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Dissipative structures and oscillations in reaction-diffusion models with or without time-delay

  • Manuel G. Velarde
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 164)

Keywords

Hopf Bifurcation Dissipative Structure Stable Limit Cycle Stable Steady State Limit Cycle Oscillation 
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References

  1. 1.
    Andronov,A.A.,E.A. Leontovich, I.I. Gordon and A.G. Maier,1973(English transl.)THEORY OF BIFURCATIONS OF DYNAMIC SYSTEMS ON A PLANE,Wiley, New York.Google Scholar
  2. 2.
    Aris,R.,1976,THE MATHEMATICAL THEORY OF DIFFUSION AND REACTION IN PERMEABLE CATALYSTS( 2 vols.),Clarendon Press,Oxford.Google Scholar
  3. 3.
    Arnold,L.,1974,STOCHASTIC DIFFERENTIAL EQUATIONS,Wiley,New york.Google Scholar
  4. 4.
    Arnold,L.,W. Horsthemke and R.Lefever,1978,Z.Phys. B29,367.Google Scholar
  5. 5.
    Arnold, L. and R.Lefever,1981(editors),STOCHASTIC NONLINEAR SYSTEMS,Springer-Verlag New York.Google Scholar
  6. 6.
    Arnold,V.,1978(English transl.) ORDINARY DIFFERENTIAL EQUATIONS,M.I.T.Press, Cambridge,Mass.Google Scholar
  7. 7.
    Arnold,V.,1980(French transl.) CHAPITRES SUPPLEMENTAIRES DE LA THEORIE DES EQUATIONS DIFFÉRENTIELLES ORDINAIRES,Mir, Moscow.Google Scholar
  8. 8.
    Balslev,I. and H. Degn,1975,J.Theor.Biol.49,173.Google Scholar
  9. 9.
    Bonilla,L.L. and M.G.Velarde,1979,J.Math.Phys.,20,2692.Google Scholar
  10. 10.
    Bonilla,L.L. and M.G.Velarde,1980,J.Math.Phys.,21,2586.Google Scholar
  11. 11.
    Bonilla,L.L. and M.G.Velarde,198l,J.interdisciplin.Cycle Res.,to appear.Google Scholar
  12. 12.
    Bonilla,L.L.,A.Fdez.Cancio and M.G.Velarde, 1981, J.interdiciplin. Cycle Resin press.Google Scholar
  13. 13.
    Bonilla,L.L.;and M.G.Velarde,1981, J.Math.Phys. submitted por publication.Google Scholar
  14. 14.
    Bonilla,L.L.,M.G.Velarde and E.Parisi,1981(report in preparation.).Google Scholar
  15. 15.
    Bonilla,L.L.,M.G.Velarde,A.Pimpale and P.Landsberg,1981(report in preparation).Google Scholar
  16. 16.
    Boyce,W.E. and R.C.DiPrima,1977,ELEMENTARY DIFFERENTIAL EQUATIONS AND BOUNDARY VALUE PROBLEMS,Wiley,New York.Google Scholar
  17. 17.
    C1ark,W.C.,D.D.Jones and C.S.Holling,in SPATIAL PATTERNS IN PLANKTON COMMUNITIES (J.H.Steele, editor) Plenum Press, New York.Google Scholar
  18. 18.
    Coddington,E.A. and N.Levinson,1955, ORDINARY DIFFERENTIAL EQUATIONS,McGraw-Hill, New York.Google Scholar
  19. 19.
    Cashing,J.M.,1977,INTEGRO-DIFFERENTIAL EQUATIONS AND DELAY MODELS IN POPULATION DYNAMICS, Springer-Verlag,New York.Google Scholar
  20. 20.
    Degn,H. and D.E.F. Harrison,1969,J.Theor.Bio1.22,238.Google Scholar
  21. 21.
    de la Rubia,J.F. and M.G.Velarde,1978,Phys.Lett.A69,304.Google Scholar
  22. 22.
    Degtyarenko,N.N.,V.F.Elesin and V.A.Furmenov,1974,Sov.Phys.Semicond. 7,1147.Google Scholar
  23. 23.
    Ebeling,W.,1976,STRUKTURBILDING BEI IRREVERSIBLEN PROZESSEN,Teubner Verlag,Leipzig.Google Scholar
  24. 24.
    Eigen,M.,1971,Die Naturwiss.,58,465.Google Scholar
  25. 25.
    Eigen,M. and P.Chuster,1979, THE HYPERCYCLE, Springer-Verlag,New York.Google Scholar
  26. 26.
    Fairén,V. and M.G.Velarde,1979,J.Math.Bio1.,8,47.Google Scholar
  27. 27.
    Fairén,V. and M.G.Velarde,1979,Prog.Theor.Phys.61,801.Google Scholar
  28. 28.
    Fairén,V. and M.G.Velarde,1979,Rep. Math. Phys.16,421.Google Scholar
