Journal of Statistical Physics

, Volume 119, Issue 5–6, pp 1069–1138 | Cite as

Bifurcations in a System of Interacting Fronts

  • A. Amann
  • E. SchöllEmail author


We show that the bifurcation scenario in a high-dimensional system with interacting moving fronts can be related to the universal U-sequence which is known from the symbolic analysis of iterated one-dimensional maps. This connection is corroborated for a model of a semiconductor superlattice, which describes the complex dynamics of electron accumulation and depletion fronts. By a suitable Poincaré section we reduce the dynamics to a low-dimensional iterated map, for which in the most elementary case the bifurcation points can be determined analytically.


Front dynamics U-sequence semiconductor superlattice 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Scott, A. eds. 2005Encyclopedia of Nonlinear ScienceRoutledgeLondonGoogle Scholar
  2. 2.
    Langer, J.S. 1980Instabilities and pattern formation in crystal growthRev. Mod. Phys.521CrossRefGoogle Scholar
  3. 3.
    Boroson, B., McCray, R., Clark, C.O., Slavin, J., Low, M.-M.M., Chu, Y.-H., Buren, D.V. 1997An interstellar conduction front within a wolf-rayet ring nebula observed with the GHRSAstrophys. J.478638Google Scholar
  4. 4.
    Zhabotinskii, A.M. 1964Periodic processes of the oxidation of malonic acid in solutionBiofizika9306Google Scholar
  5. 5.
    Kapral, R.Showalter, K. eds. 1995Chemical Waves and PatternsKluwer Academic PublishersDordrechtGoogle Scholar
  6. 6.
    Merzhanov, A.G., Rumanov, E.N. 1999Physics of reaction wavesRev. Mod. Phys.711173CrossRefGoogle Scholar
  7. 7.
    Davidenko, J.M., Pertsov, A.M., Salomonsz, R., Baxter, W., Jalife, J. 1992Stationary and drifting spiral waves of excitation in isolated cardiac muscleNature355349CrossRefPubMedGoogle Scholar
  8. 8.
    Steinbock, O., Siegert, F., M üller, S.C., Weijer, C.J. 1993Three-dimensional waves of excitation during dictyostelium morphogenesisProc. Natl. Acad. Sci.907332PubMedGoogle Scholar
  9. 9.
    Cross, M.C., Hohenberg, P.C. 1993Pattern formation outside of equilibriumRev. Mod. Phys.65851CrossRefGoogle Scholar
  10. 10.
    Mikhailov, A.S. 1994Foundations of Synergetics Vol I2SpringerBerlinGoogle Scholar
  11. 11.
    Falkovich, G., Gawdzki, K., Vergassola, M. 2001Particles and fields in fluid turbulenceRev. Mod. Phys.73913CrossRefGoogle Scholar
  12. 12.
    Gunn, J.B. 1963Microwave oscillations of current in III-V semiconductorsSol. Stat. Comm.188CrossRefGoogle Scholar
  13. 13.
    Bonch-Bruevich, V.L., Zvyagin, I.P., Mironov, A.G. 1975Domain Electrical Instabilities in SemiconductorsConsultant BureauNew YorkGoogle Scholar
  14. 14.
    Sch öll, E. 1987Nonequilibrium Phase Transitions in SemiconductorsSpringerBerlinGoogle Scholar
  15. 15.
    Shaw, M.P., Mitin, V.V., Sch öll, E., Grubin, H.L. 1992The Physics of Instabilities in Solid State Electron DevicesPlenum PressNew YorkGoogle Scholar
  16. 16.
    Peinke, J., Parisi, J., R össler, O., Stoop, R. 1992Encounter with ChaosSpringerBerlinHeidelbergGoogle Scholar
  17. 17.
    E. Sch öll, F.-J. Niedernostheide, J. Parisi, W. Prettl and H. Purwins, Formation of Spatio-temporal structures in semiconductors, in Evolution of spontaneous structures in Dissipative Continuous Systems, F.H. Busse and S.C. M üller (Springer, Berlin, 1998), pp. 446–494.Google Scholar
  18. 18.
    Aoki, K. 2000Nonlinear Dynamics and Chaos in SemiconductorsInstitute of Physics PublishingBristolGoogle Scholar
  19. 19.
