Andrianov
IV and Awrejcewicz
J (2001a), New trends in asymptotic approaches: summation and interpolation methods, Appl. Mech. Rev. 54(1), 69–92.

Andrianov
IV and Awrejcewicz
J (2001b), Asymptotic approaches to simplified boundary value problems of non-linear dynamics, Nonlinear Analysis, 47, 2261–2269.

Andrianov
IV and Manevitch
LI (1992), Asymptotology: Problems, ideas and results, J Natural Geometry 2(2), 137–150.

Andrianov IV and Manevitch LI (1994), *Asymptotology: Ideas, Methods, Results* (in Russian), Aslan, Moscow.

Barantsev RG (1989), Asymptotic versus Classical Mathematics, *Topics in Mathematical Analysis*, World Scientific, Singapore, 49–64.

Babič
VM and Buldirev
VS (1982), The art of asymptotic, Vestnik Lenigrad Univ Math 10, 227–235.

Crigton DG (1994), Asymptotics-An indispensible complement to thought, computation and experiment in applied mathematical modeling, *7th Eur Conf on Math in Industry*, A Fasano and M Primicerio (eds), BG Teubner, Stuttgart.

Friedrichs
KO (1955), Asymptotic phenomena in mathematical physics, Bull AMS 61, 485–504.

Friedrichs KO (1965), *Perturbation of Spectra in Hilbert Space*, AMS, Providence RI.

Kruskal MD (1963), Asymptotology, *Mathematical Models in Physical Science*, Prentice-Hall, Englewood Cliffs NJ, 17–48.

Kuiken HK (2001), *Practical Asymptotics*, Kluwer, Amsterdam.

Lin SS and Segel LA (1988), *Mathematical Methods Applied to Deterministic Problems in the Natural Sciences*, SIAM, Philadelphia.

Segel
LA (1966), The importance of asymptotic analysis in applied mathematics, Am. Math. Monthly 73(1), 7–14.

Segel LA and Handelman GH (1977), *Mathematics Applied to Continuum Mechanics*, McMillan, New York.

Bender
CM, Milton
KA, Moshe
Moshe, Pinsky
SS, and Simmonds
LM (1987), Logarithmic approximations to polynomic lagrangeans, Phys. Rev. Lett. 58(25), 2615–2618.

Bender
CM, Milton
KA, Moshe
Moshe, Pinsky
SS, and Simmonds
LM (1988), Novel perturbative scheme in quantum field theory, Phys. Rev. D 37(6), 1472–1484.

Bender
CM, Milton
KA, Pinsky
SS, and Simmonds
LM (1989), A new perturbative approach to nonlinear problems, J. Math. Phys. 30(7), 1447–1455.

Bender
CM, Milton
KA, and Boettcher
S (1991), A new perturbative approach to nonlinear partial differential equations, J. Math. Phys. 32(11), 3031–3038.

Bender
CM, Duncan
A, and Jones
HF (1994), Convergence of the optimited δ expansions for the connected vacuum amplitude: Zero dimensions, Phys. Rev. D 49(8), 4219–4225.

Shamrovskii AD (1997), *Asymptotic Group Analysis of the Theory of Elasticity Differential Equations*, (in Russian), Zaporozhie State Ing Academy, Zaporozhie.

Manevitch LI, Pavlenko AV, and Shamrovskii AD (1970), Application of group theory methods to the dynamical problems for orthotropic plates, *Proc VII All-Union Conf on Plates and Shells Theory (Dnepropetrovsk, 1969)* (in Russian), Nauka, Moscow, 408–412.

Shamrovskii
AD (1979), Asymptotic integration of static equation of the theory of elasticity in Cartesian coordinates with automated search of integration parameters, PMM J. Appl. Math. Mech. 43(5), 925–934.

Estrada
R (1998), The Cesaro behavior of distributions, Proc. R. Soc. London, Ser. A 454, 2425–2443.

Estrada
R and Kanwal
RP (1990), A distributional theory for asymptotic expansions, Proc. R. Soc. London, Ser. A 428, 399–430.

Estrada
R and Kanwal
RP (1993), Taylor expansions for distribution, Math. Methods Appl. Sci. 16, 297–304.

Estrada R and Kanwal RP (1994), *Asymptotic Analysis: a Distributional Approach*, Birkäuser, Boston, Basel, Berlin.

Estrada R and Kanwal RP (2002), *A Distributional Approach to Asymptotics, Theory and Applications*, Birkhäuser, Boston, Basel.

Bruning
J and Seeley
R (1985), Regular singular asymptotics, Adv. Math. 58, 133–148.

Wong
R (1980), Distributional derivation of an asymptotic expansion, Proc. AMS 80(2), 266–270.

Dmitriev
MG (1982), Differential relations for an initial jump in a singularly perturbated problems and their applications, Sov. Math. Dokl. 25(3), 730–733.

Vasil’eva AB, Dmitriev MG, Glizer VYa, and Faminskaya MV (1981), Application of regularization and singular perturbation theory to nonlinear impulse optimal control, *Prepr 8-th Triennial World IFAC Cong*, Kyoto, Japan, 3 , 177–180.

Andrianov
IV (1997), A new asymptotic method of calculation stiffened constructions with allowance for the discrete arrangement and of the width of ribs, Phys. Dokl. 42(2), 84–86.

Kalamkarov AL and Andrianov IV (1997), A new asymptotic approach to the analysis of reinforced structure, *Appl Mech in the Americas 4, Mech and Dyn of Solids*, LA Godoy, M Rijsz, and LE Snares (eds), Univ of Iowa, Iowa City IA, 155–158.

Pol B and Bremmer H (1955), *Operational Calculus Based on the Two-Sided Laplace Integral*, Cambridge UP, Cambridge.

Andrianov
IV, Bulanova
NS, and Sedin
VL (1999), Vibration of ribbed plates on elastic basis, Int. Appl. Mech., 25(1), 64–68.

Andrianov
IV, Mikolenko
VA, and Kholod
EG (1999), Nonlinear dynamics of a plane fibrous composite taking into account the width of the fiber, Mech. Solids 34(2), 71–75.

