Abstract
An asymptotic approach is rather efficient when dealing with the theory of oscillations, since one often can figure out a number of relatively simple limit cases which can be efficiently treated and completely understood. Such an approach allows the deepest possible simplification but preserves the most significant features of dynamical behavior. At the same time, one can use the expansions by small parameters characterizing the deviation of the system from the tractable limit case.
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References
Akhmeriev, N.N., Ankiewicz, A.: Solitons: Nonlinear Pulses and Beams. Chapman and Hall, London (1992)
Andrianov, I.V.: Asymptotic solutions for nonlinear systems with high degree of nonlinearity. PMM J. App. Math. Mech. 57, 941–943 (1993)
Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, New York, NY (1978)
Arnold, V.I., Afrajmovich, V.S., Il’yashenko, Yu.S., Shil’nikov, L.P.: Dynamical Systems V. Encyclopedia of Mathematical Sciences. Springer, Berlin (1994)
Arnold, V.I., Kozlov, V.V., Neishtadt, A.I., Khukhro, E.: Mathematical Aspects of Classical and Chelestial Mechanics. Springer, Berlin (2006)
Atay, F.M.: Distributed delays facilitate amplitude death of coupled oscillators. Phys. Rev. Lett. 91, 094101 (2003)
Azeez, M.A.F., Vakakis, A.F., Manevich, L.I.: Exact solutions of the problem of the vibro-impact oscillations of a discrete system with two degrees of freedom. J. App. Math. Mech. 63, 527–530 (1999)
Babitsky, V.I., Veprik, A.M.: Universal bumpered vibration isolator for severe environment. J. Sound Vib. 218, 269–292 (1998)
Berinde, V.: Iterative Approximation of Fixed Points. Springer, New York, NY (2007)
Binder, P., Abraimov, D., Ustinov, A.V., Flach, S., Zolotaryuk, Y.: Observation of breathers in Josephson ladders. Phys. Rev. Lett. 84, 745–748 (2000)
Blekhman, I.I.: Vibrational Mechanics: Nonlinear Dynamic Effects, General Approach, Applications. World Scientific, Singapore (2000)
Bluman, G.W., Kumei, S.: Symmetries and Differential Equations. Springer, New York, NY (1989)
Burns, T.J., Jones, C.K.R.T.: Mechanism for capture into resonance. Physica D. 69, 85–106 (2003)
Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. 2, pp. 136–147. Wiley, New York, NY (1962)
Den Hartog, J.P.: Mechanical Vibrations. McGraw-Hill, New York, NY (1956)
Eiermann, B., Anker, T., Albiez, M., Taglieber, M., Marzlin, K.P., Oberthaler, M.K.: Bright Bose - Einstein gap solitons of atoms with repulsive interaction. Phys. Rev. Lett., 92, 230401 (2004)
Eilbeck, J.C., Lomdahl, P.S., Scott, A.C.: The discrete self – trapping equation. Physica D. 16, 318–338 (1985)
Eisenberg, H.S., Silberberg, Y., Morandotti, R., Boyd, A.R., Aitchison, J.S.: Discrete spatial optical solitons in waveguide arrays. Phys. Rev. Lett. 81, 3383 (1998)
Feng, B.-F.: An integrable three particle system related to intrinsic localized modes in Fermi-Pasta-Ulam-beta chain. J. Phys. Soc. Jpn. 75, 014401 (2006)
Fermi, E., Pasta, J., Ulam, S.: Los Alamos Science Laboratory Report No. LA-1940 unpublished; (Reprinted in Collected Papers of Enrico Fermi, E. Segre (ed.). University of Chicago Press, Chicago, 1965), vol. 2, p. 978 (1955)
Flach, S., Ivanenchenko, M.V., Kanakov, O.V.: q-breathers in Fermi-Pasta-Ulam chains: existence, localization, and stability. Phys. Rev. E. 73, 036618 (2006)
