Nonlinear Resonance Analysis

Theory, Computation, Applications

Nonlinear Resonance Analysis

Nonlinear resonance analysis is a unique mathematical tool that can be used to study resonances in relation to, but independently of, any single area of application. This is the first book to present the theory of nonlinear resonances as a new scientific field, with its own theory, computational methods, applications and open questions. The book includes several worked examples, mostly taken from fluid dynamics, to explain the concepts discussed. Each chapter demonstrates how nonlinear resonance analysis can be applied to real systems, including large-scale phenomena in the Earth's atmosphere and novel wave turbulent regimes, and explains a range of laboratory experiments. The book also contains a detailed description of the latest computer software in the field. It is suitable for graduate students and researchers in nonlinear science and wave turbulence, along with fluid mechanics and number theory. Colour versions of a selection of the figures are available at www.cambridge.org/9780521763608.


 Reviews:

'… brings significant rewards … equipping the receptive reader to address a number of open questions.' Contemporary Physics

A. Baker . Transcendental Number Theory (Cambridge University Press, 1975).
A. Bruno and V. Edneral . Normal forms and integrability of ODE systems. Prog. Comp. Soft. 32 (3) (2006), 139–44.
A. Bruno and V. Edneral . On integrability of the Euler–Poisson equations. J. Math. Sci. 152 (4) (2008), 479–89.
A. Chorin . Vorticity and Turbulence (Springer, 1994).
A. Constantin , D. Sattinger , and W. Strauss . Variational formulations for steady water waves with vorticity. Fluid Mech. 548 (2006), 151–63.
A. Constantin , E. Kartashova , and E. Wahlén . Discrete wave turbulence of rotational capillary water waves. E-print: arXiv. 1001.1497 (2010).
A. Constantin , M. Ehrnström , and E. Wahlén . Symmetry of steady periodic gravity water waves with vorticity. Duke Math. J. 140 (3) (2007), 591–603.
A. Constantin . The trajectories of particles in Stokes waves. Invent. Math. 166 (2006), 523–35.
A. Constantin and E. Kartashova . Effect of non-zero constant vorticity on the nonlinear resonances of capillary water waves. EPL 86 (2009), 29001-1-6.
A. Constantin and W. Strauss . Exact steady periodic water waves with vorticity. Comm. Pure Appl. Math. 57 (2004), 481–527.
A. Constantin and W. Strauss . Stability properties of steady water waves with vorticity. Comm. Pure Appl. Math. 60 (2007), 911–50.
A. D. Alexandrov . Uniqueness theorems for surfaces in general. Vestnik Leningrad Univ. Math. 11 (1956), 5–17.
A. D. Bruno . Local Methods in Nonlinear Differential Equations (Springer, Berlin, 1989).
A. D. Craik . Wave Interactions and Fluid Flows (Cambridge University Press, 1985).
A. Delshamsa , R. de la Llaveb , and T. M. Seara . Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows. Advan. Math. 202 (1) (2006), 64–188.
A. E. Dolinko . From Newton's second law to Huygens's principle: visualizing waves in a large array of masses joined by springs. Eur. J. Phys. 30 (2009), 1217–28.
A. F. T. Da Silva and D. H. Peregrine . Steep, steady surface waves on water of finite depth with constant vorticity. Fluid Mech. 195 (1988), 281–302.
A. H. Nayfeh . Introduction to Perturbation Techniques (Wiley-Interscience, NY, 1981).
A. H. Nayfeh . Method of Normal Forms (Wiley-Interscience, NY, 1993).
A. I. Dyachenko , A. O. Korotkevich , and V. E. Zakharov . Weak turbulence of gravity waves. JETP Lett. 77 (10) (2003), 546–50.
A. I. Dyachenko , Y. V. Lvov , and V. E. Zakharov , Five-wave interaction on the surface of deep fluid. Physica D 87 (1995), 233–61.
A. J. Majda and A. L. Bertozzi . Vorticity and Incompressible Flow (Cambridge University Press, 2002).
A. Jarmén , L. Stenflo , H. Wilhelmsson , and F. Engelmann . Effect of dissipation on nonlinear interaction. Phys. Lett. 28A (11) (1969), 748–9.
A. L. Karuzskii , A. N. Lykov , A. V. Perestoronin , and A. I. Golovashkin . Microwave nonlinear resonance incorporating the helium heating effect in superconducting microstrip resonators. Phys. C: Supercond. 408–410 (2004), 739–40.
A. N. Ganshin , V. B. Efimov , G. V. Kolmakov , L. P. Mezhov-Deglin , and P. V. E. McClintock . Observation of an inverse energy cascade in developed acoustic turbulence in superfluid helium. Phys. Rev. Lett. 101 (2008): 065303-1-4.
A. N. Kolmogorov . On the conservation of conditionally periodic motions for a small change in Hamilton's function. Dokl. Akad. Nauk SSSR 98 (1954), 527–30.
A. N. Kolmogorov . The local structure of turbulence in incompressible viscous fluids at very large Reynolds numbers. Dokl. Akad. Nauk SSSR 30 (1941), 301–5; reprinted: Proc. R. Soc. Lond. A 434 (1991), 9–13.