Andrianov
IV, Awrejcewicz
J, and Matyash
M (2000), On application of perturbation method with a few perturbation parameters, Machine Dyn Problems, 24(3), 5–10.

Andrianov
I, Galka
A, and Tokarzewski
S (2000), Asymptotic study of geometrically nonlinear elastic strip with regular system of fibres, Eur. J. Mech. A/Solids, 19(4), 689–698.

Andrianov
IV, Ismagulov
BG, and Matyash
MV (2000), Buckling of cylindrical shells of variable thickness, loaded by external uniform pressure, Tech. Mech. 20(4), 349–354.

Andrianov
IV (1993a), Asymptotic solutions for nonlinear systems with high degrees of nonlinearity, PMM J. Appl. Math. Mech. 57(5), 941–943.

Andrianov
IV (1994), Sequential construction of the asymptotic solution in the essentially nonlinear systems, Phys. Dokl. 39(7), 532–533.

Awrejcewicz J, Andrianov IV, and Manevitch LI (1998), *Asymptotic Approaches in Nonlinear Dynamics: New Trends and Applications*, Springer-Verlag, Berlin.

Andrianov
IV (1993b), A new asymptotic method of integrating the equations of quantum mechanics for strong coupling, Phys. Dokl. 38(2), 56–57.

Vorovich II, Aleksandrov VM, and Babeshko VA (1974), *Nonclassical Mixed Problems of the Theory of Elasticity* (in Russian), Nauka, Moscow.

Egarov
YuV (1990), A contribution to the theory of generalized functions, Russ. Math. Surveys, 45(5), 1–49.

Maslov
VP and Omel’yanov
GA (1981), Asymptotic soliton-form solutions of equations with small dispersion, Russ. Math. Surveys 36(3), 73–149.

Maslov VP and Omel’yanov GA (2001), *Geometric Asymptotic for Nonlinear PDE. 1*, AMS, Providence RI.

Barenblatt
GI and Zel’dovitch
YaB (1971), Intermediate asymptotics in mathematical physics, Russ. Math. Surveys, 26(2), 115–129.

Barenblatt GI (1979), *Similarity, Self-similarity and Intermediate Asymptotics*, Plenum, New York, London.

Barenblatt GI (1987), *Dimensional Analysis*, Gordon and Breach, New York, London.

Barenblatt
GI (1993), Intermediate asymptotic, scaling laws and renormalization group in continuum mechanics, Meccanica 28, 177–183.

Zel’dovitch YaB (1987), *Foreword to Barenblatt* (1987) [50], XIX.

Bolotin VV (1961), An asymptotic method for the study of the problem of eigenvalues of rectangular regions, *Problems of Continuum Mech*., SIAM, Philadelphia, 56–58.

Bolotin VV (1984), *Random Vibration of Elastic Systems*, Martinus Nijhoff Publ, the Hague, Boston.

Andrianov
IV and Kholod
EG (1993a), Non-linear free vibration of shallow cylindrical shell by Bolotin’s asymptotic method, J. Sound Vib. 160(1), 594–603.

Andrianov
IV and Kholod
EG (1993b), Intermediate asympotics in the nonlinear dynamics of shells, Mech. Solids 28(2), 160–165.

Andrianov
IV and Krizhevsky
GA (1991), Investigation of natural vibrations of circular and sector plates with consideration of geometric nonlinearity, Mech. Solids 26(2), 143–148.

Andrianov
IV, and Krizhevsky
GA (1993), Free vibration analysis of rectangular plates with structural inhomogenity, J. Sound Vib., 162(2), 231–241.

Chen G and Zhou J (1993), *Vibration and Damping in Distributed Systems*, CRS, Boca Raton FL.

Andrianov
IV (1983), On the theory of Berger plates, PMM J Appl Math. Mech. 47(1), 142–144.

Andrianov
IV (1986), Construction of simplified equation of nonlinear dynamics of plates and shallow shells by the averaging method, PMM J. Appl. Math. Mech. 50(1), 126–129.

Andrianov
IV and Sedin
VL (1988), Composition of simplified equations of nonlinear dynamics of plates and shells on the basis of homogenization method, Z. Angew. Math. Mech. 67(7), 573–575.

Manevitch LI, Mikhlin YuV, and Philipchuk VN (1989), *Method of Normal Vibrations for Essentially Non-linear Systems* (in Russian), Nauka, Moscow.

Vakakis AF, Manevitch LI, Mikhlin YuV, Pilipchuk VN, and Zevin AA (1996), *Normal Modes and Localization in Non-linear Systems*, Wiley, New York.

Wah
T (1964), The normal modes of vibration of certain nonlinear continuous systems, Trans. ASME 31 (1).

Kantorovich LV and Krylov VI (1958), *Approximate Methods of Higher Analysis*, Noordhoff, Groningen.

Birkhoff
G (1983), Numerical fluid dynamics, SIAM Rev. 25(1), 1–24.

Bjerrum-Bohr
NEJ (2000), 1/χ expansions in nonrelativistic quantum mechanics, J. Math. Phys. 41(5), 2515–2536.

Andrianov
IV, and Danishevs’kyy
VV (2002), Asymptotic approach for nonlinear periodical vibrations of continuous structures, J. Sound Vib. 249(3), 465–481.

Andrianov
IV, and Samoilenko
OG (2001), Ishlinsky-Leibenzon method in the theory of elastic stability, Mech. Solids 36, 6.

He
Ji-Huan (1997), A new approach to nonlinear partial differential equations, Com. Non. Sc. Num. Sim. 2(4), 230–235.

He
Ji-Huan (1999), Homotopy perturbation technique, Comput. Methods Appl. Mech. Eng. 178, 257–256.

He
Ji-Huan (2000), A coupling method of homotopy technique and a perturbation technique for non-linear problems, Int. J. Non-Linear Mech. 35, 37–43.

Liao
SJ (1995), An approximate solution technique not depending on small parameters: a special example, Int. J. Non-Linear Mech. 30(3), 371–380.

Andrianov
IV, and Awrejcewicz
J (2000b), A role of initial conditions choice on the results obtained using different perturbation methods, J. Sound Vib. 236(1), 161–165.

