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References
C. Anderson and C. Greengard,SIAM J. Numer. Anal.
J. T. Beale and A. Majda,Math. Comp. 39:1–27 (1982).
J. T. Beale and A. Majda,Math. Comp. 39:29–52 (1982).
J. T. Beale and A. Majda,J. Comput. Phys. 58:188–208 (1985).
J. T. Beale and A. Majda,Contemp. Math. 28:221–229 (1984).
J. T. Beale,Math. Comp., to appear.
A. Chorin,SIAM J. Sci. Stat. Comput. 1:1–21 (1980).
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G. H. Cottet,Méthodes Particulaires pour l'Équation d'Euler dans le Plan, thèse de 3e cycle, Université P. et M. Curie, Paris, 1982.
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References
M. Benedicks and L. Carleson,On Iterations of 1−ax 2 on (−1,1), preprint, Institute Mittag Leffler (1983).
P. Collet,Ergodic Properties of Some Unimodal Mappings of the Interval, preprint, Institute Mittag Leffler (1984).
P. Collet and J. P. Eckmann,Iterated Maps of an Interval as Dynamical Systems (Birkhaüser, Basel, Boston, 1980).
P. Collet and J. P. Eckmann, in G. Velo and A. Wightman, eds.,Regular and Chaotic Motions in Dynamic Systems (Plenum Press, New York, 1985).
P. Collet and J. P. Eckmann,Commun. Math. Phys. 76:211–254 (1980).
P. Collet and J. P. Eckmann,Ergod. Theory Dyn. Syst. 3:13–46 (1983).
J. Guckenheimer and P. Holmes,Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer Verlag, Berlin/Heidelberg/New York, 1983).
M. Jakobson,Commun. Math. Phys. 81:39–88 (1981).
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A. Katok and Y. Kifer, to appear.
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F. Ledrappier,Publ. Sci. IHES. 53:163–188 (1981).
J. Milnor,On the Concept of Attractor, preprint, Institute for Advanced Studies, Princeton, New Jersey (1984).
M. Misiurewicz,Publ. Sci. IHES 53:17–52 (1981).
T. Nowicki,Symmetric S-Unimodal Mappings and Positive Lyapunov Exponents, preprint, Warsaw University (1985).
C. Preston,Iterates of Maps on an Interval, Lecture Notes in Mathematics 999 (Springer Verlag, Berlin/Heidelberg/New York, 1983).
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Bibliography
L. Tartar, in R. J. Knops, ed.,Research Notes in Mathematics, Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol.4 (Pitman Press, 1979). This paper provides an introduction to the Young measure and compensated compactness. It includes a proof of convergence of the viscosity method for a scalar conservation law using the aforementioned tools.
L. Tartar, in J. M. Ball, ed.,Systems of Nonlinear Partial Differential Equations: NATO ASI Series (Reidel, New York, 1983). This paper provides an introduction to the compensated compactness theory together with applications to convergence of the viscosity method for nondegenerated hyperbolic systems of conservation laws. Additional background information and applications to numerical methods can be found in the following paper.
R. J. DiPerna,Arch. Rat. Mech. Anal. 82:27–70 (1983). For results dealing with large-data existence and convergence of the viscosity for the isentropic equations of gas dynamics in one space dimension, we refer the reader to the following article.
R. J. DiPerna,Comm. Math. Phys. 91:1–30 (1983). For a general introduction to the weak topology in the setting of elasticity and the calculus of variations we refer the reader to the following two papers.
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Review Articles or Books
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P. Betgé, Y. Pomeau, and C. Vidal,L'Ordre dans le Chaos (Hermann, Paris, 1984).
N. B. Abraham, J. P. Gollub, and H. Swinney,Physica D 11:252 (1984).
Routes Through Biperiodism—Experiments
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J. P. Gollub and S. V. Benson,J. Fluid Mech. 100:449 (1980).
M. Dubois and P. Bergé,J. Phys. 42:167 (1981).
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A. P. Fein, M. S. Heutmaker, and J. P. Gollub,Phys. Script. T 9:79 (1985).
P. Bergé, M. Dubois,J. Phys. Lett., Mai, 1985.
Experimental Attractors
F. C. Moon and P. J. Holmes,J. Sound Vibrat. 65:275 (1979).
F. C. Moon,J. Appl. Mech. 47:638 (1980).
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J. C. Roux and H. L. Swinney,Topology of Chaos in a Chemical Reaction in “Non-Linear Phenomena in Chemical Dynamics” (C. Vidal and A. Pacault, eds.), (Springer-Verlag, Berlin, 1981), p. 38.
M. Dubois, P. Berge, and V. Croquette,J. Phys. Lett. 43:295 (1982).
V. Croquette,62:62 (1982).
P. Bergé,Phys. Script. T 1:71 (1982).
M. Dubois,Experimental Aspects of the Transition to Turbulence in Rayleigh-Bénard Convention, in “Lecture Notes in Physics”164:117 (1982).
