ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)


For citation:

Shcherbinin S. A., Goncharov P. P., Chechin G. M. Investigation of stability of nonlinear normal modes in electrical lattices. Izvestiya VUZ. Applied Nonlinear Dynamics, 2013, vol. 21, iss. 2, pp. 34-51. DOI: 10.18500/0869-6632-2013-21-2-34-51

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Language: 
Russian
Article type: 
Article
UDC: 
530.182

Investigation of stability of nonlinear normal modes in electrical lattices

Autors: 
Shcherbinin Stepan Aleksandrovich, Southern Federal University
Goncharov Petr Petrovich, Southern Federal University
Chechin Georgij Mihajlovich, Southern Federal University
Abstract: 

The problems of existence and stability of the symmetry-induced nonlinear normal modes in the electric chain of non-linear capacitors, connected to each other with linear inductors (the model described in Physica D238 (2009) 1228) are investigated. For all modes of this type, the upper limit of the stability region (in amplitude of voltage oscillations on capacitors) as a function of the chain cell number were found. Asymptotic formulas were determined at cell number tends to infinity.

Reference: 
  1. Trias E, Mazo JJ, Orlando TP. Discrete breathers in nonlinear lattices: Experimental detection in a Josephson array. Physical Review Letters. 2000;84(4):741–744. DOI:10.1103/PhysRevLett.84.741
  2. Binder P, Abraimov D, Ustinov AV, Flach S, Zolotaryuk Y. Observation of breathers in Josephson ladders. Physical Review Letters. 2000;84(4):745–748. DOI:10.1103/PhysRevLett.84.745
  3. Sato M, Hubbard BE, Sievers AJ, Ilic B, Czaplewski DA, Craighead HG. Observation of locked intrinsic localized vibrational modes in a micromechanical oscillator array. Physical Review Letters. 2003;90(4):1–4. DOI:10.1103/PhysRevLett.90.044102
  4. Sato M, Hubbard BE, Sievers AT. Nonlinear energy localization and its manipulation in micromechanical oscillator arrays. Reviews of Modern Physics. 2006;78(1):137–157. DOI:10.1103/RevModPhys.78.137
  5. Boechler N, Theocharis G, Job S, Kevrekidis PG, Porter MA, Daraio C. Discrete breathers in one-dimensional diatomic granular crystals. Physical Review Letters. 2010;104(24):244302–244304. DOI:10.1103/PhysRevLett.104.244302
  6. Afshari E, Hajimiri A. Nonlinear transmission lines for pulse shaping in silicon. Journal of Solid-state Circuits. 2005;40(3):744–752. DOI:10.1109/JSSC.2005.843639
  7. Afshari E, Bhat HS, Hajimiri A, Marsden JE. Extremely wideband signal shaping using one- and two-dimensional nonuniform nonlinear transmission lines. Journal of Applied Physics. 2006;99(5):054901–054916. DOI:10.1063/1.2174126
  8. Bhat HS, Afshari E. Nonlinear constructive interference in electrical lattices. Physical Review E. 2008;77(6):066602–066613. DOI:10.1103/PhysRevE.77.066602
  9. Bhat HS, Osting B. The zone boundary mode in periodic nonlinear electrical lattices. Physica D. 2009;238(14):1216–1228. DOI:10.1016/j.physd.2009.04.009
  10. Budinsky N, Bountis T. Stability of nonlinear models and chaotic properties of 1D Fermi-Pasta-Ulam lattices. Physica D. 1983;8(3):445–452.
  11. Sandusky KW, Page JB. Interrelation between stability of extended normal modes and the existence of intrinsic localized modes in nonlinear lattices with realistic potentials. Physical Review B. 1994;50(2):866–887. DOI:10.1103/PHYSREVB.50.866
  12. Poggi P, Ruffo S. Exact solution in the FPU oscillator chain. Physica D. 1997;103(1):251–272.
