Brief Tour of Wavelet Theory

In this chapter, the main definitions of wavelet theory are given. To explain the basic ideas of the continuous wavelet transform, we describe a transition from Fourier analysis to wavelets. Mother functions and numerical techniques for implementing the wavelet transform are described. The problem of visualising the results is considered. Finally, features of the discrete wavelet transform are discussed.

[1]  Maritan,et al.  Chaos, noise, and synchronization. , 1994, Physical review letters.

[2]  Jelena Kovacevic,et al.  Wavelets and Subband Coding , 2013, Prentice Hall Signal Processing Series.

[3]  A. Hramov,et al.  Time scale synchronization of chaotic oscillators , 2005, nlin/0602053.

[4]  V. A. Nechitailo,et al.  Wavelets and their uses , 2001 .

[5]  G. Buzsáki,et al.  Neuronal Oscillations in Cortical Networks , 2004, Science.

[6]  Grigory V. Osipov,et al.  PHASE SYNCHRONIZATION EFFECTS IN A LATTICE OF NONIDENTICAL ROSSLER OSCILLATORS , 1997 .

[7]  J. Morlet,et al.  Wave propagation and sampling theory—Part II: Sampling theory and complex waves , 1982 .

[8]  Alexey A Koronovskii,et al.  Synchronization of spectral components and its regularities in chaotic dynamical systems. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  Isao Noda,et al.  Two-dimensional correlation spectroscopy , 2002 .

[10]  J. Kurths,et al.  From Phase to Lag Synchronization in Coupled Chaotic Oscillators , 1997 .

[11]  Alexey A Koronovskii,et al.  An approach to chaotic synchronization. , 2004, Chaos.

[12]  S. Mallat.  A wavelet tour of signal processing , 1998 .

[13]  P. V. Popov,et al.  Chaotic synchronization of coupled electron-wave systems with backward waves. , 2005, Chaos.

[14]  Marie Farge,et al.  Improved predictability of two-dimensional turbulent flows using wavelet packet compression , 1992 .

[15]  L. Tsimring,et al.  Generalized synchronization of chaos in directionally coupled chaotic systems. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[16]  Eugenio Rodriguez,et al.  Studying Single-Trials of phase Synchronous Activity in the Brain , 2000, Int. J. Bifurc. Chaos.

[17]  Paul S. Addison,et al.  The Illustrated Wavelet Transform Handbook Introductory Theory And Applications In Science , 2002 .

[18]  Alexey A. Koronovskii,et al.  Experimental study of the time-scale synchronization in the presence of noise , 2010 .

[19]  P. Bergé,et al.  L'ordre dans le chaos. , 1984 .

[20]  Alexey A Koronovskii,et al.  Detecting synchronization of self-sustained oscillators by external driving with varying frequency. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  J. O'Brien,et al.  An Introduction to Wavelet Analysis in Oceanography and Meteorology: With Application to the Dispersion of Yanai Waves , 1993 .

[22]  Y. Meyer.  Wavelets and Operators , 1993 .

[23]  J. Morlet,et al.  Wave propagation and sampling theory—Part I: Complex signal and scattering in multilayered media , 1982 .

[24]  J. L. Hudson,et al.  Locking-based frequency measurement and synchronization of chaotic oscillators with complex dynamics. , 2002, Physical review letters.

[25]  M. Farge.  Wavelet Transforms and their Applications to Turbulence , 1992 .

[26]  H. L. Gray,et al.  Applied time series analysis , 2011 .

[27]  S. Mallat.  Multiresolution approximations and wavelet orthonormal bases of L^2(R) , 1989 .

[28]  Claudio R. Mirasso,et al.  Analytical and numerical studies of noise-induced synchronization of chaotic systems. , 2001, Chaos.

[29]  I. Daubechies.  Orthonormal bases of compactly supported wavelets , 1988 .

[30]  Alexey A. Koronovskii,et al.  Wavelet transform analysis of the chaotic synchronization of dynamical systems , 2004 .

[31]  Alexey A. Koronovskii,et al.  Chaotic synchronization in coupled spatially extended beam-plasma systems , 2006 .

[32]  Matthias Holschneider,et al.  Wavelets - an analysis tool , 1995, Oxford mathematical monographs.