Andrianov
IV, and Awrejcewicz
J (2000c), Construction of periodic solutions to partial differential equations with non-linear boundary conditions, Int. J. Nonlin. Sc. Num. Sim. 1(4), 327–332.

Miles
RN, and Bigelow
SP (1994), Random vibration of a beam with a stick-slip end condition, J. Sound Vib. 169(1), 445–457.

Miloserdova
JV, Novikov
AA, and Potapov
AJ (1981), Impulsive waves in one-dimensional system with nonlinear boundaries (in Russian), Waves and Diffraction 2, 118–121.

Munizyn
AJ (1998), Natural oscillations of a beam with nonlinear support (in Russian), Probl. Theor. Eng. Realiability Machines 2, 36–39.

Bourgain
J (1995), Construction of periodic solutions of nonlinear wave equations in higher dimension, Geometry Func. Anal. 5, 105–140.

Brezis
H (1983), Periodic solutions of nonlinear vibrating strings and duality principles, Bull. Am. Math. Soc. 8, 409–426.

van der Burgh
AHP (1979), On the asymptotic validity of perturbation methods for hyperbolic differential equations, Lect. Notes Math. 711, 229–240.

Chikwendu
SC (1981), Nonlinear wave propagation solutions by Fourier transform perturbation, Int. J. Non-Linear Mech. 16(1), 117–128.

Chikwendu
SC and Kevorkian
J (1972), A perturbation method for hyperbolic equations with small nonlinearities, SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 22, 235–258.

Chow
PL (1972), Asymptotic solutions of inhomogeneous initial boundary value problems for weakly nonlinear partial differential equations, SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 22, 629–647.

Craig
W and Wayne
CE (1993), Newton’s method and periodic solutions of nonlinear wave equations, Commun. Pure Appl. Math. 46, 1409–1498.

Eckhaus
W (1975), New approach to the asymptotic theory of non-linear oscillations and wave propagation, J. Math. Anal. Appl. 49, 575–611.

Lau
SL, Cheung
YK, and Chen
Chuhui (1989), An alternative perturbation procedure of multiple scales for nonlinear dynamics systems, Trans. ASME 56(3), 587–605.

van Horssen
WT (1988), An asymptotic theory for a class of initial-boundary value problems for weakly nonlinear wave equations with an application to a model of the galopping oscillations of overhead transmission lines, SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 48, 1227–1243.

van Horssen
WT (1992), Asymptotics for a class of semilinear hyperbolic equations with an application to a problem with quadratic nonlinearity, Nonl. Anal. 19, 501–530.

van Horssen
WT, and van der Burgh
AHP (1988), On initial-boundary value problems for weakly semi-linear telegraph equations. Asymptotic theory and application, SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 48, 719–736.

Keller
JB, and Kogelman
S (1970), Asymptotic of initial value problems for nonlinear partial differential equations, SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 18, 748–758.

Kevorkian J and Cole JD (1996), *Multiple Scale and Singular Perturbation Methods*, Springer-Verlag, New York.

Krol
MS (1989), On a Galerkin-averaging method for weakly nonlinear wave equations, Math. Methods Appl. Sci. 11, 649–664.

Lardner
RW (1977), Asymptotic solutions of nonlinear wave equations using the methods of averaging and two-timing, Q. Appl. Math. 35, 225–238.

Lidskij
BV and Schulman
EJ (1988), Periodic solutions of the equation u_{tt}=u_{xx}+u^{3}=0,Funct. Anal. Appl. 22, 332–333.

Luke
JC (1966), A perturbation method for nonlinear dispersive wave problems, Proc. R. Soc. London, Ser. A 292, 403–412.

Mitropolsky YU, Khoma G, and Gromak M (1997), *Asymptotic Methods for Investigating Equations of Hyperbolic Type*, Kluwer, Dordrecht.

Rabinowitch
P (1977), Free vibrations for a semilinear wave equation, Commun. Pure Appl. Math. 30, 31–68.

Wayne
CE (1990), Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Com Math Phys 127, 479–528.

Wayne
CE (1997), Periodic solutions of nonlinear partial differential equations, Not. Am. Math. Soc. 7, 895–902.

Witham
GB (1965), A general approach to linear and non-linear dispersive waves, J. Fluid Mech. 22, 273–283.

Witham GB (1974), *Linear and Nonlinear Waves*, Willey, New York.

Boertjens
GJ and van Horssen
WT (1998), On mode interactions for a weakly nonlinear beam equation, Nonlinear Dyn. 17, 23–40.

Boertjens
GJ and van Horssen
WT (2000), An asymptotic theory for a beam equation with a quadratic perturbation, SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 60, 602–632.

Eisenberger
M (1994), Vibration frequencies for beams on variable one- and two-parameter elastic foundations, J. Sound Vib. 176(5), 577–584.

Lewandowski
R (1994a), Nonlinear free vibration of beams by the finite element and continuation method, J. Sound Vib. 170, 577–593.

Lewandowski
R (1994b), Solutions with bifurcation points for free vibration of beams: an analytical approach, J. Sound Vib. 177, 239–249.

Lewandowski
R (1996a), Computational formulation for periodic vibration of geometrically nonlinear structures. Part 1: Theoretical background, Int. J. Solids Struct. 34(15), 1925–1947; Part 2: Numerical strategy and examples, *ibid.*, 1949–1964.

Lewandowski
R (1996b), On beams, membranes and plates vibration backbone curves in cases of internal resonance, Meccanica 31, 323–346.

Pereira
DC (1990), Existence, uniqueness and asymptotic behavior for solutions of the nonlinear beam equation, Nonl Anal-Theory, Meth Appl 14(8), 613–623.

Arnold
VI (1965), Small denominators I: Mappings of the circumference onto itself, Am. Math. Soc. Trans. 2(46), 213–284.

Arnold VI (1978), *Mathematical Methods of Classical Mechanics*, Springer-Verlag, Berlin.

Arnold VI (1988), *Geometrical Methods in the Theory of Ordinary Differential Equations*, Springer-Verlag, New York.

Arnold VI, Kozlov VV, and Neishtadt AI (1997), *Mathematical Aspects of Classical and Celestial Mechanics*, Springer-Verlag, New York.