    Sch öll, E. 2001Nonlinear Spatio-temporal Dynamics and Chaos in SemiconductorsVol. 10 (Cambridge University PressNonlinear Science SeriesGoogle Scholar
  20. 20.
    Cantalapiedra, I.R, Bergmann, M.J., Bonilla, L.L., Teitsworth, S.W. 2001Chaotic motion of space charge wave fronts in semiconductors under time independent voltage biasPhys.Rev.E63056216CrossRefGoogle Scholar
  21. 21.
    Bonilla, L.L., Cantalapiedra, I.R. 1997Universality of the Gunn effect, Self-sustained oscillations mediated by solitary wavesPhys.Rev.E563628CrossRefGoogle Scholar
  22. 22.
    Wacker, A. 2002Semiconductor superlattices: A model system for nonlinear transportPhys.Rep.3571CrossRefGoogle Scholar
  23. 23.
    Bonilla, L.L. 2002Theory of nonlinear charge transport, wave propagation, and self-oscillations in semiconductor superlatticesJ. Phys.: Condens. Matter14R341CrossRefGoogle Scholar
  24. 24.
    Schomburg, E., Scheuerer, R., Brandl, S., Renk, K.F., Pavel’ev, D.G., Koschurinov, Y., Ustinov, V., Zhukov, A., Kovsh, A., Kop’ev, P.S. 1999InGaAs/InAlAs superlattice oscillator at 147 GHzElectron. Lett.351491CrossRefGoogle Scholar
  25. 25.
    Schlesner, J., Amann, A., Janson, N.B., Just, W., Sch öll, E. 2003Self-stabilization of high frequency oscillations in semiconductor superlattices by time–delay autosynchronizationPhys.Rev.E68066208CrossRefGoogle Scholar
  26. 26.
    Schlesner, J., Amann, A., Janson, N.B., Just, W., Sch öll, E. 2004Self-stabilization of chaotic domain oscillations in superlattices by time–delayed feedback controlSemicond.Sci.Technol.19S34CrossRefGoogle Scholar
  27. 27.
    Faist, J., Capasso, F., Sivco, D.L., Sirtori, C., A.L, Hutchinson, Cho, A.Y. 1994Quantum cascade laserScience264553Google Scholar
  28. 28.
    Gmachl, C., Capasso, F., Sivco, D.L., Cho, A.Y. 2001Recent progress in quantum cascade lasers and applicationsRep.Prog.Phys.641533CrossRefGoogle Scholar
  29. 29.
    Fromhold, T.M., Patane, A., Bujkiewicz, S., Wilkinson, P.B., Fowler, D., Sherwood, D., Stapleton, S.P., Krokhin, A.A., Eaves, L., Henini, M., Sankeshwar, N.S., Sheard, F.W. 2004Chaotic electron diffusion through stochastic webs enhances current flow in superlatticesNature428726CrossRefPubMedGoogle Scholar
  30. 30.
    Chase, C., Serrano, J., Ramadge, P.J. 1993Periodicity and chaos from switched flow systems: contrasting examples of discretely controlled continuous flow systems, IEEE TransAutomat Control3870CrossRefGoogle Scholar
  31. 31.
    Steuer, H., Wacker, A., Sch öll, E., Ellmauer, M., Schomburg, E., Renk, K.F. 2000Thermal breakdown, bistability, and complex high-frequency current oscillations due to carrier heating in superlatticesAppl.Phys.Lett.762059CrossRefGoogle Scholar
  32. 32.
    Grüner, G. 1988The dynamics of charge-density wavesRev. Mod. Phys.601129CrossRefGoogle Scholar
  33. 33.
    Keener, J.P., Sneyd, J. 1998Mathematical physiologySpringerNew YorkGoogle Scholar
  34. 34.
    Flach, S., Zolotaryuk, Y., Kladko, K. 1999Moving lattice kinks and pulses: An inverse methodPhys.Rev.E596105CrossRefGoogle Scholar
  35. 35.
    Carpio, A., Bonilla, L.L. 2003Depinning transitions in discrete reaction-diffusion equationsSIAM J.Appl.Math.631056CrossRefGoogle Scholar
  36. 36.
    Bonilla, L.L., Kindelan, M., Moscoso, M., Venakides, S. 1997Periodic generation and propagation of travelling fronts in dc voltage biased semiconductor superlatticesSIAM J.Appl.Math.571588CrossRefGoogle Scholar
  37. 37.
