References
Books
Arnold, V. I., The Theory of Singularities And Its Applications . Accademia Nazionale Dei Lincei, Pisa, 1991.
Arnold, V. I., S. M. Gusein-Zade, A. N. Varchenko, Singularities of differentiable maps I : "The classification of critical points, caustics and wave fronts". Nauka, Moscow, 1982. § 21 (english translation : Birkhäuser), Boston, 1985.
Bai-lin, Hao, Elementary Symbolic Dynamics and Chaos in Dissipative Systems., 1989.
Beardon, A. F., Iteration of Rational Functions, Springer-Verlag, 1991.
Collet, P., J-P. Eckmann, Iterated Maps On The Interval As Dynamical Systems. Birkhäuser, Boston, 1980.
Gardini, L., A. Barugola, J.-C. Cathala, and C. Mira, Chaotic Dynamics. Two-dimensional Endomorphisms, World Scientific, to appear.
Guckenheimer, J. and P. H. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer, New York 1983.
Gumowski and C. Mira, Dynamique Chaotique. Transition ordre - desordre; Cepadues editeur,Toulouse, 1980.
Gumowski and C. Mira, Recurrences and Discrete Dynamic Systems, Lecture Notes in Mathematics 809, Springer-Verlag, 1980).
McCracken, J., M. Marsden, The Hopf Bifurcation,: Lecture Notes in Mathematics, Springer, 1976.
Mira, Christian, Chaotic Dynamics : from the one-dimensional endomorphism to the two-dimensional diffeomorphism, World Scientific, 1987.
Articles
Abraham, R. H., J. P. Gollub, and H. L. Swinney, "Testing nonlinear dynamics", Physica 11D, pp 252-264, 1984.
Abraham, R. H., and H. B. Stewart, "A chaotic blue sky catastrophe in forced relaxation oscillations", Physica 21D, pg 394, 1986.
D.G., Aronson, , M. A. Chory, , G. R. Hall and R. P. McGehee, "Resonance phenomena for two parameter families of maps of the plane: uniqueness and nonuniqueness of rotation numbers", in Nonlinear Dynamics and Turbulence, ed. Barenblatt, G. I., Iooss, G., and Joseph, D.D., Ch. 2 pp. 35-47, Pitman Advanced, Boston, 1983.
Aronson, D. G., M. A. Chory, G. R. Hall and R. P. McGehee, "Bifurcations from an invariant circle for two-parameter families of maps of the plane : a computer assisted study", Communs. Math. Phys, vol. 83, pp 303-354, 1982.
Barugola, A., "Determination de la frontiere d'une zone absorbante relative a une recurrence du deuxieme ordre a inverse non unique", C.R. Acad. Sc., Paris Serie B, vol. 290, p. 257-260, 1980.
Barugola, A., "Colloque sur la theorie de l'Iteration et ses Applications", C.N.R.S, p. 83, 1982.
Barugola, A., "Quelques proprietes des lignes critiques d'une recurrence du second ordre a inverse non unique. Determi nation d'une zone absorbante", R.A.I.R.O., Analyse Numerique, vol. 18, p. 137-151, 1984.
Barugola, A., "On some properties of an absorptive area and a chaotic area for a R2 endomorphism", Lecture Notes in Mathematics, p. 1163, 1985.
Barugola, A., "Sur certaines zones absorbantes et zones chao tiques d'un endomorphisme bidimensionnel", Int. J. Nonlinear Mech., vol. 21, p. 165-168, 1986.
Barugola, Alexandre, "Sur Certaines Zones Chaotique", International Journal of Non-Linear Mechanics, vol. 23, pp. 355-359, 1988.
Barugola, A. and J. C. Cathala, "Sur les zones absorbantes et les zones chaotiques d'un endomorphisme bidimensionnel", Ann. Telecommun., vol. 42, pp. 1-8, 1987.
Barugola, A. and J. C. Cathala, "Endomorphismes de dimension deux et attracteurs etranges", in Traitement Numerique des Attracteurs Etranges, ed. M. Cosnard, pp. 81-106, Editions du Centre National de la Recherche Scientifique, Paris 1987.
Barugola, A. and J. C. Cathala, "Sur les zones absorbantes et les zones chaotiques d'un endomorphisme bidimensionnel", Ann. Telecommun. vol. 42, pp. 255-262, 1987.