Pustyl’nikov
LD (1997), Infinite dimensional non-linear ordinary differential equations and the KAM theory, Russ. Math. Surveys 52(3), 551–604.

Siegel CL and Moser JK (1971), *Lectures on Celestial Mechanics*, Springer, New York.

Pilipchuk
VN (1985), The calculation of strongly nonlinear systems close to vibro-impact systems, PMM J Appl Mech 49(5), 744–752.

Pilipchuk
VN (1988), A transformation of vibrating systems based on a nonsmooth periodic pain of functions, Dokl. Akad. Nauk SSSR, Ser. A A(4), 37–40 (in Russian).

Pilipchuk
VN (1996a), Calculation of mechanical systems with pulsed excitation, ASME J. Appl. Mech. 60(2), 217–226.

Pilipchuk
VN (1996b), Analytical study of vibrating systems with strong non-linearities by employing saw-tooth time transformations, J. Sound Vib. 192(1), 43–64.

Pilipchuk
VN (1999), Application of special nonsmooth temporal transformations to linear and nonlinear systems under discontinuous and impulsive excitation, Nonlinear Dyn. 18, 203–234.

Pilipchuk
VN (2000), Non-smooth spation-temporal transformation for impulsively forced oscillators with rigid barriers, J. Sound Vib. 237(5), 915–919.

Pilipchuk
VN (2001), Impact modes in discrete vibrating systems with rigid barriers, Non-lin Mech 36, 999–1012.

Pilipchuk
VN, Vakakis
AF, and Azeez
MAF (1997), Study of a class of subharmonic motions using a nonsmooth temporal transformation (NSTT), Physica D 100, 145–164.

Wei
Y (1999), Basic vibrations and anharmonic analysis of a vibration, Int. J. Non-Linear Mech. 34, 1061–1069.

Zhuravlev
VPh (1976), A method for analysing vibration—impact systems by means of special functions, Mech. Solids 11(2), 30–34.

Zhuravlev
VPh (1977), Investigation of some vibro-impact systems by the method of non-smooth transformations, Mech. Solids, 12(6), 24–28.

Zhuravlev VPh and Klimov DM (1988), *Applied Methods in the Theory of Oscillations* (in Russian), Moscow, Nauka.

Dimentberg MF (1988), *Statistical Dynamics of Nonlinear and Time-Varying Systems*, John Wiley & Sons, New York.

Awrejcewicz J and Andrianov IV (2000), *Asymptotic Methods and their use in the Theory of Shells* (in Polish), Wydawnictwo Naukowo-Techniczne, Warsaw.

Rosenberg
RM (1963), The Ateb(h)-functions and their properties, Q. J. Mech. Appl. Math. 21(1), 37–47.

Babuska
I (1976), Homogenization approach in engineering, Lect Notes Econ. Math. Syst. 134, 137–153.

Bakhvalov N and Panasenko G (1989), *Averaging Processes in Periodic Media. Mathematical Problems in Mechanics of Composite Materials*, Kluwer, Dordrecht.

Bensoussan A, Lions J-L, and Papanicolaou G (1978), *Asymptotic Methods in Periodic Structures*, North-Holland, Amsterdam.

Berdichevsky VL (1983), *Variational Principles of the Continuum Mechanics* (in Russian), Nauka, Moscow.

Berdichevsky V, Jikov V, and Papanicolaou G (eds) (1999), *Homogenization*, Worlds Scientific, Singapore.

Mei Chiang C, Auriault J-L, and Ng Chin-on (1996), Some application of the homogenization theory, *Adv Appl Mech*32 , Academic Press, New York et al. 278–348.

Olejnik OA, Yosifyan GA, and Shamaev AS (1992), *Mathematical Problems in Elasticity and Homogenization*, North-Holland, Amsterdam.

Andrianov IV, Manevitch LI, and Oshmyan VO (2001), *Mechanics of Periodic Structures*, Springer-Verlag, Berlin.

Andrianov
IV, Zarubinskaya
MA, and Paschenik
AN (2001), An asymptotic analysis of stress-strain state of a strip reinforced with ribs, PMM J. Appl. Math. Mech. 65(1), 119–124.

Van Dyke M (1975), *Perturbation Methods in Fluid Mechanics*, Parabolic Press, Stanford CA.

Savin GN and Guz AN (1967), *Stress around Curvilinear Holes in Shells*, Nat2 Areon Lab, Bangalore.

Grigolyuk EI and Phyl’shtinsky LA (1970), *Perforated Plates and Shells* (in Russian), Nauka, Moscow.

Andrianov
IV, Konashenko
SI, and Sedin
VL (1995), Design of plate with wide ribs, Int. Appl. Mech. 31(3), 229–237.

Andrianov
IV, Shevchenko
VV, and Kholod
EG (1995), Asymptotic methods in the statics and dynamics of perforated plates and shells with periodic structures, Tech. Mech. 15(2), 141–157.

Andrianov IV, Lesnichaya VA, and Manevitch LI (1985), *Homogenization Method in Statics and Dynamics of Reinforced Shells* (in Russian), Nauka, Moscow.

Andrianov
IV and Manevitch
LI (1975), Calculation for the strain-stress state in an orthotropic strip stiffened by ribs, Mech. Solids 10(4), 125–129.

Andrianov
IV and Manevitch
LI (1983), Homogenization method in the theory of shells, Advanced in Mechs 6(3/4), 3–29 (in Russian).

Andrianov
IV, and Piskunov
VI (1997a), Stability of ribbed plates with allowance for the discrete arrangement, Mech. Solids 32(6), 135–141.

Andrianov
IV and Piskunov
VI (1997b), An asymptotic investigation of the dynamics of eccentrically stiffened plates, PMM J. Appl. Maths. Mechs. 61(2), 329–331.

Artola
M, and Duvaut
G (1977), Homogenisation d’une plaque renforcee, CR Acad. Sci. Paris Ser. A 284(12), 707–710.

Caillerie
D (1984), Thin elastic and periodic plates, Math. Methods Appl. Sci. 6, 151–191.