A. N. Pushkarev . On the Kolmogorov and frozen turbulence in numerical simulation of capillary waves. Eur. J. Mech. – B/Fluids 18(3) (1999), 345–51.
A. N. Pushkarev and V. E. Zakharov . Turbulence of capillary waves – theory and numerical simulation. Physica D 135(1–2) (2000), 98–116.
A. Newell , S. Nazarenko and L. Biven . Wave turbulence and intermittency. Phys. D, 152–153 (2001), 520–50.
A. Pikovsky , M. Rosenblum , and J. Kurths . Sinchronization: A Universal Concept in Nonlinear Sciences (Cambridge University Press, 2001).
A. Pikovsky and M. Rosenblum . Self-organized partially synchronous dynamics in populations of nonlinearly coupled oscillators. Physica D 238 (2009), 27–37.
A. Pikovsky and Yu. Maistrenko . Synchronization: Theory and Application NATO Science Series II: Mathematics, Physics and Chemistry 109 (Kluwer Academic Publishers, Dodrecht, Boston, London, 2003).
A. S. Besicovitch . On the linear independence of fractional powers of integers. J. Lond. Math. Soc. 15 (1) (1940), 3–6.
B. B. Kadomtsev . Plasma Turbulence (Acad. Press, London, 1965).
B. L. van der Waerden . Modern Algebra I (Springer, 2003).
B. M. Bredichin . Free number semigroups of power densities. Mat. Sborn. 46 (88) (1958), 143–58 [in Russian].
B. M. Lake and H. C. Yuen . A note on some water-wave experiments and the comparision of data with the theory. Fluid Mech. 83 (1977), 75–81.
B. R. Safdi and H. Segur . Explosive instability due to four-wave mixing. Phys. Rev. Lett. 99 (2007), 245004-1-4.
B. V. Chirikov . A universal instability of many-dimensional oscillator systems. Phys. Rep. 52 (5) (1979), 263–379.
C. C. Chow , D. Henderson , and H. Segur . A generalized stability criterion for resonant triad interactions. Fluid Mech. 319 (1996), 67–76.
C. Connaughton , S. V. Nazarenko , and A. C. Newell , Dimensional analysis and weak turbulence. Physica D 184 (2003), 86.
C. Falcon , E. Falcon , U. Bortolozzo , and S. Fauve . Capillary wave turbulence on a spherical fluid surface in low gravity. EPL 86 (2009), 14002-1-6.
C. Jacobi . Fundamenta Nova Theoriae Functionum Ellipticarum (Koenigsberg, 1829) [in Latin].
C. R. Menyuk , H. H. Chen , and Y. C. Lee . Restricted multiple three-wave interactions: integrable cases of this system and other related systems. J. Math. Phys. 24 (1983), 1073–9.
C. R. Menyuk , H. H. Chen , and Y. C. Lee . Restricted multiple three-wave interactions: Panlevé analysis. Phys. Rev. A 27 (1983), 1597–611.
C. Sulem and P. -L. Sulem . Nonlinear Schrödinger Equations: Self-Focusing and Wave Collapse (App. Math. Sci. 139, New York, Springer, 1999).
C. Vedruccio , E. Mascia , and V. Martines . Ultra high frequency and microwave non-linear interaction device for cancer detection and tissue characterization, a military research approach to prevent health diseases. Int. Rev. Armed Forces Med. Serv. 78 (2) (2005), 120–30.
CENREC, http://cenrec.risc.uni-linz.ac.at/portal/
Ch. A. C. Cunningham and I. F. De Albuquerque Cavalcanti . Intraseasonal modes of variability affecting the South Atlantic convergence zone. Int. J. Climatology 26 (2006), 1165–80.
Ch. Doench , E. Kartashova , and L. Tec . Construction of optimal set of Manley–Rowe constants for resonance clustering in wave turbulent regimes. In preparation (2010).
D. A. Kovriguine and G. A. Maugin . Multiwave nonlinear couplings in elastic structures. Math. Prob. Engin. (2006), doi:10.1155/MPE/2006/76041.
D. C. Schmidt . Model-driven engineering. IEEE Comp. 39(2) (2006), 25–31.
D. Coutand and S. Shkoller . Well-posedness of the free-surface incompressible Euler equations with or without surface tension. J. Amer. Math. Soc. 20 (2007), 829–930
D. Cumin and C. Unsworth . Generalising the Kuramoto model for the study of neuronal synchronisation in the brain. Physica D 226 (2007), 181–96.
D. Hilbert . Mathematical problems. Bull. Amer. Math. Soc. 8 (1902), 437–79.
D. M. Henderson , M. S. Patterson , and H. Segur . On the laboratory generation of two-dimensional, progressive, surface waves of nearly permanent form on deep water. Fluid Mech. 559 (2006), 413–37.
E. A. Kartashova , L. I. Piterbarg , and G. M. Reznik . Weakly nonlinear interactions between Rossby waves on a sphere. Oceanology 29 (1990), 405–11.
E. A. Kartashova and G. M. Reznik . Interactions between Rossby waves in bounded regions. Oceanology 31 (1992), 385–89.
E. Falcon , C. Laroche , and S. Fauve . Observation of gravity-capillary wave turbulence. Phys. Rev. Lett. 98 (2007), 094503-1-4.