    Carpio, A., Bonilla, L.L., Wacker, A., Schöll, E. 2000Wavefronts may move upstream in semiconductor superlatticesPhys.Rev.E614866CrossRefGoogle Scholar
  38. 38.
    Amann, A., Wacker, A., Bonilla, L.L., Schöll, E. 2001Dynamic scenarios of multi-stable switching in semiconductor superlatticesPhys.Rev.E63066207CrossRefGoogle Scholar
  39. 39.
    Carpio, A., Bonilla, L.L., Dell’Acqua, G. 2001Motion of wave fronts in semiconductor superlatticesPhys.Rev.E64036204CrossRefGoogle Scholar
  40. 40.
    Kastrup, J., Hey, R., Ploog, K.H., Grahn, H.T., L.L, Bonilla, Kindelan, M., Moscoso, M., Wacker, A., Galán, J. 1997Electrically tunable GHz oscillations in doped GaAs- AlAs superlatticesPhys.Rev.B552476CrossRefGoogle Scholar
  41. 41.
    Bonilla, L.L., J.Galán, , Cuesta, J.A., Martínez, F.C, Molera, J.M. 1994Dynamics of electric field domains and oscillations of the photocurrent in a simple superlattice modelPhys.Rev.B508644CrossRefGoogle Scholar
  42. 42.
    A. Amann, J. Schlesner, A. Wacker, and E. Sch öll, Self-generated chaotic dynamics of field domains in superlattices, in Proc. of 26th International Conference on the Physics of Semiconductors (ICPS-26), (Edinburgh 2002), ed. J.H. Davies and A.R Long (2003).Google Scholar
  43. 43.
    Bonilla, L.L., Cantalapiedra, I.R., Gomila, G., Rubí, J.M. 1997Asymptotic analysis of the Gunn effect with realistic boundary conditionsPhys.Rev.E561500CrossRefGoogle Scholar
  44. 44.
    Sánchez, D., Moscoso, M., Bonilla, L.L., Platero, G., Aguado, R. 1999Current self-oscillations, spikes and crossover between charge monopole and dipole waves in semiconductor superlatticesPhys.Rev.B604489CrossRefGoogle Scholar
  45. 45.
    Kastrup, J., Prengel, F., Grahn, H.T., Ploog, K., Sch öll, E. 1996Formation times of electric field domains in doped GaAs- AlAs superlatticesPhys.Rev.B531502CrossRefGoogle Scholar
  46. 46.
    Bulashenko, O.M., Bonilla, L.L. 1995Chaos in resonant-tunneling superlatticesPhys.Rev.B527849CrossRefGoogle Scholar
  47. 47.
    Alekseev, K.N., Berman, G.P., Campbell, D.K., Cannon, E.H, Cargo, M.C. 1996Dissipative chaos in semiconductor superlatticesPhys.Rev.B5410625CrossRefGoogle Scholar
  48. 48.
    Bonilla, L.L., Bulashenko, O.M., Galán, J., Kindelan, M., Moscoso, M. 1996Dynamics of electric-field domains and chaos in semiconductor superlatticesSol.State El.40161CrossRefGoogle Scholar
  49. 49.
    Bulashenko, O.M., Luo, K.J., Grahn, H.T., Ploog, K.H., Bonilla, L.L. 1999Multifractal dimension of chaotic attractors in a driven semiconductor superlatticePhys.Rev.B605694CrossRefGoogle Scholar
  50. 50.
    Cao, J.C., Lei, X.L. 1999Synchronization and chaos in miniband semiconductor superlatticesPhys.Rev.B601871CrossRefGoogle Scholar
  51. 51.
    Zhang, Y., Kastrup, J., Klann, R., Ploog, K.H., Grahn, H.T. 1996Synchronization and chaos induced by resonant tunneling in GaAs/ AlAs superlatticesPhys.Rev.Lett.773001CrossRefPubMedGoogle Scholar
  52. 52.
    Luo, K.J., Grahn, H.T., Ploog, K.H., Bonilla, L.L. 1998Explosive bifurcation to chaos in weakly coupled semiconductor superlatticesPhys.Rev.Lett.811290CrossRefGoogle Scholar
  53. 53.