  29. 29.
    Faraday Symposia (Chemical Society),1974,n29,PHYSICAL CHEMISTRY OF OSCILLATORY PHENOMENA,London.Google Scholar
  30. 30.
    Fife,P.C.,1979,MATHEMATICAL ASPECTS OF REACTING AND DIFFUSING SYSTEMS, Springer-Verlag,New York.Google Scholar
  31. 31.
    Frank,G.M. et al.,1967 (editors),OSCILLATORY PROCESSES IN BIOLOGICAL AND CHEMICAL SYSTEMS,Nauka,Moscow(in Russian).Google Scholar
  32. 32.
    Fujisaka,H. and T.Yamada,1977,Prog.Theor.Phys.55,734.Google Scholar
  33. 33.
    Gihman,I.I. and A.V.Skorohod,1972 (English transl.) STOCHASTIC DIFFERENTIAL EQUATIONS,Springer-Verlag,New York.Google Scholar
  34. 34.
    Gross,E.F.,D.A.Permogorov,V.V.Travnikov and A.V.Silin,1972,Sov.Phys.Solid-St. 14,1193.Google Scholar
  35. 35.
    Haken,H.,1977,SYNERGETICS (2nd. edition),Springer-Verlag, New York.Google Scholar
  36. 36.
    Hale,J.,1977,THEORY OF FUNCTIONAL DIFFERENTIAL EQUATIONS,Springer-Verlag,New York.Google Scholar
  37. 37.
    Hale,J.,1978, in NONLINEAR OSICLLATIONS IN BIOLOGY,Lect.Appl.Math.,17,157.Google Scholar
  38. 38.
    Hanusse,P.,1972,Comptes Rendus Acad.Sci.Paris,C274,1247.Google Scholar
  39. 39.
    Harrison,D.E.F. and S.J.Pirt,1967,J.Gen.Microbiol.46,193.Google Scholar
  40. 40.
    Harrison,D.E.F. and H.H.Topiwala,1974,Adv.Biochem.Eng.,vol.3,Springer-Verlag,Berlin.Google Scholar
  41. 41.
    Hemmer,P.C. and M.G.Velarde,1977,Z.Phys.B 31,111.Google Scholar
  42. 42.
    Higgins,J.,1967,Ind.Eng.Chem.59,19.Google Scholar
  43. 43.
    Hirsch,M.W. and S.Smale,1974,DIFFERENTIAL EQUATIONS,DYNAMICAL SYSTEMS AND LINEAR ALGEBRA, Academic Press,New York.Google Scholar
  44. 44.
    Horsthemke,W. and M. Malek-Mansour,1976,Z.Phys.B24,307.Google Scholar
  45. 45.
    Horsthemke,W.and L.Brenig,1977,Z.Phys.B27,341.Google Scholar
  46. 46.
    Horsthemke,W. and R.Lefever,1977,Phys.Lett.A64,19.Google Scholar
  47. 47.
    Hulin,D.,A.Mysyrowicz and C.Benoit,1980,Phys.Rev.Lett.45,1980.Google Scholar
  48. 48.
    Hutchinson,G.E.,1948,Ann.N.Y.Acad.Sci.,50,221.Google Scholar
  49. 49.
    Ibañez,J.L.,V.Fairén and M.G.Velarde,1976,Phys.Lett.A58,364.Google Scholar
  50. 50.
    Ibañez,J.L.and M.G.Velarde,1977,J.Physique Lett.38,L465.Google Scholar
  51. 51.
    Ibañez,J.L. and M.G.Velarde,1978,Phys.Rev.A18,750.Google Scholar
  52. 52.
    Ibañez,J.L. and M.G.Velarde,1978,J.Non-equilib-.Therm.,3,63.Google Scholar
  53. 53.
    Iooss G. and D.D. Joseph,1980,ELEMENTARY BIFURCATION THEORY,Springer-Verlag,New York.Google Scholar
  54. 54.
    Jetschke,G.,1979,Phys.Lett.A72,265.Google Scholar
  55. 55.
    Jetschke,G.,1980,Report N.80.17 F.Schiller Univ.Jena (in German).Google Scholar
  56. 56.
    Klingenstein,W. and W.Schmid,1979,Phys.Rev.B20,3285.Google Scholar
  57. 57.
    Kuramoto,Y. and T.Tsuzuki,1976,Prog.Theor.Phys.55,356.Google Scholar
  58. 58.
    Kuramoto,Y and T.Yamada,1976,Prog.Theor.Phys.56,679.Google Scholar
  59. 59.
    Kuramoto,Y. 1977,: in SYNERGETICS (H.Haken,editor)Springer-Verlag,New York.Google Scholar
  60. 60.
    Kurtz,T.G.,1971,J.Appl.Prob.8,344.Google Scholar
  61. 61.
    Landauer,R.,1962,J.Appl.Phys.33,2209.Google Scholar
  62. 62.
    Landsberg.,P.T. and A.Pimpale,1976,J.Phys.C9,148.Google Scholar
  63. 63.
    Landsberg,P.T.,1978,THERMODYNAMICS AND STATISTICAL MECHANICS, Clarendon Press, Oxford.Google Scholar
  64. 64.
    Landsberg,P.V.,1980,European J.Phys.1,31.Google Scholar