Andrianov
IV, Diskovsky
AA, and Kholod
EG (1998), Homogenization method in the theory of corrugated plates, Tech. Mech. 18(2), 123–133.

Hoffmann
K-H and Botkin
ND (2000), Homogenization of von Kármán plates exited by piezoelectric patches, Z. Angew. Math. Mech. 80(9), 579–590.

Kalamkarov AL and Kolpakov AG (1997), *Analysis, Design and Optimization of Composite Shells*, Wiley, New York.

Parton VZ and Kudryavtsev BA (1993), *Engineering Mechanics of Composite Materials*, CRC Press, Boca Raton FL.

Woźniak C and Wierzbicki E (2000), *Averagning Techniques in Thermomechanics of Composite Solids. Tolerance Averaging Versus Homogenazition*, Czȩstochowa, Politechnika.

Dal Mazo (1993), *An Introduction to Γ Convergence*, Birkhäuser, Boston.

Cioranescu D and Donato P (1999), *An Introduction to Homogenization*, Oxford UP, Oxford.

Kalamkarov AL (1992), *Composite and Reinforced Elements of Construction*, Wiley, Chichester.

Khruslov EYa (1995), Homogenized modelling of strongly inhomogeneous media, *Int Congr Math*2 , Basel, Birkhäuser, 270–278.

Kozlov V, Maz’ya V and Movchan A (1999), *Asymptotic Analysis of Fields in Multi-Structures*, Oxford UP, Oxford.

Sanchez-Palencia E (1980), *Non-Homogeneous Media and Vibration Theory*, Springer-Verlag, Berlin.

Allaire
G (1992), Homogenization and two-scale convergence, SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 23, 1482–1518.

Conča C, Planchard J, and Varrninatham M (1995), *Fluids and Periodic Structures*, Collection Ram 38 , John Wiley, Masson, Paris.

Conča
C, and Lund
F (1999), Fourier homogenisation method and the propagation of acoustic waves through a periodic vortex array, SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 59(5), 1573–1581.

Brewster
M and Beylkin
G (1995), A multiresolution strategy for numerical homogenisation, Appl. Comput. Harmon. Anal. 2, 327–349.

Woźniak
C (1987), A non-standard method of modelling of thermoelastic periodic composites, Int. J. Eng. Sci. 5, 483–499.

Molotkov LA (1984), *Matrix Methods in the Theory of Wave Spreading in Layered Elastic and Liquid Media* (in Russian), Nauka, Leningrad.

Pilipchuk
VN and Starushenko
GA (1997), On one variant of nonsmooth transformations of variables for 1-D elastic systems of a periodic structure, PMM J. Appl. Mech. 61(2), 267–274.

Pilipchuk
VN and Vakakis
AF (1998), Study of the oscillations of a nonlinearly supported string using a nonsmooth transformation, J. Vibr. Acoust. 120(2), 434–440.

Salenger
GD and Vakakis
AF (1998a), Discreteness effects in the forced dynamics of a string an a periodic array of non-linear supports, Int. J. Non-Linear Mech. 33(4), 659–673.

Salenger
GD and Vakakis
AF (1998b), Localized and periodic waves with discreteness effects, Mech. Res. Commun. 25(1), 97–104.

Vedenova
EG, Manevitch
LI, and Philipchuk
VN (1985), Normal oscillations of a string with concentrated masses on non-linearly supports, PMM J Appl Mech 49(2), 572–578.

Cioranescu
D and Paulin
JSJ (1986), Reinforced and honey-comb structure, J Math. Pures Appl. 65, 403–422.

Cioranescu D and Paulin JSJ (1999), *Homogenization of Reticulated Structures*, Springer-Verlag, Berlin.

Altenbach
H (1998), Theories of laminated and sandwich plates-a review, Mech. Compos. Mater. 34(3), 333–348.

Ciarlet PG (1990), *Plates and Junctions in Elastic Multi-Structures*, Masson, Paris.

Lewinski T and Telega JJ (2000), *Plates, Laminates and Shells*, World Scientific, Singapore et al.

Boutin
C (1996), Microstructural effects in elastic composites, Int. J. Solids Struct. 33(7), 1023–1051.

Boutin
C (2000), Study of permeability by periodic and self consistent homogenization, Eur. J. Mech. A/Solids, 19, 603–632.

Castro
C and Zuazua
E (2000), Low frequency asymptotic analysis of a string with rapidly oscillating density, SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 60(4), 1205–1233.

Manevitch
LI and Oshmyan
VG (1999), An asymptotic study of the linear vibrations of a stretched beam with concentrated masses and discrete elastic support, J. Sound Vib. 223(5), 679–691.

Neuss-Radu
M (2000), A result on the decay of the boundary layers in the homogenisation theory, Asymptotic Anal. 23, 313–328.

Nazarov SA (2001), *Asymptotic Theory of Thin Plates and Shells*, Nauchnaya kniga (“Scientific Book”), Novosibirsk (in Russian).

Filimonov
AM (1996), Some unexpected results in the classical problem of vibrations of the string with N beads. The case of multiple frequencies, CR Acad. Sci. Paris Ser. I 315(8), 957–961.

Filimonov
AM (1996), Continuous approximations of difference operators, J. Diff. Eqns. 7(4), 411–422.

Filimonov
AM and Myshkis
AD (1998), Asymptotic estimate of solution of one mixed difference differential equation of oscillation theory, J. Diff. Eqns. 2(1), 13–16.

Filimonov
AM, Kurchanov
PF, and Myshkis
AD (1991), Some unexpected results in the classical problem of vibrations of the string with n beads when n is large, CR Acad. Sci. Paris Ser. I 313, 961–965.

Maslov VP (1976), *Operational Methods*, Moscow, Mir.

Wattis
JAD (1993), Approximations to solitary waves on lattices, II: Quasi-continuum approximations for fast and slow waves, J. Phys. A 26, 1193–1209.

Wattis
JAD (2000), Quasi-continuum approximations to lattice equations arising from the discrete nonlinear telegraph equation, J. Phys. A 33, 5925–5944.

Duncan
DB, Eilbeck
JC, Fedddersen
H, and Wattis
JAD (1993), Solitous on lattices, Physica D 68, 1–11.