Barugola, A. and J. C. Cathala, "An extension of the notion of chaotic area in two-dimensional endomorphisms", ECIT (European Conference on Iteration Theory), Batschuns, Austria, 1992.
Barugola, A., J.C. Cathala, and C. Mira, "Annular chaotic areas", Nonlinear Analysis, vol. 10, p. 1223-1236, 1986.
Beddington, J. R., C. A. Free, J. H. Lawton, "Dynamic complexity in predator-prey models framed in difference equations", Nature, Lond., vol. 255, pp 58-60, 1975.
Benettin, G., L. Galgani, J-M. Strelcyn, "Kolmogorov entropy and numerical experiments", Phys. Rev. A, vol. 14, pg 2338, 1976.
Bernussou, J., Hsu Liu, and C. Mira, "Quelques exemples de solutions stochastiques bornees dans les recurrences du 2e ordre", Actes du Coloque, Transformations ponctuelles et applications, pp. 194-220, 1973.
Cathala, J.C., "Colloque sur la Theorie de l'Iteration et ses Applications", C.N.R.S, p. 91, 1982.
Cathala, J.C., "Sur la dynamique complexe et la determination d'une zone absorbante pour un systeme a donnees echantillonnees decrit par une recurrence du second ordre", R.A.I.R.O. Automatique, vol. 16, p. 175-193, 1982.
Cathala, J.C., "Absorptive area and chaotic area in two-dimensional endomorphisms", Nonlinear Analysis, vol. 7, p. 147-160, 1983.
Cathala, J.C., "On the bifurcation between a chaotic area of Tk and a chaotic area of T", Lecture Notes in Mathematics, vol. 1163, 1985.
Cathala, J.C., "Bifurcations occurring in absorptive and chaotic areas", Int. J. Systems Sci., vol. 18, pp. 339-349, 1987.
Cathala, J.C., "Multi-connected chaotic areas in second order endomorphisms", Int. J. Systems Sci., vol. 21, pp. 863-887, 1990.
Cathala, "Absortive and chaotic areas", Internal report, March, 1991.
Cathala, J. C., L. Gardini, and C. Mira, "Contact bifurcations", preprint, 1992.
Cathala, J. C., H. Kawakami, and C. Mira, "Singular points with two multipliers in the bifurcation curves of maps", Int. J. of Bifurcation and Chaos, vol. 2, no. 4, pp. 1001-1004, 1992.
Collet, P., J-P. Eckmann, H. Koch, "Period doubling bifurcations for families of maps on Rn", J. Stat. Phys., vol. 25, pg 1, 1981.
Crutchfield, J. P. and K. Kaneko, "Phenomenology of apatio-temporal chaos", in Directions in Chaos, World Scientific, Singapore 1987.
Farmer, D., "Order within chaos", Thesis, University of California, Santa Cruz, 1981.
Fatou, M. P., "Sur les equations fonctionelles", Bulletin de la Societe Mathematiques de France, vol. 47 pp 161-271, 1919, vol. 48, pp 33-94,208-314, 1920.
Feigenbaum, M. J., "Quantitative universality for a class of nonlinear transformations", J. Stat. Phys., vol. 19, 1978.
Feigenbaum, M. J., "Universal behaviour in nonlinear systems", in Nonlinear Dynamics and Turbulence, ed. Barenblatt, G. I., Iooss, G., and Joseph, D.D., Ch. 7 pp. 101-138, Pitman Advanced, Boston, 1983.
Feudel, U. and W. Jansen, "CANDSYS/QA - A Software System For Qualitative Analysis Of Nonlinear Dynamical Systems", Int. J. Bifurcation and Chaos, vol. 2 no. 4, pp. 773-794, World Scientific, 1992.
Gallegati, M., and L. Gardini, "A nonlinear model of the business cycle with money and finance", Metroeconomica vol. 42, pp. 1-32, 1991.
Gardini, Laura, R. Lupini, C. Mammana, M. G. Massia, "Bifurcations and transitions to chaos in the three-dimensional Lotka-Volterra map", SIAM J. Appl. Math., vol. 47, pg 455, 1987.
Gardini, Laura, "Linee critiche in una mappa bidimensionale di interesse economico", Quaderno n. 8904, Gruppo Nationale M.U.R.S.T., Modelli Nonlineari in Economia e Dinamiche Complesse , Urbino 1990.