Gendelman, O.: Transition of energy to a nonlinear localized mode in a highly asymmetric system of two oscillators. Nonlinear Dyn. 25, 237–253 (2001)
Gendelman, O.V.: Bifurcations of nonlinear normal modes of linear oscillator with strongly nonlinear damped attachment. Nonlinear Dyn. 37, 115–128 (2004)
Gendelman, O.V.: Modeling of inelastic impacts with the help of smooth functions. Chaos Solitons Fractals. 28, 522–526 (2006)
Gendelman, O.V., Gorlov, D.V., Manevitch, L.I., Musienko, A.I.: Dynamics of coupled linear and essentially nonlinear oscillators with substantially different masses. J. Sound Vib. 286, 1–19 (2005)
Gendelman, O., Manevitch, L.I., Vakakis, A.F., M’Closkey, R.: Energy pumping in nonlinear mechanical oscillators I: dynamics of the underlying hamiltonian systems. ASME J. App. Mech. 68, 34–41 (2001)
Gendelman, O.V., Meimukhin, D.: Response regimes of integrable damped strongly nonlinear oscillator under impact periodic forcing. Chaos Solitons Fractals. 32(2), 405–414 (2007)
Gendeman, O.V., Starosvetsky, Y., Feldman, M.: Attractors of harmonically forced linear oscillator with attached nonlinear energy sink I. Description of response regimes. Nonlinear Dyn. 51, 31–46 (2008)
Goldreich, P., Peale, S.: Spin-orbit coupling in the solar system. Astron. J. 71, 425–438 (1966)
Gourdon, E., Lamarque, C.H.: Energy pumping with various nonlinear structures: numerical evidences. Nonlinear Dyn. 40, 281–307 (2005)
Grosberg, Y.A., Khokhlov, A.R.: Statistical Physics of Macromolecules. Nauka, Moscow (1989) [in Russian]
Guckenheimer, J., Hoffman, K., Weckesser, W.: Bifurcations of relaxation oscillations near folded saddles. Int. J. Bifurcat. Chaos. 15, 3411–3421 (2005)
Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York, NY (2002)
Guckenheimer, J., Wechselberger, M.: Lai-Sang Young: chaotic attractors of relaxation oscillators. Nonlinearity. 19, 701–720 (2006)
Jackson, E.A.: Perspectives of Nonlinear Dynamics, vol. 1. Cambridge University Press, Cambridge (1991)
Kauderer, H.: Nichtlineare Mechanik. Springer, Berlin (1958)
Kevorkian, J., Cole, J.D.: Multiple Scale and Singular Perturbation Methods. Springer, Berlin/NewYork (1996)
Khasnutdinova, K.R., Pelinovsky, D.E.: On the exchange of energy in coupled Klein-Gordon equations. Wave Motion. 38, 1–10 (2003)
Korn, G.A., Korn, T.M.: Mathematical Handbook for Scientists and Engineers, 2nd edn. Dover Publications, New York, NY (2000)
Kosevitch, A.M., Kovalyov, A.S.: Introduction to Nonlinear Dynamics. Naukova Dumka, Kiev (1989) [in Russian]
Landa, P.S.: Nonlinear Oscillations and Waves in Dynamical Systems. Springer, Berlin (1996)
Landau, L.D., Lifshits, E.M.: Mechanics. Butterworth-Heinemann, Boston, MA (1976)
Lin, W.A., Reichl, L.E.: External field induced chaos in an infinite square well potential. Physica D. 19, 145–152 (1986)
Lyapunov, A.: The General Problem of the Stability of Motion. Princeton University Press, Princeton, NJ (1947)
Machida, M., Koyama, T.: Localized rotating-modes in capacitively coupled intrinsic Josephson junctions: systematic study of branching structure and collective dynamical instability. Phys. Rev. B. 70, 024523 (2004)
Manevich, A.I., Manevitch, L.I.: The Mechanics of Nonlinear Systems with Internal Resonances. Imperial College Press, London (2005)
