Fractal Geometry and Chaotic Phenomena

This course is taught since 2007 at Supelec (France). It is split into 5 lectures (3 hours each) and one oral exam.

Lecture 1: Fractal Geometry

B. Mandelbrot, The Fractal Geometry of Nature (San Francisco, Freeman, 1982).
J.-F. Gouyet, Physics and Fractal Structures (Springer Verlag Gmbh, 1996)
J. Feder, Fractals (New York, Plenum Press, 1988).
K. J. Falconer, The Geometry of Fractal Sets (Cambridge, England Cambridge University Press, 1986).
B. Sapoval, Universalité et fractales (Paris, Flammarion, coll. Champs., 2001).
P. Bak, How Nature Works: The Science of Self-Organized Criticality (New York: Copernicus, 1996).

Lecture 2: Brownian motion

J. Klafter and I. M. Sokolov, First Steps in Random Walks: From Tools to Applications (Oxford University Press, 2011).
W. Feller, An Introduction to Probability Theory and Its Applications, Volumes I and II, Second Edition (John Wiley & Sons, New York, 1971).
P. Levy, Processus stochastiques et movement brownien (Paris, Gauthier-Villard, 1965).
S. Redner, A Guide to First-Passage Processes (Cambridge University Press, Cambridge, England, 2001).

Lecture 3: Chaotic Dynamics

E. Lorenz, The Essence of Chaos (University of Washington Press, 1996).
M. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer-Verlag, 1990).
S. H. Kellert, In the Wake of Chaos: Unpredictable Order in Dynamical Systems (University of Chicago Press, 1993).
P. Collet and J.-P. Eckmann, Iterated Maps on the Interval as Dynamical Systems (Birkhauser, 1980).
R. Kautz, Chaos: The Science of Predictable Random Motion (Oxford University Press, 2011)
F. Moon, Chaotic and Fractal Dynamics (Springer-Verlag, 1990).
P. A. Davidson, Turbulence: An Introduction for Scientists and Engineers (Oxford University Press, 2004).
J. Mathieu and J. Scott, An Introduction to Turbulent Flow (Cambridge University Press, 2000).
J. Cardy, G. Falkovich and K. Gawedzki, Non-equilibrium statistical mechanics and turbulence (Cambridge University Press, 2008).
T. Bohr, M. H. Jensen, G. Paladin and A. Vulpiani, Dynamical Systems Approach to Turbulence (Cambridge University Press, 1998)
P. A. Durbin and B. A. Pettersson Reif, Statistical Theory and Modeling for Turbulent Flows (Johns Wiley & Sons, 2001).

Lecture 4: Growth Phenomena

T. Vicsek, Fractal Growth Phenomena, 2nd ed. (Singapore, World Scientific, 1992).
R. B. Banks, Growth and Diffusion Phenomena: Mathematical Frameworks and Applications (Springer Science & Business Media, 1994)
P. L. Krapivsky, S. Redner, E. Ben-Naim, A Kinetic View of Statistical Physics (Cambridge, Cambridge University Press, 2010).

Lecture 5a: Percolation

D. Stauffer and A. Aharony, Introduction to Percolation Theory (2nd Ed., Taylor and Francis, London, 1994)
M. Sahimi, Applications of Percolation Theory (Taylor & Francis, London, 1994)
D. Ben-Avraham and S. Havlin, Diffusion and Reactions in Fractals and Disordered Systems (Cambridge University Press, 2000)

Lecture 5b: Wavelets

I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, 1992)
S. Mallat, A wavelet tour of signal processing (2nd Ed., Academic Press, 1999)
Y. Meyer, Wavelets and Operators (Cambridge Studies in Advanced Mathematics, Vol. 37, 1993)
Other references
Your appreciation can be sent to denis.grebenkov@polytechnique.edu
Last modified 18/10/2014

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