Kosevich AM (1999), *Crystal Lattice: Phonons, Solitons, Dislocations*, Wiley-VCH, Berlin, New York.

Obraztsov IF, Nerubaylo BV, and Andrianov IV (1991a), *Asymptotic Methods in the Structural Mechanics of Thin-Walled Structures*, (in Russian), Mashinostroyenie, Moscow.

Obraztsov
IF, Andrianov
IV, and Nerubaylo
BV (1991b), Continuum approximation for high-frequency oscillations of a chain and composite equations, Sov. Phys. Dokl. 336(7), 522.

Andrianov
IV (2002), Feature of limiting passage from the discrete to continuous media, PMM J. Appl. Math. Mech. 66(2), 261–265.

Agostini L and Bass J (1950), Les theories de la Turbulence, *Pub Sci Tech du Ministre de l’ais 237*.

Ulam SM (1960), *A Collection of Mathematical Problems*, Interscience, New York.

Filimonov
AM, Mao
X, and Maslov
S (2000), Splash effect and erogodic properties of solution of the classic difference-differential equation, J. Diff. Eqns. 6, 319–328.

Miċkens RE (1990), *Difference Equations. Theory and Applications*, Second Edition, Chapman and Hall, New York, London.

Trefethen
LN (1998), Maxims about Numerical Mathematics, Computers Science and Life, SIAM News 31(1), Jan/Feb, 4.

Nayfeh AH (1981), *Introduction to Perturbation Techniques*, John Wiley and Sons, New York.

Sedov LI (1993), *Similarity and Dimensional Methods in Mechanics*, CRC Press, Boca Raton FL.

Gol’denviezer AL (1961), *Theory of Elastic Thin Shells*, Pergamon Press, New York, Oxford, London, Paris.

Guckenheimer J and Holmes P (1983), *Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields*, Springer-Verlag, New York.

Evkin
AYu (1986), A new approach to the asymptotic integration of the equations of shallow convex shells in the post-critical stage, PMM J. Appl. Math. Mech. 53(1), 92–96.

Evkin
A and Kalamkarov
A (2001a), Analysis of large deflection equilibrium states of composite shells of revolution, Part 1: General model and singular perturbation analysis, Int. J. Solids Struct. 38(50–51), 8961–8974.

Evkin
A and Kalamkarov
A (2001b), Analysis of large deflection equilibrium states of composite shells of revolution, Part 2: Application and numerical results, Int. J. Solids Struct. 38(50–51), 8975–8987.

Koiter WT (1945), On the stability of elastic equilibrium, PhD Thesis, Univ Delft, the Netherlands; (English Transl: Tech rep AFFDL-TR-70-25, Air Force Flight Dyn Lab, 1970).

Pogorelov AV (1988), *Bending of Surface and Stability of Shells*, AMS, Providence RI.

Manevitch LI and Pavlenko AV (1971), Asymptotic analysis of excentrically stiffened cylindrical shells theory equations, *Theory of Plates and Shells* (in Russian), Nauka, Moscow, 185–190.

Manevitch LI and Pavlenko AV (1991), *Asymptotic Method in Micromechanics of Composite Materials* (in Russian), Naukova Dumka, Kiev.

Manevitch LI, Pavlenko AV, and Koblik SG (1982), *Asymptotic Methods in the Theory of Orthotropic Body Elasticity* (in Russian), Visha Shkola, Kiev-Donezk.

Muskhelishvili NI (1953), *Some Basic Problems in the Mathematical Theory of Elasticity*, Noordhoff, Groningen.

Lekhnitskii SG (1968), *Anisotropic Plates*, 2 Edition, Gordon and Breach, New York.

Ting TCT (1996), *Anisotropic Elasticity: Theory and Applications*, Oxford UP, Oxford.

Kuhn P (1956), *Stresses in Aircraft Shell Structures*, McGraw-Hill, New York.

Balabuch LI, Kolesnikov KC, Zarubin VS, Alfutov NA, Usyukin VI, and Chizhov VF (1969), *Foundations of Rockets Structural Mechanics* (in Russian), Vischa Shkola (High School), Moscow.

Christensen RM (1979), *Mechanics of Composite Materials*, John Wiley and Sons, New York.

Everstine
GC and Pipkin
AC (1971), Stress channelling in transversally isotropic elastic composites, Z. Angew. Math. Phys. 22, 225–230.

Everstine
GC and Pipkin
AC (1973), Boundary layers in fiber-reinforced materials, ASME J. Appl. Mech. 40, 512–518.

Spencer
AJM (1974), Boundary layers in highly anisotropic plane elasticity, Int. J. Solids Struct. 10, 1103–1112.

Kosmodamianskii AS (1976), *Stress State of the Anisotropic Media with Holes and Cavities* (in Russian), Vischa Schola, Kiev-Dorezk.

Bogan
YuA (1983), One singular perturbed boundary value problem in plane elasticity, Dynamics of Continuous Media 61, 13–24 (in Russian).

Bogan
YuA (1987), On class of singular perturbated problems in two-dimensional theory of elasticity, J. Appl. Mech. Tech. Phys. 28(2), 138–143.

Kagadii TS (1998) *Perturbation Method in Mechanics of Elastic (Viscoelastic) Anisotropic and Composite Materials* (in Russian), National Mining Academy of Ukraine, Dnepropetrovsk.

Kagadii
TS, Mossakovskaya
LV, and Pavlenko
AV (1992), Perturbation method in three-dimensional problem of linear viscoelasticity of anisotropic bodies, PMM J Appl Math Mech 56(5), 167–171.

Andrianov IV, Lesnichaya VA, Loboda VV, and Manevitch LI (1986), *Investigation of Engineering Structures Reinforced Shells* (in Russian), Vischa Shkola, Kiev-Donezk.

Andrianov
IV, Kholod
EG, and Olevsky
VI (1996), Approximate non-linear boundary value problems of reinforced shell dynamics, J. Sound Vib. 194(3), 369–387.

Bauer SM, Filippov SB, Smirnov AL, and Tovstik PE (1993), asymptotic methods in mechanics with applications to thin shells and plates, Asymptotic Methods in Mechanics, Vaillancourt R and Smirnov AL (eds), AMS, Providence RI, 3–140.