Gardini, Laura and C. Mira, "A procedure to determine chaotic and absorbing areas in two-dimensional endomorphisms", Quaderno n. 9002, Gruppo Nazionale M.U.R.S.T., Modelli Nonlineari in Economia e Dinamiche Complesse, Urbino 1990.
Gardini, Laura, "On a model of financial crisis. Critical lines as new tools of global analysis", Nonlinear Dynamics in Economics and Social Sciences, Gori & Geronazzo eds., Springer-Verlag, NY, 1993.
Gardini, Laura, "Global Analysis and Bifurcations in Two-Dimensional Endomorphisms by Use of Critical Curves" in European Conference on Iteration Theory, eds. J. P. Lampreia, J. Llibre, C. Mira, J. Sousa Ramos, Gy. Targonski, World Scientific, Singapore, 1991.
Gardini, Laura, "Bifurcations of invariant sets", 1991.
Gardini, Laura, "Insieme invarianti", 1991.
Gardini, Laura, "On the global bifurcation of two-dimensional endomorphisms by use of critical lines", Nonlinear Analysis, vol. 18, no. 4, pp. 361-399, 1991.
Gardini, Laura, "Homoclinic orbits of saddles", 1992.
Gardini, Laura, "Homoclinic bifurcations", 1992.
Gardini, Laura, "Absorbing areas and their bifurcations", 1992.
Gardini, Laura, "Some Global Bifurcations of Two-Dimensional Endomorphisms by Use of Critical Lines", Nonlinear Analysis, Theory, Methods & Applications, Vol. 18, No. 4, pp. 361-399, 1992.
Gardini, Laura and Renzo Lupini, "One-dimensional chaos", 1992.
Gardini, Laura, Cristian Mira, and Daniele Fournier-Prunaret, "Properties of invariant areas", 1992.
Gardini, Laura, "Definitions", 1992, preprint, Urbino.
Gardini, Laura, "Absorbing areas", Quaderno 16, Urbino.
Gardini, Laura, Daniele Fournier-Prunaret, Ralph H. Abraham, and Ronald J. Record, "A Double Logistic Map", Intern. J. Bifurcation and Chaos, vol. 4, no. 1, pp. 145-176, 1994.
Giraud, A., "Application des recurrences a l'etude de certains systemes de commande", These d' Ingenieur-Docteur, Faculte des Sciences de l'Universite de Toulouse, 21 April 1969.
Grevy, 1892, 1894, 1897
Guckenheimer, J, "Endomorphisms of the Riemann sphere", Proc. Sympos. Pure Math., S.S. Chern and S. Smale, eds, vol. 14, pp. 95-123, American Mathematical Society, Providence, 1970.
Guckenheimer, J. and G. Oster, "Bifurcation Behavior of Population Models, in The Hopf Bifurcation" (Marsden & McCracken, eds.), Springer, 1976.
Guckenheimer, J, "Bifurcations of maps of the interval", Inventiones Math., vol. 39, pp. 165-178, 1977.
Guckenheimer, J., G. Oster, and A. Ipaktchi, "The dynamics of density dependent population models", Journal of Mathematical Biology, vol. 4, pp. 101-147, 1977.
Guckenheimer, J, "Sensitive dependence on initial conditions for one-dimensional maps", comm. Math. Phys., vol. 70, pp. 133-160, 1979.
Guckenheimer, J., "Computational environments for exploring dynamical systems", Int. J. Bifurcation and Chaos, vol. 1, pp. 269-276, World Scientific, 1991.
Gumowski, I. and C. Mira, "Sur un algorithme de determination du domaine de stabilite d'un point double d'une recurrenct non lineaire du deuxieme ordre a varialbles reeles", C.R. Acad. Sc., groupe 2, vol. 260, pp. 6524-6527, 1965.
Gumowski, I. and C. Mira, "Sensitive problems related to certain bifurcations in nonlinear recurrence relation", Automatica, vol. 5, pp. 303-317, 1969.
Gumowski, I. and C. Mira, "Solutions chaotiques bornees d'une recurrence ou tranformation ponctuelle du 2eme ordre a inverse non unique", C.R. Acad. Sc., Paris, Serie A , vol. 285, p. 477-480, 1977.
Gumowski, I. and C. Mira, "Bifurcation destabilisant une solution chaotique d'un endomorphisme du deuxieme ordre", C.R Acad. Sc., Paris, Serie A , vol. 286, p. 427, 1978.