M. Sano and Y. Sawada, Chaos and Statistical Mechanics, in Proceedings of the Kyoto Summer Institute (Y. Kuramoto, ed.) (Springer, Berlin, 1983).
Characterization of Chaos
P. Grassberger and I. Procaccia,Phys. Rev. Lett. 150:346 (1983).
P. Grassberger and I. Procaccia,Physica D 9:189 (1983).
P. Grassberger and I. Procaccia,Phys. Rev. A 28:2591 (1983).
J. D. Farmer, E. Ott, and J. Yorke,Physica D 7:153 (1983).
Y. Termonia,Phys. Rev. A 29:1612 (1984).
A. Wolf, J. Swift, H. Swinney, and J. Vastano,Physica D (1984).
G. Paladin and A. Vulpiani,Lett. Nuovo Cimento 41:82 (1984).
Dimension Measurements from Experimental Data
B. Malraison, P. Atten, P. Bergé, and M. Dubois,J. Phys. Lett. 44:897 (1983).
J. Guckenheimer and G. Buzyna,Phys. Rev. Lett. 51:1438 (1983).
A. Brandstäter, J. Swift, H. Swinney, A. Wolf, D. Farmer, E. Jen, and J. P. Crutchfield,Phys. Rev. Lett. 51:1442 (1983).
P. Atten, J. P. Caputo, B. Malraison, and Y, Gagne,J. Mécanique.
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References
P. Collet, J.-P. Eckmann, and O. E. Lanford III,Comm. Math. Phys. 76:211–254 (1980).
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G. Roepstorff, private communication.
J. Ecalle, Publications Mathématiques d'Orsay No. 67-7409, Théorie des Invariants Holomorphes.
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References
V. Franceschini and C. Tebaldi,J. Stat. Phys. 21:707–726 (1979).
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Books
P. Collet and J.-P. Eckmann,Iterated Maps on the Interval as Dynamical Systems (Birkhäuser, Basel, 1980).
P. Cvitanovic,Universality on Chaos (Adam Hilger Ltd., 1984).
J. Guckeneheimer and P. Holmes,Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer-Verlag, Berlin, 1983).
G. Iooss and D. D. Joseph,Elementary Stability and Bifurcation Theory (Springer-Verlag, Berlin, 1980).
C. Sparrow,The Lorenz Equations: Bifurcations, Chaos and Strange Attractors (Springer-Verlag, Berlin, 1982).
References
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References Books, Review Papers, and Crucial Research Papers
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Annotated bibliography A. Principal References
R. Thomas,Kinetic Logic, Lecture Notes in Biomathematics29 (1979). There are several papers by Thomas in this volume on kinetic logic and its application, and two papers by Van Ham. These can be regarded as the immediate progenitors of the study of Boolean delay equations.
D. Dee and M. Ghil, “Boolean Difference Equations I: Formulation and Dynamic Behavior,”SIAM J. Appl. Math. 43:1019 (1983). This paper initiates the study of BDEs qua dynamical systems and studies a system with solutions of increasing complexity.
M. Ghil and A. Mullhaupt, “Boolean Delay Equations II: Periodic and Aperiodic Solutions,”J. Stat. Phys. 41 (to appear). This surveys what is known about BDEs. It contains several classifications of BDEs, and estimates for period length, complexity, effect of resonance, etc. as well as a characterization of structural stability. This paper introduces the concept of asymptotic simplification and gives an algebraic method for studying this phenomenon.
A. Mullhaupt,Boolean Delay Equations: A Class of Semi-Discrete Dynamical Systems, Ph.D. dissertation, New York University (1984). This contains some results in common with Ghil and Mullhaupt,(3) with detailed proofs. In addition, two appendices are given, one explaining various numerical algorithms for computing BDEs and their relative merits, the other a survey of diophantine approximation, which considers the question of resonances for BDEs.
S. Golomb,Shift-Register Sequences (Holden-Day, San Francisco, 1967). This volume studies linear scalar shift register sequences algebraically and statistically. Some numerical work on cycle length of nonlinear shift register sequences is given.
M. Ghil, A. Mullhaupt, and P. Pestiaux, “Deep Water Formation and Quaternary Glaciations,” unpublished preprint. This paper contains two models for the ice ages using BDEs.
B. Related Material
S. Wolfram,Rev. Mod. Phys. 55 (1983). This is a survey of cellular automata, a field closely related to BDEs, in which all delays are equal to each other.
V. I. Arnold,Geometric Methods in the Theory of Ordinary Differential Equations (Springer-Verlag, New York, 1983).
J. Guckenheimer and P. Holmes,Nonlinear Oscillations and Dynamical Systems and Bifurcations of Vector Fields (Springer-Verlag, New York, 1983). Together, these two books provide an excellent survey of dynamical systems. The first book emphasizes theoretical aspects, the second emphasizes more concrete examples and applications.