E. Kartashova , C. Raab , Ch. Feurer , G. Mayrhofer , and W. Schreiner , Symbolic computations for nonlinear wave resonances. In Extreme Ocean Waves, eds. E. Pelinovsky and Ch. Kharif (Springer, 2008), 97–128.
E. Kartashova , S. Nazarenko , and O. Rudenko , Resonant interactions of nonlinear water waves in a finite basin. Phys. Rev. E 98 (2008), 0163041-1-9.
E. Kartashova . A model of laminated turbulence. JETP Lett. 83 (2006), 341–45.
E. Kartashova . Applicability of weakly nonlinear theory for planetary-scale flows. Scientic report WR 95-03, KNMI, 1995, 1–29.
E. Kartashova . Clipping – a new investigation method for PDEs in compact domains. Theor. Math. Phys. 99 (1994), 675–80.
E. Kartashova . Discrete wave turbulence. EPL 87 (2009), 44001-1–5.
E. Kartashova . Exact and quasi-resonances in discrete water-wave turbulence. Phys. Rev. Lett. 98 (2007), 214502-1-4.
E. Kartashova . Fast computation algorithm for discrete resonances among gravity waves. Low Temp. Phys. 145 (2006), 286–95.
E. Kartashova . Nonlinear resonances of water waves. DCDS Series B 12(3) (2009), 607–21.
E. Kartashova . On large-scale dynamics of weakly nonlinear wave systems. In Advanced Series in Nonlinear Dynamics 7, eds. A. Mielke and K. Kirchgaessner (World Scientific, 1995), 282–90.
E. Kartashova . On properties of weakly nonlinear wave interactions in resonators. Physica D 54 (1991), 125–34.
E. Kartashova . Partitioning of ensembles of weakly interacting dispersing waves in resonators into disjoint classes. Physica D 46 (1990), 43–56.
E. Kartashova . Resonant interactions of the water waves with discrete spectra. In Proc. of Nonlinear Water Waves Workshop, ed. D. H. Peregrine (University of Bristol, UK, 1992), 43–53.
E. Kartashova . Towards H-mode discharge explanation? In Current Topics in Astrophysical and Fusion Plasma, eds. M. F. Heyn , W. Kernbichler , and K. Biernat (Verlag der Technische Universität Graz, 1994), 179–84.
E. Kartashova . Wave resonances in systems with discrete spectra. In Nonlinear Waves and Weak Turbulence, ed. V. E. Zakharov (AMS Trans. 2, 1998), 95–129.
E. Kartashova . Weakly nonlinear theory of finite-size effects in resonators. Phys. Rev. Lett. 72 (1994), 2013–16.
E. Kartashova and A. Kartashov . Exact and quasi-resonances among gravity-capillary waves. (In preparation, 2010).
E. Kartashova and A. Kartashov . Laminated wave turbulence: generic algorithms I. Int. J. Mod. Phys. C 17 (2006), 1579–96.
E. Kartashova and A. Kartashov . Laminated wave turbulence: generic algorithms II. Comm. Comp. Phys. 2 (2007), 783–94.
E. Kartashova and A. Kartashov . Laminated wave turbulence: generic algorithms III. Physica A: Stat. Mech. Appl. 380 (2007), 66–74.
E. Kartashova and A. Kartashov . Resonance clustering of rotational capillary waves. Comm. Comp. Phys. submitted (2010).
E. Kartashova and G. Mayrhofer . Cluster formation in mesoscopic systems. Physica A: Stat. Mech. Appl. 385 (2007), 527–42.
E. Kartashova and M. D. Bustamante . Resonance clustering in wave turbulent regimes: integrable dynamics. E-print: arXiv:1002.4994 (2010).
E. Kartashova and V. S. L'vov . A model of intraseasonal oscillations in Earth's atmosphere. Phys. Rev. Lett. 98 (2007), 198501-1-4.
E. Kartashova and V. S. L'vov . Cluster dynamics of planetary waves. Europhys. Lett. 83 (2008), 50012-1-6.
E. Landau . Handbuch für Lehre von der Verteilung der Primazahlen. II (Teubner, Leipzig, 1909).
E. T. Whittaker . A Treatise on the Analytical Dynamics of Particles and Rigid Bodies (Cambrige University Press, 1937).
E. V. Kozik and B. V. Svistunov . Kelvin-wave cascade and decay of superfluid turbulence. Phys. Rev. Lett. 92 (2004), 035301-1-4.
E. Wahlén . A Hamiltonian formulation of water waves with constant vorticity. Lett. Math. Phys. 79 (2007), 303–15.
E. Wahlén . On rotational water waves with surface tension. Philos. Trans. Roy. Soc. Lond. Ser. A 365 (2007), 2215–25.
Elastic pendulum. http://academic.reed.edu/physics/courses/phys 100/Lab Manuals/Nonlinear Pendulum/nonlinear.pdf
F. Almonte , V. K. Jirsa , E. W. Large , and B. Tuller . Integration and segregation in auditory streaming. Physica D 212 (1–2) (2005), 137–59.