  13. Chechin GM, Ryabov DS. Stability of nonlinear normal modes in the FPU-β chain in the thermodynamic Limit. Physical Review E. 2012;85(5):056601. DOI: 10.1103/PhysRevE.85.056601
  14. Yoshimura K. Modulational instability of zone boundary mode in non-linear lattices: rigorous result. Physical Review E. 2004;70(1):016611–016617. DOI:10.1103/PhysRevE.70.016611
  15. Chechin GM, Zhukov KG. Stability analysis of dynamical regimes in nonlinear systems with discrete symmetries. Phys. Rev. E. 2006;73(3):36216. DOI: 10.1103/PhysRevE.73.036216
  16. Zhukov KG, Chechin GM. Group-theoretical methods for simplification of stability analysis of dynamical regimes in nonlinear systems with discrete symmetry. Izvestiya VUZ. Applied Nonlinear Dynamics. 2008;16(4):147-166. DOI: 10.18500/0869-6632-2008-16-4-147-166.
  17. Chechin GM, Novikova NV, Abramenko AA. Bushes of vibrational modes for Fermi–Pasta–Ulam chains. Physica D. 2002;166(3-4):208–238. DOI:10.1016/S0167-2789(02)00430-X
  18. Chechin GM, Ryabov DS, Zhukov KG. Stability of low dimensional bushes of vibrational modes in the Fermi–Pasta–Ulam chains. Physica D. 2005;2033):121–166. DOI:10.1016/J.PHYSD.2005.03.009
  19. Bezuglova GS, Goncharov PP, Gurov JV, Chechin GM. Discrete breathers in scalar dynamical models on the plane square lattice. Izvestiya VUZ. Applied Nonlinear Dynamics. 2011;19(3):89-103. DOI: 10.18500/0869-6632-2011-19-3-89-103.
  20. Bezuglova GS, Chechin GM, Goncharov PP. Discrete breathers on symmetrydetermined invariant manifolds for scalar models on the plane square lattice. Physical Review E. 2011;84(3):036606. DOI:10.1103/PhysRevE.84.036606
  21. Rosenberg RM. The normal modes of nonlinear n-degree-of-freedom systems. J. Appl. Mech. 1962;29:7–14. DOI:10.1115/1.3630071
  22. Rink B. Symmetric invariant manifolds in the Fermi–Pasta–Ulam lattice. Physica D. 2003;175(1-2):31–42. DOI:10.1016/S0167-2789(02)00694-2
  23. Chechin GM, Ryabov DS, Sakhnenko VP. Bushes of normal modes as exact excitations in nonlinear dynamical systems with discrete symmetry. Nonlinear phenomena research perspectives. Ed. by C. W. Wang. New-York: Nova Science Publishers; 2007. 225 p.
  24. Bountis T, Chechin CM, Sakhnenko VP. Discrete symmetries and stability in Hamiltonian dynamics. International J. of Bifurc. Chaos. 2011;21(6):1539–1582. DOI:10.1142/S0218127411029276
  25. Chechin GM, Sakhnenko VP, Stokes HT, Smith AD, Hatch DM. Non-linear normal modes for systems with discrete symmetry. Int. J. Non-Linear Mech. 2000;35(3):497–513. DOI:10.1016/S0020-7462(99)00037-2
  26. Sakhnenko VP, Chechin GM. Symmetry selected rules in nonlinear dynamics of atomic systems. Dokl. Akad. Nauk. 1993;330(3):308–310.
  27. Sakhnenko VP, Chechin GM. Groups of modes and normal oscillations for nonlinear dynamical systems with discrete symmetry. Dokl. Math. 1994;39(9):625–628.
  28. Chechin GM, Sakhnenko VP. Interaction between normal modes in nonlinear dynamical systems with discrete symmetry. Exact results. Physica D. 1998;117(1-4):43–76. DOI: 10.1016/S0167-2789(98)80012-2
  29. Petrashen MI, Trifonov ED. Application of group theory in quantum mechanics. Moscow: Nauka; 1967. 308 p. (In Russian).
  30. Abramowitz M, Stigan I. Handbook of special functions with formulas, graphs and mathematical tables. Moscow: Nauka; 1979. 832 p. (In Russian).
Received: 
19.07.2012
Accepted: 
19.07.2012
Published: 
31.07.2013
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