Gol’denviezer
AL (1982), The asymptotic method in the theory of shells, Adv in Mech 5(1/2), 137–182.

Kaplunov JD, Kossovich LYa, and Nolde EV (1998), *Dynamics of Thin Walled Elastic Bodies*, Academic Press, San Diego CA.

Ambartsumian
SA (1962), On a general theory of anisotropic shells and plates, Appl. Mech. Rev. 15, 146–158.

Andrianov
IV and Pasechnik
AN (1993), Asymptotic study of the normal-mode vibrations of a cylindrical shells, Int Appl Mech 29(11), 930–934.

Andrianov
IV and VerbonolVerbonol’
VM (1990), Investigation of stringer shells stability with taking into account prebuckling state momentous (in Russian), Adv. in Mech. 13(3/4), 59–88.

Berger
HM (1955), A new approach to the analysis of large deflections of plates, ASME J. Appl. Mech. 22(4), 465–472.

Timoshenko SP and Woinowsky-Krieger S (1959), *Theory of Plates and Shells*, McGraw Hill, New York, Toronto, London.

Jones
R (1975), Remarks on the approximate analysis of the nonlinear behavior of shallow shells, J. Struct. Mech. 3(2), 157.

Nash WA and Modeer JR (1960), Certain approximate analysis of the nonlinear behavior of plates and shallow shells, *Proc Symp on the Theory of Thin Elastic Shells*, North-Holland, Amsterdam, 331–354.

Nowinski
J and Ismail
IA (1964), Certain approximate analysis of large deflections of cylindrical shells, Z. Angew. Math. Mech. 15(5), 449–455.

Grigolyuk EI and Kulikov GM (1981), About an approximate method for solving nonlinear problems in the theory of elastic plates and shells, *Some Appl Problems in the Theory of Plates and Shells* (in Russian), Moscow Univ, 91–121.

Kamiya
N (1978), Berger’s method and its applications, Res Repts Fac Eng Mie Univ 3, 67.

Ramachandran
J (1974), Vibration of shallow spherical shell of large amplitudes, ASME J. Appl. Mech. 41(3), 84–92.

Bucco
D, Jones
R, and Mazumdar
J (1978), The dynamic analysis of shallow spherical shells, ASME J. Appl. Mech. 45(3), 690–691.

Schmidt
R and DaDeppo
DA (1975), A new approach to the analysis of shells, plates and membranes with finite deflection, Int. J. Non-Linear Mech. 9(5), 409–421.

Prathar
G (1979), On the Berger approximation: a critical re-examination, J. Sound Vib. 66(2), 149–154.

Maslov VP and Nazaikinskii VE (1988), *Asymptotics of Operator and Pseudo-Differential Equations*, Consultant Bureau, New York, London.

McRae
SM and Vrskay
ER (1997), Perturbation theory and the classical limit of Quantum Mechanics, J. Math. Phys. 38(6), 2899–2921.

Nayfeh AH (1973), *Perturbation Methods*, John Wiley and Sons, New York.

Mathews J and Walker RL (1964), *Mathematical Methods of Physics*, WA Benjamin, New York, Amsterdam.

Chen
SH and Cheung
YK (1996), An elliptic perturbation method for certain strongly non-linear oscillators, J. Sound Vib. 192(2), 453–464.

Turbiner
AV (1984), Spectral problem in Quantum Mechanics and nonlinearization procedure, Soviet Phys. Vspekhu 141, 1, 35–78.

Fernández FM (2000), *Introduction to Perturbation Theory in Quantum Maechanics*, CRC Press, Boca Raton FL.

Fernández
FM (2001), Perturbation theory free from secular terms for the equations of motions of anharmonic oscillators, J. Math. Phys. 42(10), 1–10.

Forest
E and Murray
D (1994), Freedom in minimal normal forms, Physica D, D74, 181.

Hennemann
RHG and Montroll
EW (1974), On a nonlinear perturbation theory without secular terms: 1. Classical coupled anharmonic oscillators, Physica (Utrecht) 74, 22–32.

Kahn
PB, Murray
D, and Zarmi
Y (1993), Freedom in small parameter expansion for nonlinear perturbations, Proc. R. Soc. London, Ser. A, A443, 83–94.

Kahn
PB and Zarmi
Y (1991), Minimal normal forms in harmonic oscillations with small nonlinear perturbations, Physica D, D54, 65–74.

Kahn
PB and Zarmi
Y (1993), Radius renormalization in limit cycles, Proc. R. Soc. London, Ser. A A440, 189–199.

Kahn PB and Zarmi Y (1998), *Nonlinear Dynamics: Exploration Through Normal Forms*, Wiley, New York.

Kummer
M (1971), How to avoid secular terms in classical and quantum mechanics, Nuovo Cimento Soc. Ital. Fis., B B1, 123–126.

Manevitch LI (1999), Complex representation of dynamics of coupled nonlinear oscillators, *Math Models of Non-Lin Excitations Transfer, Dyn and Control in Condensed Syst and Other Media*, Kluwer, New York, 269–300.

Fedoryuk MV (1989), Aymptotic methods in analysis, *Analysis 1*, Integral Represas Math, RV Gamkrelidze ed. SWAP Enc Math Sc 13 , Springer, Berlin, New York.

Malinetskii GG (1997), *Chaos, Structures, Numerical Simulation: Introduction to Nonlinear Dynamics* (in Russian), Nauka, Moscow.

Sanders JA and Verhulst F (1985), *Averaging Methods in Nonlinear Dynamical Systems*, Springer-Verlag, New York.

Schult
DA (1999), Matched asymptotic expansions and the closure problem for combustion waves, SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 60(1), 136–155.

Bruno AD (1989), *Local Methods in Nonlinear Differential Equations*, Springer-Verlag, New York.

Bruno AD (2000a), *Power Geometry in Algebraic and Differential Equations*, Elsevier, Amsterdam.

Bruno
AD (2000b), Self-similar solutions and power geometry, Russ Math Sur 55(1), 1–42.