F. Bashforth and J. C. Adams . An Attempt to Test the Theories of Capillary Action (Cambridge University Press, 1883).
F. Calogero . Classical Many-Body Problems Amendable to Exact Treatments (LNP: Monographs 66, Springer, 2001).
F. Calogero . Isochronous Systems (Oxford University Press, 2008).
F. D. Campello , J. M. B. Saraiva , and N. Krusche . Periodicity of atmospheric phenomena occurring in the extreme south of Brazil. Atm. Sci. Lett. 5 (2004), 65–76.
F. Diacu and P. Holmes . Celestial Encounters: The Origins of Chaos and Stability (Princeton University Press, 1996).
F. Lindemann . Über die Zahl π. Math. Annal. 20 (1882), 213–25.
F. Verheest . Integrability of restricted multiple three-wave interactions. II: Coupling constants with ratios 1 and 2. J. Math. Phys. 29 (1988), 2197–201.
F. Verheest . Proof of integrability for five-wave interactions in a case with unequal coupling constants. Phys. A: Math. Gen. 21 (1988), L545–49.
F. Verhulst . Hamiltonian normal forms. Scholarpedia 2 (8) (2007), 2101.
F. W. Bessel . Untersuchungen über die Länge des einfachen Secundenpendels. Abh. Berlin Akad. (Berlin, 1828; reprinted by H. Bruns, Leipzig, 1889).
F.-F. Jin and M. Ghil . Intraseasonal oscillations in the extratropics: Hofp bifurcation and topographic instabilities. Atm. Sci. 47 (1990), 3007–22.
G. B. Whitham . Lectures on Wave Propagation (Springer for TATA Institute of fundamental research, Bombay, 1979).
G. B. Whitham . Linear and Nonlinear Waves (Wiley Series in Pure and Applied Mathematics, 1999).
G. Galileo . Discorsi e Dimostrazioni Matematiche, Intorno a' due Nuove Scienze (Elsevier, Leiden, 1638).
G. Kolmakov , A. Levchenko , M. Braznikov , L. Mezhov-Deglin , A. Slichenko , and P. McClintock , Formation of a direct Kolmogorov-like cascade of second-sound waves in He II. Phys. Rev. Lett. 93 (2004), 074501-1-4.
G. P. Berman and F. M. Israilev . The Fermi-Pasta-Ulam problem: fifty years of progress. Chaos 15 (1) (2005), 015104-1-18.
G. W. Brandstator . A striking example of the atmosphere's leading traveling pattern. J. Atm. Sci. 44 (1987), 2310–23.
H. Davenport . Multiplicative Number Theory (Markham Publishing Company, Chicago, 1967).
H. L. Grant , R. W. Stuart , and A. Moilliet . Turbulence spectra from tidal channel. Fluid. Mech. 12 (1961), 241–63.
H. Ocamoto and M. Shoji . The Mathematical Theory of Permanent Progressive Water Waves (World Scientific, 2001).
H. Poincaré . Oeuvres (Paris, 1951).
H. Punzmann , M. G. Shats , and H. Xia . Phase randomization of three-wave interactions in capillary waves. Phys. Rev. Lett. 103 (2009), 064502-1-4.
H. Segur , D. Henderson , J. Hammack , C. -M. Li , D. Pheiff , and K. Socha . Stabilizing the Benjamin–Feir instability. Fluid Mech. 539 (2005), 229–71.
H. Segur and D. M. Henderson . The modulation instability revisited. Euro. Phys. J. – Spec. Topics 1147 (2007), 25–43.
H. W. Wyld . Formulation of the theory of turbulence in an incompressible fluid. Ann. Phys. 14 (1961), 134–65.
H. Wilhelmsson , L. Stenflo , and F. Engelmann . Explosive instabilities in the well-defined phase description. J. Math. Phys. 11 (1970), 1738–42.
H. Wilhelmsson and V.P. Pavlenko . Five-wave interaction – a possibility for enhancement of optical or microwave radiation by nonlinear coupling of explosively unstable plasma waves. Phys. Scripta 7 (1972), 213–16.
I. G. Jonsson . Wave–current interactions. In The Sea (Wiley, New York, 1990), 65–120.
I. M. Vinogradov . The Foundations of Number Theory (Pergamon, London, 1955).
I. Newton . Philosophiae Naturalis Principia Mathematica I–III. Royal Soc. Lond. (1687).
I. S. Grigoriev and E. Z. Meilikhov (eds.) Fisicheskie Velichiny (Physical Quantities) (Energoatomizdat, Moscow, 1991) [in Russian].
I. Silberman . Planetary waves in atmosphere. Meteorology 11 (1954), 27–34.
Jü. Pöschel . Nonlinear partial differential equations, Birkhoff normal forms and KAM theory Progr. Math. 169 (1998), 167–86.
J. A. Sanders , F. Verhulst , and J. Murdock . Averaging Methods in Nonlinear Dynamical Systems. Appl. Math. Sci. 59 (Springer, 2007).
J. Barrow-Green . Poincare and the Three Body Problem (American Mathematical Society, London Mathematical Society, 1997).
J. K. Engelbrecht , V. E. Fridman , and E. N. Pelinovsky . Nonlinear Evolution Equations (Pitman Res. Not. Math. Ser. 180, Longman, London, 1988)
J. L. Hammack , D. M. Henderson , and H. Segur . Progressive waves with persistent, two-dimensional surface patterns in deep water. Fluid Mech. 532 (2005), 1–51.
