a periodic orbit for a strongly resonant reversible autonomous vector field”, SIAM J. Math. Anal., 24:6 (1993), 1577–1596 | DOI | MR

[25] Sevryuk M. B., Reversible systems, Lecture Notes in Math., 1211, Springer, Berlin, 1986, 319 pp. | MR | Zbl

[26] Shilnikov L. P., Shilnikov A. L., Turaev D. V., Chua L., Metody kachestvennoi teorii v nelineinoi dinamike, V 2-kh tt., «Regulyarnaya i khaoticheskaya dinamika», M.–Izhevsk, 2003, 442 pp.; 2010, 548 pp.

[27] Shilnikov L. P., “Ob odnom novom tipe bifurkatsii mnogomernykh dinamicheskikh sistem”, Dokl. AN SSSR, 182:1 (1969), 53–56

[28] Shilnikov A., Nicolis G., Nicolis S., “Bifurcation and predictability analysis of a low-dimensional atmospheric circulation model”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 5:6 (1995), 1701–1711 | DOI | MR

[29] Takens F., “Forced oscillations and bifurcations”, Applications of global analysis (Utrecht, 1973), v. 1, Comm. Math. Inst. Rijksuniv. Utrecht, 3, Utrecht, Math. Inst. Rijksuniv. Utrecht, 1974, 1–59 ; Global analysis of dynamical systems, ред. H. W. Broer, B. Krauskopf, G. Vegter, Bristol, IOP Publ., 2001, 2–61 | MR | MR

[30] Teixeira M. A., “Singularities of reversible vector fields”, Phys. D, 100:1-2 (1997), 101–118 | DOI | MR | Zbl

[31] Tkhai V. N., “Obratimye mekhanicheskie sistemy”, Nelineinaya mekhanika, eds. V. M. Matrosov, V. V. Rumyantsev, A. V. Karapetyan, Fizmatlit, M., 2001, 131–146

[32] Vanderbauwhede A., Fiedler B., “Homoclinic period blow-up in reversible and conservative system”, Z. Angew. Math. Phys., 43:2 (1992), 291–318 | DOI | MR

[33] Wang L.-J., “Homoclinic and heteroclinic orbits for the $0^2$ or $0^2i\omega$ singularity in the presence of two reversibility symmetries”, Quart. Appl. Math., 67:1 (2009), 1–38 | MR | Zbl