Kolokoltsov VN and Maslov VP (1997), *Idempotent Analysis and Applications*, Kluwer, Dordrecht.

Litvinov
GL, Maslov
VP, and Shpiz
GB (1998), Linear functionals on idempotent spaces: an algebraic approach, Dokl. Math. 58, 389–391.

Maslov VP and Samborskii SN (eds) (1992), *Idempotent Analysis*, Adv in Sov Math 13, AMS, Providence RI.

Barantsev RG (1965b), *Lectures in Transonic Gasdynamics*, (in Russian), LSU, Leningrad.

Barantsev
RG, Mikhailova
IA, and Isitelov
IM (1961), On determination of orders of perturbating functions in the method of small perturbations (in Russian), Ing J 1, 2, 69–81.

Barantsev RG and Engelgart VN (1987), *Asymptotic Methods in Fluid Dynamics* (in Russian), LSU, Leningrad.

Barantsev RG (1981), Asymptotic methods in rarefied gas dynamics, *Results in Sci and Technology. Fluid Mech.*14 , VINITI, Moscow, 3–65 (in Russian).

Barantsev RG (1985), Analytical studies of gas-surface interaction, *Rarefied Gas Dynamics, Proc 13th Int Symp in Novosibirsk*1 , Plenum Press, NY, 645–652.

Alexeeva EV and Barantsev RG (1976), *Local method of the aerodynamic calculation in rarefied gas* (in Russian), LSU, Leningrad.

Barantsev RG and Fedorova VM (1989), Singularities of the local theory for thin axisymmetric bodies (in Russian), *Vestnik Leningr Univ*, Ser 1, 2, 40–42.

Barantsev RG (1976), On asymptotology (in Russian), *Vestnik Leningr Univ*1 , 69–77.

Alexeeva EV, Barantsev RG, and Pashkevich DA (1996), Using of the Padé approximants to temperature calculation in a hypersonic boundary layer (in Russian), *Heat-Mass-Exchange*, Minsk, 1 , 1, 114–118.

Alexeeva EV, Barantsev RG, and Shatrov AV (1996), Combination of temperature asymptotics in the boundary layer, *Vestnik SPb Univ. Ser 1*, 2, 96–99 (in Russian).

Barantsev RG, Pashkevich DA, and Shatrov AV (1999), Combination of asymptotics in the boundary layer of reacting gas mixture, *5th Conf on Dynamical Systems Theory and Applications*, Łódź 137–140.

Anolik MV and Barantsev RG (1998), Combination of asymptotics in the Knudsen layer, II: Testing, *Rarefied Gas Dynamics, Proc 21st Int Symp Book of Abstracts*, Marseille, 2 , 75–76.

Nazarov SA and Plamenevsky BA (1994), *Elliptic Problems in Domain with Picewise Smooth Boundaries*, Walter de Gruyder, Berlin.

Il’in AM (1992), *Matching of Asymptotic Expansions of Solutions of Boundary Value Problems*, AMS, Providence RI.

Goldenfeld N, Martin O, and Oono Y (1991), Asymptotics of partial differential equations and the renormalisation group, *Asymp Beyond All Orders*, M Segur et al. (eds) Plenum Press, NY.

Pilipchuk
VN and Ibrahim
RA (1999), Application of the Lie group transfromations to nonlinear dynamical systems, ASME J. Appl. Mech. 66, 439–447.

Zhuravlev
VPh (1986), The application of monomial Lie groups to the problem of asymptotically integrating equations of mechanics, PMM J. Appl. Math. Mech. 50, 349–352.

Marchuk GI, Agoshkov VI, and Shutyaev VP (1996), *Adjoint Equations and Perturbation Algorithms in Nonlinear Problems*, CRC Press, Boca Raton FL.

Berdichevsky
VL (1979), Variational-asymptotic method of constructing a theory of shells, PMM J. Appl. Math. Mech. 43(4), 711–736.

Bosley
DL (1996a), A technique for the numerical verification of asymptotic expansions, SIAM Rev. 38(1), 128–135.

Bosley
DL (1996b), An improved matching procedure for transient resonance layers in weakly nonlinear oscillatory systems, SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 56(2), 420–445.

Morrison
JA (1966), Comparison of the modified method of averaging and the two variable expansion procedure, SIAM Rev. 8, 66–70.

Zwillinger D (1989), *Handbook of Differential Equations*, Academic Press, New York.

Babič VM and Buldirev VS (1991), *Short-Wavelength Diffraction Theory-Asymptotic Methods*, Springer-Verlag, Berlin.

Feshchenko SF, Shkil’Nl, and Nikolenko LD (1967), *Asymptotic Methods in the Theory of Linear Differential Equations*, Elsevier, New York.

Fedoryuk MV (1993), *Asymptotic Analysis. Linear Ordinary Differential Equations*, Springer, Berlin.

Lomov SA (1992), *Introduction to the General Theory of Singular Perturbations*, AMS, Providence RI.

Trenogin
VA (1970), The development and application of the asymptotic method of Lyusternik and Vishik, Russ. Math. Surveys 25(4), 119–156.

Vishik
MI and Lyusternik
LA (1960), The asymptotic behavior of solutions of linear differential equations with large or quickly changing coefficients and boundary conditions, Russ. Math. Surveys 15(4), 23–91.

Vishik
MI and Lyusternik
LA (1962), Regular degeneration and boundary layer for linear differential equations with small parameters, Am. Math. Surv. Transl. 2(20), 239–364.

Eckhaus W (1979), *Asymptotic Analysis of Singular Perturbations*, North-Holland, Amsterdam.

Eckhaus
W (1994), Fundamental concepts of matching, SIAM Rev. 36(3), 431–439.

Berg Van den
I (1987), Nonstandard Asymptotic Analysis, Lect Notes Math 1249, 810–884.

Jones DS (1997), *Introduction to Asymptotics*, World Scientific, Singapore etc.

Lutz R and Gose M (1981), *Nonstandard Analysis: A Practical Guide with Applications*, Springer-Verlag, Berlin.

Eckhaus
W (1983), Relaxation oscillations, including a standard chase on French ducks, Lect. Notes Math. 985, 449–494.