J. L. Hammack and D. M. Henderson . Experiments on deep water waves with two-dimensional surface patterns. J. Offshore Mech. & Artic Eng., 125 (2003), 48–53.
J. L. Hammack and D. M. Henderson . Resonant interactions among surface water waves. Ann. Rev. Fluid Mech. 25 (1993), 55–96.
J. Lagrange . Oeuvres 6 (Gauthier-Villars, Paris, 1873).
J. Lighthill . Waves in Fluids (Cambridge University Press, 1978).
J. M. Basilla . On the solution of x2 + dy2 = m. Proc. Japan. Acad. Series A 80 (10) (2004), 40–1.
J. M. Manley and H. E. Rowe . Some general properties of non-linear elements – Part 1: General energy relations. Proc. Inst. Rad. Engrs. 44 (1956), 904–13.
J. M. Tuwankotta and F. Verhulst . Symmetry and resonance in Hamiltonian system. Preprint, Utrecht University, 2000, 1–21.
J. Moser . On invariant curves of area preserving mappings of an annulus. Nachr. Akad. Wiss. Goett., Math. Phys. Kl. (1962), 1–20.
J. Murdock . Normal Forms and Unfoldings for Local Dynamical Systems (Springer-Verlag, New York, 2003).
J. Pedlosky . Geophysical Fluid Dynamics (Springer-Verlag, New York, 1987).
K. Hasselmann . A criterion for nonlinear wave stability. Fluid Mech. 30 (1967), 737–39.
K. Hasselmann . On nonlinear energy transfer in gravity-wave spectrum. Part 1: General theory. Fluid Mech. 12 (1962), 481–500.
K. Horvat , M. Miskovic , and O. Kuljaca . Avoidance of nonlinear resonance jump in turbine governor positioning system using fuzzy controller. Industr. Techn. 2 (2003), 881–5.
K. M. Watson , B. J. West , and B. I. Cohen . Coupling of surface and internal gravity waves: a mode coupling model. Fluid Mech. 77 (1976), 185–208.
K. M. Weickmann , G. R. Lussky , and J. E. Kutzbach . Intraseasonal (30–60 day) fluctuations of ongoing longwave radiation and 250 mb stream function during northern winter. Mon. Wea. Rev. 113 (1985), 941–61.
K. Rajendran and A. Kitoh . Modulation of tropical intraseasonal oscillations by ocean–atmosphere coupling. J. Climate 19 (2006), 366–91.
K. Takaya and H. Nakamura . Geographical dependence of upper–level blocking formation associated with intraseasonal amplification of the Siberian high. Atmos. Sci. 62 (2005), 4441–9.
K. Watson and J. Bride . Excitation of capillary waves by longer waves. Fluid. Mech. 250 (1993), 103–19.
L. A. Ostrovskii , S. A. Rybak , and S. L. Tsimring . Negative energy waves in hydrodynamics. Sov. Phys. Uspekhi 29 (11) (1986), 1040–52.
L. Biferale and I. Procaccia . Anisotropy in turbulent flows and in turbulent transport. Phys. Rep. 414 (2006), 43–164.
L. Biven , S. Nazarenko , and A. Newell . Breakdown of wave turbulence and the onset of intermittency. Phys. Lett. A 280 (2001), 28–32.
L. Bourouiba . Discreteness and resolution effects in rapidly rotating turbulence. Phys. Rev. E 78 (5) (2008), 056309-1-12.
L. Euler . Theoria motuum planetarum et cometarum. Opera 25 (2) (1744), 105–251.
L. F. McGoldrick . Resonant interactions among capillary-gravity waves. Fluid Mech. 21 (1967), 305–31.
L. I. Piterbarg . Hamiltonian formalism for Rossby waves. In Nonlinear Waves and Weak Turbulence, ed. V. E. Zakharov (American Mathematical Society Trans. 2, 1998), 131–66.
L. Merkine and L. Shtilman . Explosive instability of baroclinic waves. Proc. R. Soc. Lond. A 395 (1984), 313–39.
L. R. Wilberforce . On the vibrations of a loaded spiral spring. Phil. Magaz. 38 (1896), 386–92.
L. Stenflo , J. Weiland , and H. Wilhelmsson . A solution of equations describing explosive instabilities. Phys. Scr. 1 (1970), 46.
L. Stenflo . Resonant three-wave interactions in plasmas. Phys. Scr. T50 (1994), 15–9.
L. Stenflo and H. Wilhelmsson . Stabilization of nonlinear instabilities by means of dissipation. Phys. Lett 29A (5) (1969), 217–18.
L. V. Abdurakhimov , Y. M. Brazhnikov , G. V. Kolmakov , and A. A. Levchenko . Study of high-frequency edge of turbulent cascade on the surface of He-II. J. Phys.: Conf. Ser. 150 (3) (2009), 032001.
M. Abramovitz and I. A. Stegun . Handbook of Mathematical Functions (Dover Publications, 1972).
M. Brazhnikov , G. Kolmakov , A. Levchenko , and L. Mezhov-Deglin . Observation of capillary turbulence on the water surface in a wide range of frequencies. Europhys. Lett. 58 (2002), 510–15.