    Amann, A., Schlesner, J., Wacker, A., Sch öll, E. 2002Chaotic front dynamics in semiconductor superlatticesPhys.Rev.B65193313CrossRefGoogle Scholar
  54. 54.
    Or-Guil, M., Kevrekidis, I.G., Bär, M. 2000Stable bound states of pulses in an excitable mediumPhysicaD135154Google Scholar
  55. 55.
    Wolf, A., Swift, J., Swinney, H., Vastano, J. 1985Determining Lyapunov exponents from a time seriesPhysicaD16285Google Scholar
  56. 56.
    J. Schlesner and A. Amann,Superlattice bifurcation scenarios(2003), private communication.Google Scholar
  57. 57.
    Amann, A., Peters, K., Parlitz, U., Wacker, A., Sch öll, E. 2003A hybrid model for chaotic front dynamics: From semiconductors to water tanksPhys.Rev.Lett.91066601CrossRefPubMedGoogle Scholar
  58. 58.
    Alur, R., Courcoubetis, C., Halbwachs, N., Henzinger, T.A., Ho, P.-H., Nicollin, X., Olivero, A., Sifakis, J., Yovine, S. 1995The algorithmic analysis of hybrid systemsTheoretical Computer Science1383CrossRefGoogle Scholar
  59. 59.
    Katzorke, I., Pikovsky, A. 2000Chaos and complexity in a simple model of production dynamicsDiscrete Dyn. Nature Soc.5179Google Scholar
  60. 60.
    Sch ürmann, T., Hoffmann, I. 1995The entropy of strange billards inside n-simplexesJ.Phys.A285033Google Scholar
  61. 61.
    Peters, K., Parlitz, U. 2003Hybrid systems forming strange billardsInt.J.Bifur.Chaos132575CrossRefGoogle Scholar
  62. 62.
    Carretero-Gonzdflez, R., Arrowsmith, D.K., Vivaldi, F. 2000One-dimensional dynamics for traveling fronts in coupled map latticesPhys.Rev.E611329CrossRefGoogle Scholar
  63. 63.
    Torcini, A., Vulpiani, A., Rocco, A. 2002Front propagation in chaotic and noisy reaction-diffusion systems: a discrete-time map approachEur.Phys.J.B25333CrossRefGoogle Scholar
  64. 64.
    Kleinrock, L. 1975Queueing SystemsWileyNew YorkGoogle Scholar
  65. 65.
    Rudzick, O., Pikovsky, A., Scheffczyk, C., Kurths, J. 1997Dynamics of chaos-order interface in coupled map latticesPhysica D103330Google Scholar
  66. 66.
    Brucks, K.M., Misiurewicz, M., Tresser, C. 1991Monotonicity properties of the family of trapezoidal mapsCommun. Math Phys.1371Google Scholar
  67. 67.
    Glass, L., Zeng, W. 1994Bifurcations in flat-topped maps and the control of cardiac chaosInt. J. Bif. Chaos41061CrossRefGoogle Scholar
  68. 68.
    Wagner, C., Stoop, R. 2002Renormalization approach to optimal limiter control in 1-d chaotic systemsJ.Stat.Phys.10697CrossRefGoogle Scholar
  69. 69.
    P. Cvitanović, R. Artuso, R. Mainieri, G. Tanner, and G. Vattay, Chaos: Classical and Quantum (Niels Bohr Institute, Copenhagen,, 2003).Google Scholar
  70. 70.
    Li, T.-Y., Yorke, J.A. 1975Period three implies chaosAm. Math Monthly82985Google Scholar
  71. 71.
    Metropolis, N., Stein, M.L., Stein, P.R. 1973On finite limit sets for transformations of the unit intervalJ. Comb Theo.1525CrossRefGoogle Scholar
  72. 72.
    Sarkovskii, A.N. 1964Co-existence of cycles of a continuous mapping of a line onto itselfUkr. Math. Z.1661Google Scholar
  73. 73.
    Badii, R., Brun, E., Finardi, M., Flepp, L., Holzner, R., Parisi, J., Reyl, C., Simonet, J. 1994Progress in the analysis of experimental chaos through periodic orbitsRev. Mod. Phys.661389CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Institut für Theoretische PhysikTechnische Universität BerlinBerlinGermany
  2. 2.Tyndall National InstituteLee MaltingsIreland

Personalised recommendations