D. Knuth,The Art of Computer Programming, Vols. 1, 2, and 3, 2nd ed. (Addison-Wesley, Reading, Massachussetts, 1971). The first and third volumes are relevant to the more straightforward numerical methods for BDEs. The second volume contains an excellent survey of shift register sequences as well as a very useful section on continued fractions.
W. Schmidt,Diophantine Approximation (Springer-Verlag, New York, 1980). One of the most complete books on the theorems of diphantine approximation.
G. H. Hardy,Ramanujan (Chelsea, New York, 1959). Chapter five of this book solves a problem of diophantine approximation that is closely related to the study of intermittency in BDEs.
M. Ghil, in M. Ghil, R. Benzi, and G. Parisi, eds.,Turbulence and Predictability in Geophysical Fluid Dynamics (North-Holland, Amsterdam, 1985). This article surveys the question of climate variability, in particular the quaternary glaciations from the point of view of experimental data (proxy records of temperature history), as well as theory (radiation-energy balance, coupling of the atmosphere-ocean-ice-crust mantle, and celestial mechanics). Limitations on the predictability and reconstruction of climate history are discussed.
References
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V. Shankar and S. Osher,AIAA J. 21:1262–1267 (1983).
V. Shankar, K. Y. Szema, and S. Osher,A Conservative Type-Dependent Full Potential Method for the Treatment of Supersonic Flows with Embedded Subsonic Regions, AIAA Paper #83-1887.
S. R. Chakravarthy and S. Osher,Computing with High Resolution Upwind Schemes for Hyperbolic Equations, to appear in Proceedings of AMS-SIAM, 1983 Summer Seminar, La Jolla, California.
B. Van Leer,J. Comp. Phys. 32:101–136 (1979).
S. R. Chakravarthy,Relaxation Methods for Unfactored Implicit Upwind Schemes, AIAA Paper 84-0165, Reno, Nevada (1984).
M. Brio,Upwind Schemes for the MHD Equations, Ph.D. Thesis, Mathematics, UCLA (1984).
References
U. Frisch,In Varenna Summer School (1984).
U. Frisch and G. Parisi,In Varenna Summer School (1984).
R. Benzi, G. Paladin, G. Parisi, and A. Vulpiani,J. Phys. A 14:3521 (1984).
P. Grassberger and I. Procaccia,Physica D 13 (1984).
R. Benzi, G. Paladin, G. Parisi, and A. Vulpiani, Rome Preprint N. 429 (1985).
P. Grassberger,Phys. Lett. A 97:227 (1983).
G. Paladin and A. Vulpiani,Lett. Nuovo Cim. 4:82 (1984).
B. Derrida,Phys. Rep. 103:29 (1984).
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A. Einstein,Ann. Phys. 17:549 (1905).
E. Nelson,Dynamical Theories of Brownian Motion (Princeton University Press, New Jersey, 1967), Chap. 4.
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V. V. Anshelevich, Ya. G. Sinai, K. M. Khanin,Commun. Math. Phys. 85:449–470 (1982).
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G. C. Papanicolaou and S. R. S. Varadhan,Boundary Value Problems with Rapidly Oscillation Coefficients, in J. Fritz et al., eds.,Random Fields, Rigorous Results (North-Holland, Amsterdam, 1981), p. 835–873.
C. Kipnis and S. R. S. Varadhan,Central Limit Theorems for Additive Functionals of Reversible Markov Processes and Application to Simple Exclusion, preprint, NYU, New York, 1984.
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P. Ferrari, S. Goldstein, and J. L. Lebowitz,Diffusion, Mobility, and Einstein Relation, in J. Fritz, A. Jaffe, D. Szász, eds.,Statistical Physics and Dynamical Systems, Rigorous Results (Birkhaüser, Boston, 1985), p. 405.
J. Hájek and Z. Sidák,Theory of Rank Tests (Academia, Prague, 1967), Chap. VI, Sect. 1.
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Bibliography
J. Guckenheimer and Ph. Holmes,Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Fields (Springer, New York, 1983). (An easy introduction to differential dynamical systems, oriented toward chaos).
R. Bowen,Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, 470 (Springer-Verlag, Berlin, 1975). (A more advanced introduction, stressing the ergodic theory of hyperbolic systems).
L. S. Young,Physica A 124:639–646 (1984). (A brief, but excellent exposition of the inequalities for entropy and dimension).
P. Bergé, Y. Pomeau, and Ch. Vidal,L'Ordre dans le Chaos (Herman, Paris, 1984). (A very nice physics-oriented introduction, to be translated into English).
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Gallavotti, G., Beale, J.T., DiPerna, R.J. et al. Abstracts from the international conference on mathematical problems from the physics of fluids. J Stat Phys 44, 1007–1067 (1986). https://doi.org/10.1007/BF01011921
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DOI: https://doi.org/10.1007/BF01011921