M. Cheney . Tesla Man out of Time (Dorset Press, 1989).
M. D. Bustamante and E. Kartashova . Dynamics of nonlinear resonances in Hamiltonian systems. EPL 85 (2009), 14004-1-5.
M. D. Bustamante and E. Kartashova . Effect of the dynamical phases on the nonlinear amplitudes' evolution. EPL 85 (2009), 34002-1-6.
M. Ehrnström . Deep-water waves with vorticity: symmetry and rotational behaviour. DCDS Series A 19(3) (2007), 483–91.
M. Ehrnström . Uniquenes of steady symetric deep-water waves with vorticity. Nonlin. Math. Phys. 12(1) (2005), 27–30.
M. Francius and C. Kharif . Three-dimensional instabilities of periodic gravity waves in shallow water. Geoph. Res. Abst. 7 (2005), 08757.
M. Ghil , D. Kondrashov , F. Lott , and A. W. Robertson . Intraseasonal oscillations in the mid-latitudes: observations, theory, and GCM results. In Proceedings ECMWF/CLIVAR Workshop on Simulations and Prediction of Intra-Seasonal Variability (Reading, UK, 2004).
M. Ghil . Intra-seasonal oscillations in the extra-tropical atmosphere: observations, theory, and GCM experiments. In Proc. of Eighth Conference on Atmospheric and Oceanic Waves and Stabilities (American Meteorological Society, Boston, MA, 1992).
M. Ghil and K. S. Mo . Intraseasonal oscillations in the global atmosphere. Part I: Northern hemisphere and tropics. Atmos. Sci. 48 (1991), 752–79.
M. Guzzo , Z. Knezević , and A. Milani . Probing the Nekhoroshev stability of asteroids. Cel. Mech. Dyn. Astr. 83 (2002), 121–40.
M. Guzzo and G. Benettin . A spectral formulation of the Nekhoroshev theorem and its relevance for numerical and experimental data analysis. DCDS Series B 1 (2001), 1–28.
M. Kundu and D. Bauer . Nonlinear resonance absorption in the laser-cluster interaction. Phys. Rev. Lett. 96 (2005), 123401-1-4.
M. Perlin and W. M. Schultz , 2000. Capillary effects on surface waves. Annu. Rev. Fluid. Mech. 32, 241–74.
M. R. Matthews , C. F. Gauld , and A. Stinner (eds.). The Pendulum Scientific, Historical, Philosophical and Educational Perspectives (Springer, 2005).
M. Rosenblum and A. Pikovsky . Self-organized quasiperiodicity in oscillator ensembles with global nonlinear coupling. Phys. Rev. Lett. 98 (2007), 064101-1-4.
M. S. Longuet-Higgins and A. E. Gill . Resonant interactions between planetary waves. Proc. Roy. Soc. Lond. A299 (1967), 120–40.
M. Shats , H. Punzmann , and H. Xia . Capillary rogue waves. Phys. Rev. Lett. 104 (2010), 104503-1-4.
M. Stiassnie , and L. Shemer . On the interactions of four water waves. Wave motion 41 (2005), 307–28.
M. Tanaka . On the role of resonant interactions in the short-term evolution of deep-water ocean spectra. J. Phys. Oceanogr. 37 (2007), 1022–36.
M. Tanaka and N. Yokoyama . Effects of discretization of the spectrum in water-wave turbulence. Fluid Dyn. Res. 34 (2004), 199–216.
MathBroker II: Brokering Distributed Mathematical Services. Reseach Institute for Symbolic Computation (RISC), 2007. http://www.risc.uni-linz.ac.at/projects/mathbroker2.
MONET – Mathematics on the Web. The MONET Consortium, 2004. http://monet.nag.co.uk.
N. Gold , A. Mohan , C. Knight , and M. Munro . Understanding service-oriented software. IEEE Software 21(2) (2004), 71–7.
N. N. Nekhoroshev . An exponential estimate of the time of stability of nearly integrable Hamiltonian systems. Russ. Math. Surv. (Usp. Mat. Nauk) 32 (6) (1977), 1–65.
N. W. McLachlan . Theory and Application of Mathieu Functions (Clarendon Press, Oxford, 1947).
O. M. Phillips . On the dynamics of unsteady gravity waves of infinite amplitude. Fluid Mech. 9 (1960), 193–217.
O. M. Phillips . Theoretical and experimental studies of gravity wave interactions. Proc. Roy. Soc. Lond. A299 (1967), 104–19.
O. M. Phillips . Wave interactions – evolution of an idea. Fluid Mech. 106 (1981), 215–27.
O. Rudenko . Nonlinear wave resonances. Wolfram Demonstrations Project, 2008; http://demonstrations.wolfram.com/NonlinearWaveResonances/.
P. B. Kahn and Y. Zarmi . Nonlinear Dynamics: Exploration Through Normal Forms (Wiley, New York, 1998).
P. Bergé , Y. Pomeau , and Ch. Vidal . Order within Chaos: Towards a Deterministic Approach to Turbulence (Wiley-Interscience, 1987).
P. Denissenko , S. Lukaschuk , and S. Nazarenko , Gravity surface wave turbulence in a laboratory flume. Phys. Rev. Lett. 99 (2007), 014501-1-4.