Gumowski, Igor and Christian Mira, "Bifurcation destabilisantune solution chaotique d'un endomorphisme du 2nd ordre", Comptes Rendus Acad. Sc., Paris, Serie A, vol. 286, pp. 427-431, 1978.
Henon, M., "A Two Dimensional mapping with a strange attractor", Communications in Mathematical Physics, vol. 50, pg 69, 1976.
Julia, G., "Sur l'iteration des fonction rationnelles", Journal de Mathematiques Pures et Appliques, pp. Series 8, volume 1, pp 47-245, 1918.
Julia, G., "Rational substitutions", C. R. Acad. Sc. Paris, vol. 164, pp. 1098-1100, 1917, vol. 166, pp. 61-64,153-156,599-601, 1918, vol. 168, pp. 147-149, 1919, vol. 173, pp. 690-693, 1921.
Kawakami, H. and K. Kobayashi, "Computer experiments on chaotic solutions of ", Bull. Fac. of Engin., Tokushima University, vol. 16, pp. 29-46, 1979.
Lauwerier, H. A., "Part II - Iterations, Ch. 3, 4 One and Two dimensional iterative maps" in Chaos, ed. Arun V. Holden, Princeton University Press, Princeton, 1986.
Li, T., and J. A. Yorke, "Period three implies chaos", Amer. Math. Monthly, vol. 82, pp. 985-992, 1975.
Li, T., and J. A. Yorke, "Ergodic transformations from an interval into itself", Trans. Am. Math. Soc., vol. 235, pp. 183-192, 1978.
Lopez-Ruiz, R. and C. Perez-Garcia, "Dynamics of Two Logistic Maps With a Multiplicative Coupling", Intern. J. Bifurcation and Chaos, vol. 2 no. 2, pp. 421-425, World Scientific, 1992.
Lorenz, E., "Computational Chaos - A prelude to computational Instability", Physica D, pp. volume 35, pp 299-317, 1989.
Lorenz, E., "Deterministic non-periodic flows", J. Atmos. Sci., vol. 20, pp. 130-141, 1963.
Lyapunov, A. M.., Collected Papers, U.S.S.R. Acad. of Science,
Moscow 1948.
MacDonald, R. R., N. Whitehead, "A chaotic mapping that displays its own homoclinic structure", Phyica D, vol. 13, pg 401, 1984.
Marotto, F. R., "Snap-back repellors imply chaos in Rn ", J. Math. Anal. Appl., vol. 63, pg. 199, 1978.
May, R. M., "Simple mathematical models with very complicated dynamics", Nature 261, pp. 645-647, 1974.
May, R. M., "Biological populations with nonoverlapping generations: stable points, stable cycles, and chaos", Science, vol. 186, pp. 459-467, 1976.
May, R. M., "Bifurcations and dynamic complexity in ecological systems", Ann. N. Y. Acad. Sci., vol. 316, pp. 517-529, 1979.
Metropolis, N., M. L. Stein, P. R. Stein, "On finite limit sets for transformations on the unit interval", J. Combinatorical Theory, vol. 15, 1973.
Mira, C., "Determination pratiques du domain de stabilite d'un point d'equilibre d'une recurrence non lineaire du deuxieme order a variable reelles", Comptes Rendus Hebdomadaires des Seances de l'Academie des Sciences, Paris, pp. Serie A, groupe 2, volume 261, pp 5314-5317, 1965.
Mira, C., "Determination pratique du domaine de stabilite d'un point d'equilibre d'une recurrence non lineaire du deuxieme ordre a variables reeles", International Pulse Symposium, vol. D43-7/11, p. 1, 1968.
Mira, C., "Sur les cas d'exception pour une recurrence, ou transformation ponctuelle, du deuxieme ordre", C. R. Acad. Sc. Paris, t. 270, pp. 466-469, Serie A, 1970.
Mira, C., "Dynamique complexe dans les endomorphismes a deux dimensions", Proc. of the VIII International Conference on Non-linear Oscillations, 1978.
Mira, C., "Complex dynamics in two-dimensional endomorphisms", Nonlinear Analysis, vol. 4, p. 1167-1187, 1980.
Mira, C. and I. Gumowski, "Sur un algorithme de determination du domain de stabilite d'un point double d'une recurrence nonlineaire du deuxieme order a variables reelles".a, Comptes Rendus Hebdomadaires des Seances de l'Academie des Sciences, Paris, pp. Serie A, groupe 2, volume 260, pp 6524-6527, 1965.