[33]  Dmitry E. Postnov,et al.  SYNCHRONIZATION OF CHAOS , 1992 .

[34]  Jürgen Kurths,et al.  Phase Synchronization in Regular and Chaotic Systems , 2000, Int. J. Bifurc. Chaos.

[35]  A. Grossmann,et al.  DECOMPOSITION OF HARDY FUNCTIONS INTO SQUARE INTEGRABLE WAVELETS OF CONSTANT SHAPE , 1984 .

[36]  L. Glass,et al.  From Clocks to Chaos: The Rhythms of Life , 1988 .

[37]  P. Flandrin,et al.  Some Aspects of Non-Stationary Signal Processing with Emphasis on Time-Frequency and Time-Scale Methods , 1989 .

[38]  A. Balanov,et al.  Synchronization: From Simple to Complex , 2008 .

[39]  Ingrid Daubechies,et al.  The wavelet transform, time-frequency localization and signal analysis , 1990, IEEE Trans. Inf. Theory.

[40]  J. Kurths,et al.  Phase Synchronization of Chaotic Oscillators by External Driving , 1997 .

[41]  H. Saunders,et al.  Book Reviews : APPLIED TIME SERIES ANALYSIS VOLUME 1. BASIC TECHNIQUES R.K. Otnes and L. Enochson John Wiley & Sons, New York, NY 1978, $33.50 , 1981 .

[42]  Yukihiro Ozaki,et al.  Two-Dimensional Correlation Spectroscopy: Applications in Vibrational and Optical Spectroscopy , 2002 .

[43]  Jürgen Kurths,et al.  Synchronization - A Universal Concept in Nonlinear Sciences , 2001, Cambridge Nonlinear Science Series.

[44]  Nikolai F. Rulkov,et al.  Synchronous chaotic behaviour of a response oscillator with chaotic driving , 1994 .

[45]  Bruno Torrésani,et al.  Practical Time-Frequency Analysis , 1998 .

[46]  Y. Pomeau,et al.  L'ordre dans le chaos : vers une approche déterministe de la turbulence , 1988 .

[47]  Jürgen Kurths,et al.  Synchronization: Phase locking and frequency entrainment , 2001 .

[48]  R Quian Quiroga,et al.  Performance of different synchronization measures in real data: a case study on electroencephalographic signals. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[49]  S. Boccaletti,et al.  Synchronization of chaotic systems , 2001 .

[50]  Alexey A Koronovskii,et al.  Detection of synchronization from univariate data using wavelet transform. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[51]  G. Battle.  A block spin construction of ondelettes. Part I: Lemarié functions , 1987 .

[52]  David A. Yuen,et al.  Geophysical Applications of Multidimensional Filtering with Wavelets , 2002 .

[53]  Kurths,et al.  Phase synchronization of chaotic oscillators. , 1996, Physical review letters.

[54]  Carroll,et al.  Synchronization in chaotic systems. , 1990, Physical review letters.

[55]  C. Torrence,et al.  A Practical Guide to Wavelet Analysis. , 1998 .

[56]  E. Dowell,et al.  Chaotic Vibrations: An Introduction for Applied Scientists and Engineers , 1988 .

[57]  J Kurths,et al.  Quantitative analysis of chaotic synchronization by means of coherence. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[58]  R. Gencay,et al.  An Introduction to Wavelets and Other Filtering Methods in Finance and Economics , 2001 .

[59]  Y. Meyer,et al.  Ondelettes et bases hilbertiennes. , 1986 .

[60]  Alexander E. Hramov,et al.  Sleep spindles and spike–wave discharges in EEG: Their generic features, similarities and distinctions disclosed with Fourier transform and continuous wavelet analysis , 2009, Journal of Neuroscience Methods.

[61]  Jürgen Kurths,et al.  Analysing Synchronization Phenomena from Bivariate Data by Means of the Hilbert Transform , 1998 .

[62]  Hamann,et al.  Transition from chaotic to nonchaotic behavior in randomly driven systems. , 1992, Physical review letters.

[63]  Alexey A. Koronovskii,et al.  Synchronization of chaotic oscillator time scales , 2005, nlin/0504049.