P. J. Olver . Applications of Lie Groups to Differential Equations (Graduated texts in Mathematics 107, Springer, 1993).
P. Lochak and C. Meunier . Multiphase Averaging for Classical Systems (Appl. Math. Sci. Series 72, Springer, 1988)
P. Lynch . On resonant Rossby–Haurwitz triads. Tellus 61A (2009), 438–45.
P. Lynch . Resonant motions of the three-dimensional elastic pendulum. Int. J. Nonl. Mech. 37 (2002), 258–64.
P. Lynch . The swinging spring: a simple model of amospheric balance. in Large-Scale Atmosphere-Ocean Dynamics. Vol II: Geometric Methods and Models, eds. J. Norbury and I. Roulstone (Cambridge University Press, 2002), 64–108.
P. Lynch and C. Houghton . Pulsation and precession of the resonant swinging spring. Physica D 190 (2004), 38–62.
P. W. Anderson and H. Suhl . Instability in the motion of ferromagnets at high microwave power levels. Phys. Rev. 100 (1955), 1788–9.
R. A. Madden and P. R. Julian . Description of global-scale circulation cells in the tropics with a 40–50 day period. Atm. Sci. 29 (1972), 1109–23.
R. A. Madden and P. R. Julian . Detection of a 40–50 day oscillation in the zonal wind in the tropical Pacific. Atm. Sci. 28 (1971), 702–8.
R. A. Nelson and M. G. Olsson . The pendulum – rich physics from a simple system. Am. J. Phys. 54 (2) (1986), 112–21.
R. Abramov , G. Kovac , and A. J. Majda . Hamiltonian structure and statistically relevant conserved quantities for the truncated Burgers–Hopf equation. Comm. Pure App. Math. 56 (1) (2003), 1–46.
R. Baraka and W. Schreiner . Semantic querying of mathematical web service descriptions. LNCS 4184 (2006), 73–87.
R. E. Berg and T. S. Marshall . Wilberforce pendulum oscillations and normal modes. Am. J. Phys. 59 (1) (1991), 32–7.
R. E. Kelly . The stability of an unsteady Kelvin–Helmholz flow. Fluid. Mech. 22 (1965), 547–60.
R. Grimshaw and Y. Skyrnnikov . Long-wave instability in a three-layer stratified shear flow. Stud. Appl. Math. 108 (2002), 77–88.
R. Pavlović and M. Guzzo . Fulfillment of the conditions for the application of the Nekhoroshev theorem to the Koronis and Veritas asteroid families. Mon. Not. R. Astron. Soc. 384 (2008), 1575–82.
R. S. Johnson . A Modern Introduction to the Mathematical Theory of Water Waves (Cambridge University Press, 1997).
R. Treumann and W. Baumjohann . Advanced Space Plasma Physics (Imperial College Press, London, 2001).
R. Z. Sagdeev and A. A. Galeev . Nonlinear Plasma Theory (Benjamin, New York, 1969).
S. Kuksin . Analysis of Hamiltonian PDEs (Oxford University Press, 2000).
S. Kuksin . Fifteen years of KAM for PDE, AMS Transl. 2 (212) (2004), 237–58.
S. Kuksin . Hamiltonian PDEs. In Handbook on Dynamical Systems 1B, eds. B. Hasselblatt and A. Katok (Elsevier, 2005), 1087–133.
S. L. Musher , A. M. Rubenchik and V. E. Zakharov . Hamiltonian approach to the description of nonlinear plasma phenomena. Phys. Rep. 129 (1985), 285–366.
S. Nazarenko . Sandpile behaviour in discrete water-wave turbulence. J. Stat. Mech.: Theor. Exp. (2006), L02002, doi:10.1088/1742-5468/2006/02/L02002.
S. Nazarenko . Wave Turbulence (In preparation, 2010).
S. Strogatz . From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D 143 (2000), 1–20.
S. Strogatz . Nonlinear Dynamics and Chaos (Reading, MA: Addison Wesley, 1994).
S. V. Kuznetsov . The motion of the elastic pendulum. Reg. Chaot. Dyn. 4 (3) (1999), 3–12.
Sh. E. Zimring . Special Functions and Definite Integrals, Algorithms, Mini-computer Programs (Radio i svjaz, Moscow, 1988).
T. B. Benjamin . The threshhold classification of unstable disturbances in flexible surfaces bounding inviscid flows. Fluid Mech. 16 (1963), 436–50.
T. B. Benjamin and J. E. Feir . The disintegration of wavetrains in deep water, Part 1. Fluid Mech. 27 (1967), 417–31.
T. Murakami . Intraseasonal atmospheric teleconnection patterns during the Nothern Hemisphere winter. Climate 1 (1988), 117–31.
T. P. Weissert . The Genesis of Simulations in Dynamics: Pursuing the Fermi-Pasta-Ulam Problem (Springer, 1997).
T. R. N. Jansson , M. P. Haspang , K. H. Jensen , P. Hersen , and T. Bohr . Polygons on a rotating fluid surface. Phys. Rev. Lett. 96 (2006), 174502-1-4.
Tacoma video: http://www.youtube.com/watch?v=3mclp9QmCGs.