Manevitch, L.I.: Complex representation of dynamics of coupled nonlinear oscillators. In: Uvarova, L., Arinstein, A., Latyshev, A. (eds.) Mathematical Models of Non-Linear Excitations, Transfer Dynamics and Control in Condensed Systems and Other Media, pp. 269–300. Kluwer Academic Publishers, Dordrecht (1999)
Manevitch, L.I.: The description of localized normal modes in a chain of nonlinear coupled oscillators using complex variables. Nonlinear Dyn. 25, 95–109 (2001)
Manevitch, L.I.: New approach to beating phenomenon in coupled nonlinear oscillatory chains. In: Avrejcewicz, J. (ed.) Dynamical Systems:Theory and Applications, vol. 1, pp. 119–137. Lodz, Poland (2005)
Manevitch, L.I.: New approach to beating phenomenon in coupled nonlinear oscillatory chains. Arch. Appl. Mech. 77, 301–312 (2007)
Manevitch, L.I., Azeez, M.A.F., Vakakis, A.F.: Exact solutions for a discrete systems undergoing free vibro – impact oscillations. In: Babitsky, V.I. (ed.) Dynamics of Vibro – Impact Systems, Proceedings of the Euromech Colloquium 15–18 September 1998. Springer, New York, NY (1998)
Manevitch, L.I., Gourdon, E., Lamarque, C.-H.: Towards the design of an optimal energetic sink in a strongly inhomogeneous two-degree-of-freedom system. ASME J. App. Mech. 74, 1078–1086 (2007)
Manevitch, L.I., Mikhlin, Yu.V., Pilipchuk, V.N.: The Method of Normal Vibrations for Essentially Nonlinear Systems. Nauka, Moscow (1989) [in Russian]
Meirovitch, L.: Principles and Techniques of Vibrations. Prentice Hall, Upper Saddle River, NJ (2000)
Melnikov, V.K.: On the stability of the center for time periodic perturbations. Trans. Moscow Math. Soc. 12, 1–57 (1963)
Nayfeh, A.H.: Perturbation Methods. Wiley, New York, NY (2000)
Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York, NY (1995)
Neishtadt, A.I.: Passage through a separatrix in a resonance problem with slowly varying parameter. J. Appl. Math. Mech. 39, 594–605 (1975)
Pikovsky, A., Rosenblum, M., Kurtz, J.: Synchronization: Universal Concept in Nonlinear Sciences. Cambridge University Press, Cambridge (2003)
Pilipchuk, V.N.: A transformation of vibrating systems based on a non-smooth periodic pair of functions. Dokl. Akad. Nauk UkrSSR (Ukrainian Acad. Sci. Rep.). A(4), 37–40 (1988) [in Russian]
Pilipchuk, V.N.: Application of special non-smooth temporal transformations to linear and nonlinear systems under discontinuous and impulsive excitation. Nonlinear Dyn. 18, 203–234 (1999a)
Pilipchuk, V.N.: Strongly nonlinear vibrations of damped oscillators with two non-smooth limits. J. Sound Vib. 302, 398–402 (1999b)
Pilipchuk, V.N.: Impact modes in discrete vibrating systems with rigid barriers. Int. J. Non Linear Mech. 36, 999–1012 (2001)
Quinn, D.D., Rand, R.H., Bridge, J.: The dynamics of resonance capture. Nonlinear Dyn. 8, 1–20 (1995)
Rand, R.H.: The dynamics of resonance capture. In: Guran, A. (ed.) Proceedings of the First International Symposium on Impact and Friction of Solids, Structures and Intelligent Machines, pp. 91–94. World Scientific, Ottawa, ON (1998)
Rand, R.H.: Lecture Notes on Nonlinear Vibrations. The Internet-First University Press, Cornell University. http://ecommons.library.cornell.edu/handle/1813/62 (2009). Accessed 7 Aug 2009
Rand, R.H., Quinn, D.D.: Resonant capture in a system of two coupled homoclinic oscillators. J. Vibr. Control. 1, 41–56 (1995)
Rosenberg, R.M.: Normal modes in nonlinear dual-mode systems. J. App. Mech. 27, 263–268 (1960)
Rosenberg, R.M.: The normal modes of nonlinear n-degree-of-freedom systems. J. App. Mech. 29, 7–14 (1962)