Mira,C., and I. Gumoski, "Sensitivity Problems related to certain bifurcations in nonlinear recurrence relation", Automatica, volume 5 pp 303-317, 1969.
Mira, C. and F. Roubellat, "Cas ou le domaine de stabilite d'un ensemble limite attractif d'une recurrence du deuxieme ordre n'est pas simplement connexe", C.R. Acad. Sc., Serie A., vol. 268, pp. 1657-1660, 1969.
Mira, C., Daniele Fournier-Prunaret, Laura Gardini, Hiroshi Kawakami, and J-C. Cathala, "Basin bifurcation of two-dimensional noninvertible maps. Fractalization of basins", Int. J. of Bifurcation and Chaos, vol. 4, no. 2, 1994.
Mira, C., "Some Results On Basins Generated By Two-Dimensional Noninvertible Maps", Conference on Differential Equations: Bifurcation and Chaos, Katsively, Crimea, Ukraine, 3-14 May 1994.
Mira, C., and C. Rauzy, "Fractal Aggregation of Basin Islands in Two-Dimensional Quadratic Noninvertible Maps", preprint, 1994.
Myberg, P. J., "Iteration der realen Polynome zweiten Grades III", Ann. Acad. scientiarium Fennicae, vol. Ser. A 336, pp. 1-18, 1963.
Narayaninsamy, T., "On the Fractional Iteration of Endomorphisms" in European Conference on Iteration Theory, eds. J. P. Lampreia, J. Llibre, C. Mira, J. Sousa Ramos, Gy. Targonski, World Scientific, Singapore, 1991.
Neimark, Y., Doklady Akademi Nauk SSSR, pp. Volume 129, pp 736-739, 1959.
Oster, R. and G. May, "Bifurcations and Dynamic Complexity in Simple Ecological Models", Amer. Natur, 1975.
Picard, E., "Lecons sur quelques problemes aux limites de la theorie des equations differentielles", Paris, 1930.
Poincaré's Thèse (1879):
Poincaré, H., "Sur les propriétés des fonctions définies par les équations aux différences partielles" in Oeuvres de Henri Poincaré, Tome I. Gauthier-Villars, paris, 1951, XLIX-CXXIX..
Roubellat, F., "Contributions a l'etude des solutions des recurrences non lineaires et applications aux systemes a donnees echantillonnees", These de Doctorat es Sciences, 1969.
Sacker, Communications in Pure and Applied Mathematics, Volume 18, pp 717-732, 1965.
Schröder, E., "Uber iterierte Funktionen", Math. Ann., vol. 2, pp. 296-322, 1871.
Smale, S., "Diffeomorphisms with Many Periodic Points", Differential and Combinatorial Topology (S. Cairns, ed.), p. 6380, 1965.
Stein, N. , M. Metropolis, and P. Stein,, "Stable states of a nonlinear transformation", Numerische Mathematik, vol. 10, pp. 1-9, 1967.
Takahashi, Y., and G. Oster, "Models for age specific interactions in a periodic environment", Ecological Monographs, vol. 44, pp. 483-501, 1974.
Takens, F. and D. Ruelle, "On the nature of turbulence", Comm. Math. Phys., vol. 20, pp. 167-192, 1971.
Ulam, S. M. and J. v. Neumann, "On combination of stochastic and deterministic processes", Bull. Amer. Math. Soc., vol. 53, pg. 1120, 1947.
Ulam, S. M. and P. R. Stein, "Non-linear transformation studies on electronic computers", Rozprawy Metamatyczne, vol. 39, pp. 401-484, 1964.
Volterra, V., "Lecons sur les fonctions de lignes", Gauthier-Villars, Paris, 1913.
Whitney, H., On Singularities of Mappings of Euclidean Spaces I: "Mappings of the Plane Into the Plane", Ann. Math. 62 (1955), 374-410.
Wolf, A., Part V - Measuring Chaos, Ch. 13 "Quantifying chaos with Lyapunov exponents" in Chaos, ed. Arun V. Holden, Princeton University Press, Princeton, 1986.
Yorke, J. A., "DYNAMICS an Interactive Program for PC's and Unix Computers", Institute for Physical Science and Technology, University of Maryland, Preprint 1991.
Back to Dr. Record's Resume or proceed to
Dr. Record's Home Page