U. Frisch . Turbulence (Cambridge University Press, 1995).
U. Köpf . Wilberforce's pendulum revisited J. Am. Phys. 58 (9) (1990), 833–7.
U. Omar , 2001. http://bulbphotography.com/pendulum/gallery.php
V. A. Dubinina , A. A. Kurkin , E. N. Pelinovsky , and O. E. Poluhina . Resonance three-wave interactions of Stokes edge waves. Izvestiya Atm. Ocean. Phys. 42(2) (2006), 254–61.
V. E. Zakharov , A. O. Korotkevich , A. N. Pushkarev , and A. I. Dyachenko . Mesoscopic wave turbulence. JETP Lett. 82 (2005), 487–91.
V. E. Zakharov , ed. Nonlinear Waves and Weak Wave Turbulence (American Mathematical Society Trans. 2, 182, 1998).
V. E. Zakharov , V. S. L'vov , and G. Falkovich . Kolmogorov Spectra of Turbulence (Series in Nonlinear Dynamics, Springer-Verlag, New York, 1992).
V. E. Zakharov . Stability of periodic waves of finite amplitude on the surface of a deep fluid. Zh. Prikl. Mekh. Tekh. Fiz. 9 (1968), 86–94.
V. E. Zakharov . Statistical theory of gravity and capillary waves on the surface of a finite-depth fluid. Eur. J. Mech. B: Fluids 18 (1999), 327–44.
V. E. Zakharov and N. N. Filonenko . Weak turbulence of capillary waves. Appl. Mech. Tech. Phys. 4 (1967), 500–15.
V. I. Arnold , V. V. Kozlov , and A. I. Neishstadt . Mathematical Aspects of Classical and Celestial Mechanics (Springer, 1997).
V. I. Arnold . Proof of a theorem by A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian. Russ. Math. Surveys 18 (1963), 9–36.
V. M. Kamenkovich and A. S. Monin . Small fluctuations in the ocean. In Physics of the Ocean. Vol. 2: Hydrodynamics of the Ocean (Nauka, Moscow, 1978) [in Russian].
V. M. Kamenkovich and G. M. Reznik . Rossby waves. In Physics of the Ocean. Vol. 2: Hydrodynamics of the Ocean (Nauka, Moscow, 1978) [in Russian].
V. N. Oraevsky and R. Z. Sagdeev . On stability of steady longitudinal oscillations in plasma. Zh. Tekh. Fiz. 32 (1962), 1291–96 [in Russian].
V. N. Tsytovich . Nonlinear Effects in Plasma (Plenum, New York, 1970).
V. P. Krasitskii . On reduced equations in the Hamiltonian theory of weakly non-linear surface waves. Fluid Mech. 272 (1994), 1–20.
V. Shrira , V. Voronovich , and N. Kozhelupova . Explosive instability of vorticity waves. Phys. Oceanogr. 27 (1997), 542–54.
V. Zakharov , F. Dias , and A. Pushkarev . One-dimensional wave turbulence. Phys. Rep. 398 (2004), 1–65.
W. B. Wright , R. Budakian , and S. J. Putterman . Diffusing light photography of fully developed isotropic ripple turbulence. Phys. Rev. Lett. 76 (1996), 4528–31.
W. Craig , D. M. Henderson , M. Oscamou , and H. Segur . Stable three-dimensional waves of nearly permanent form on deep water. Math. Comp. Simul. (2006), doi:10.1016/j.matcom.
W. H. Meeks . The topology and geometry of embedded surfaces of constant mean curvature. J. Differential Geometry 27 (3) (1988), 539–52.
W. Kluzniak . Quasi-periodic oscillations and the possibility of an observational distinction between neutron and quark stars. Acta Phys. Polon. B 37 (4) (2006), 1361–65.
W. Lee , G. Kovacic and D. Cai . Renormalized resonance quartets in dispersive wave turbulence. Phys. Rev. Lett. 103 (2009), 024502-1-4.
W. M. Schmidt . Diophantine Approximations (Math. Lec. Not. 785, Springer, Berlin, 1980).
W. Schreiner . Web Service for computing of nonlinear resonances http://www.risc.uni-linz.ac.at/people/schreine/Wave.
W. Wang and J. -J. E. Slotine . On partial contraction analysis for coupled nonlinear oscillators. Biol. Cybern. 92 (2005), 38–53.
Y. Choi , Y. Lvov , and S. Nazarenko . Wave turbulence. Recent Res. Devel. Fluid Dynamics 5 (2004), 1–33.
Y. Kuramoto . Chemical Oscillations, Waves and Turbulence (Springer, New York, 1984).
Y. V. Lvov , S. Nazarenko and B. Pokorni . Discreteness and its effect on water-wave turbulence. Physica D 218 (2006), 24–35.
Yu. I. Manin and A. Panchishkin . Introduction to Modern Number Theory: Fundamental Problems, Ideas and Theories (Springer, 2005).
Yu. V. Matijasevich . Hilbert's Tenth Problem (MIT Press, Cambridge, MA, 1993).
Z. Knezević and R. Pavlović . Application of the Nekhoroshev theorem to the real dynamical system. Novi Sad J. Math. 38 (3) (2008), 181–8.