Rosenberg, R.M.: On nonlinear vibrations of systems with many degrees of freedom. Adv. Appl. Mech. 9, 155–242 (1966)
Saaty, T.L.: Modern Nonlinear Equations. Dover Publications Inc., New York, NY (1981)
Salenger, G., Vakakis, A.F., Gendelman, O.V., Andrianov, I.V., Manevitch, L.I.: Transitions from strongly- to weekly-nonlinear motions of damped nonlinear oscillators. Nonlinear Dyn. 20, 99–114 (1999)
Sato, M., Hubbard, B.E., Sievers, A.J., Ilic, B., Craighead, H.G.: Optical manipulation of intrinsic localized vibrational energy in cantilever arrays. Europhys. Lett. 66, 318 (2004)
Sato, M., Hubbard, B.E., Sievers, A.J., Ilic, B., Czaplewski, D.A., Craighead, H.G.: Observation of locked intrinsic localized vibrational modes in a micromechanical oscillator array. Phys. Rev. Lett. 90, 044102 (2003)
Schwarz, U.T., English, L.Q., Sievers, A.J.: Experimental generation and observation of intrinsic localized spin wave modes in an antiferromagnet. Phys. Rev. Lett. 83, 223 (1999)
Scott, A.S., Lomdahl, P.S., Eilbeck, J.C.: Between the local-mode and normal-mode limits. Chem. Phys. Lett. 113, 29–36 (1985)
Sen, A.K., Rand, R.H.: A numerical investigation of the dynamics of a system of two time-delay coupled relaxation oscillators. Commun. Pure App. Anal. 2, 567–577 (2003)
Shaw, S.W., Pierre, C.: Nonlinear normal modes and invariant manifolds. J. Sound Vib. 150, 170–173 (1991)
Shaw, S.W., Pierre, C.: Normal modes for nonlinear vibratory systems. J. Sound Vib. 164, 85–124 (1993)
Sievers, A.J., Takeno, S.: Intrinsic localized modes in anharmonic crystals. Phys. Rev. Lett. 61, 970–973 (1988)
Sokolov, I.J., Babitsky, V.I., Halliwell, N.A.: Hand-held percussion machines with low emission of hazardous vibration. J. Sound Vib. 306, 59–73 (2007)
Starosvetsky, Y., Gendelman, O.V.: Attractors of harmonically forced linear oscillator with attached nonlinear energy sink II: Optimization of a nonlinear vibration absorber. Nonlinear Dyn. 51, 47–57 (2008)
Swanson, B.I., Brozik, J.A., Love, S.P., Strouse, G.F., Shreve, A.P., Bishop, A.R., Wang, W.Z., Salkola, M.I.: Observation of intrinsically localized modes in a discrete low-dimensional material. Phys. Rev. Lett. 82, 3288 (1999)
Uzunov, I.M., Muschall, R., Gölles, M., Kivshar, Y.S., Malomed, B.A., Lederer, F.: Pulse switching in nonlinear fiber directional couplers. Phys. Rev. E. 51, 2527–2537 (1995)
Vakakis, A.F., Gendelman, O.: Energy pumping in nonlinear mechanical oscillators II: resonance capture. ASME J. App. Mech. 68, 42–48 (2008)
Vakakis, A.F., Gendelman, O.V., Bergman, L.A., McFarland, D.M., Kerchen, G., Lee, Y.-S.: Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems. Springer, Berlin (2008)
Vakakis, A.F., Manevitch, L.I., Mikhlin, Yu.V., Pilipchuk, V.N., Zevin, A.A.: Normal Modes and Localization in Nonlinear Systems. Wiley, New York, NY (1996)
Wirkus, S., Rand, R.H.: The Dynamics of two coupled van der pol oscillators with delay coupling. Nonlinear Dyn. 30, 205–221 (2002)
Zhuravlev, V.F., Klimov, D.M.: Applied Methods in Vibration Theory. Nauka, Moscow (1988)
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Manevitch, L.I., Gendelman, O.V. (2011). Discrete Finite Systems. In: Tractable Models of Solid Mechanics. Foundations of Engineering Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15372-3_2
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