  65. 65.
    La Salle,J. and S.Lefchetz,1961,STABILITY BY LYAPUNOV'S DIRECT METHOD WITH APPLICATIONS,Academic Press,New York.Google Scholar
  66. 66.
    MacDonald,N.,1978,TIME LAGS IN BIOLOGICAL MODELS,Springer-Verlag,New York.Google Scholar
  67. 67.
    Mahr,H. and C.L.Táng,1972,J.Appl.Phys.43,1818.Google Scholar
  68. 68.
    Margalef,R.,1980,LA BIOSFERA,Omega,Barcelona.(in Spanish).Google Scholar
  69. 69.
    May,R.M.,1976,Nature(London)459.Google Scholar
  70. 70.
    Maynard-Smith,J.,1974,MODELS IN ECOLOGY,University Press, Cambridge.Google Scholar
  71. 71.
    Minorsky,N.,1962,NON-LINEAR OSCILLATIONS,Van Nostrand,New York.Google Scholar
  72. 72.
    Murray,J.D.,1977,LECTURES ON NONLINEAR-DIFFERENTIAL-EQUATION MODELS IN BIOLOGY, Clarendon Press,Oxford.Google Scholar
  73. 73.
    Nicolis,G. and I.Prigogine,1977,SELF-ORGANIZATION IN NONEQUILIBRIUM SYSTEMS, Wiley,New York.Google Scholar
  74. 74.
    Normand,Ch.,Y.Pomeau and M.G.Velarde,1977,Rev.Mod.Phys.,49,581.Google Scholar
  75. 75.
    Okubo,A.,1980,DIFFUSION AND ECOLOGICAL PROBLEMS,Springer-Verlag,New York.Google Scholar
  76. 76.
    Oppenheim,I.,K.E.Shuler and G.H.Weiss,1977(editors) STOCHASTIC PROCESSES IN CHEMICAL PHYSICS,M.I.T.Press,Cambridge,Mass.Google Scholar
  77. 77.
    Pikovsky,A.S.,1981,Phys.Lett.,A85,13.Google Scholar
  78. 78.
    Pimpale,A. and P.Landsberg,1977,J.Phys.C10,1447.Google Scholar
  79. 79.
    Pimpale,A.,P.Landsberg,L.L.Bonilla and M.G.Velarde,1981,J.Phys.Chem.Solids,42,873.Google Scholar
  80. 80.
    Pomeau,Y.J.C. Roux,A.Rossi,S.Bachelart and C.Vidal,1981,J.Phys.(Paris)Lett.42,L271.Google Scholar
  81. 81.
    Prigogine,I.,1967,(published in 1969), in FROM THEORETICAL PHYSICS TO BIOLOGY(1st. Int. Symposium;M.Marois,editor)North Holland,Amsterdam.Google Scholar
  82. 82.
    Prigogine,I.,1974,(Spanish transl.) INTRODUCCION A LA TERMODINAMICA DE LOS PROCESOS IRREVERSIBLES, Seleciones Cientificas,Madrid.Google Scholar
  83. 83.
    Prigogine,I. and I.Stengers,1979,LA NOUVELLE ALLIANCE,Gallimard,Paris.(in French).Google Scholar
  84. 84.
    Rabinovich,M.I.,1978,Sov.Phys.Usp. 21,443.Google Scholar
  85. 85.
    Rayleigh,Lord, 1921,Proc.Roy.Soc. A99,372.Google Scholar
  86. 86.
    Sel'kov,E.E.,1968,European J.Biochem.,4,79.Google Scholar
  87. 87.
    Turner,J.S.,J.C.Roux,W.D.McCornick and H.L.Swinney,1981,Phys.Lett.A85,9.Google Scholar
  88. 88.
    Tyson,J.J.,1973,THE BELOUSOV-ZHABOTINSKII REACTION,Springer-Verlag,New York.Google Scholar
  89. 89.
    Van Kampen,N.G.,1976,Adv.Chem.Phys.,34,245.Google Scholar
  90. 90.
    Vasil'ev,V.A.,Yu.M.Romanovsky and V.G.Yakhno,1979,Sov.Phys.Usp.22,615.Google Scholar
  91. 91.
    Velarde,M.G.,1981,in NONLINEAR PHENOMENA AT PHASE TRANSITIONS AND INSTABILITIES (T.Riste,editor),Plenum Press,New York.Google Scholar
  92. 92.
    Waltman,P.,1974,DETERMINISTIC THRESHOLD MODELS IN THE THEORY OF EPIDEMICS, Springer-Verlag,New York.Google Scholar
  93. 93.
    Winfree,A.T.,1980,THE GEOMETRY OF BIOLOGICAL TIME,Springer-Verlag,New York.Google Scholar
  94. 94.
    Yamada,T. and Y.Kuramoto,1976,Prog.Theor.Phys.,56,681.Google Scholar
  95. 95.
    Zhabotinskii,A.M.,1964,Doklady Akad.Nauk USSR,157,392.Google Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • Manuel G. Velarde
    • 1
  1. 1.UNED-Física Fundamental Apdo. CorreosMadridSpain

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