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TECHNICAL PAPERS

Existence and Stability of Localized Oscillations in 1-Dimensional Lattices With Soft-Spring and Hard-Spring Potentials

[+] Author and Article Information
Panagiotis Panagopoulos

School of Applied Mathematical and Physical Sciences, National Technical University of Athens

Tassos Bountis

Department of Mathematics and Center for Research and Applications of Nonlinear Systems (CRANS), University of Patras, GR-26500 Patras, Greece

Charalampos Skokos

Department of Mathematics and Center for Research and Applications of Nonlinear Systems, (CRANS), University of Patras, GR-26500 Patras, Greece and Research Center for Astronomy and Applied Mathematics, Academy of Athens, Soranou Efesiou 4 GR-11527, Athens, Greece

J. Vib. Acoust 126(4), 520-527 (Dec 21, 2004) (8 pages) doi:10.1115/1.1804997 History: Received August 01, 2003; Revised October 01, 2003; Online December 21, 2004
Copyright © 2004 by ASME
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Figures

Grahic Jump Location
The log-log evolution of SALI with respect to the number N* of iterations for the stable breather solution of Fig. 2(a) with soft-spring potential containing 21 particles at C(1)=8, α=0.15275, K=2 [curve (a)] and for the same orbit with a perturbation Δu0=0.2207 in the initial position of the central particle [curve (b)].
Grahic Jump Location
Part of the homoclinic tangle around the origin of the map of Eq. (16) at C(1)=3. The stable and unstable manifolds are the curves emerging out of the (0,0) saddle point and are indicated by the incoming and outgoing arrows, respectively. Some points of two homoclinic orbits at their intersections (corresponding to two different breather solutions) are shown by dark and gray dots, respectively. They provide very accurate estimates for the oscillation amplitudes An of the particles of the lattice, n∊Z.
Grahic Jump Location
The log-log evolution of the SALI with respect to the number N* of iterations for a stable breather, like the one in Fig. 2(a), with hard-spring potential containing 21 particles at C(1)=8, α=0.15275, K=2 [curve (a)] and for the orbit with a perturbation Δu0=1.3 in the initial position of the central particle [curve (b)].
Grahic Jump Location
(a) Variation of energy per particle versus frequency ω, for the simple breather with “soft-spring” potential shown in Fig. 2(a). (b) Variation of energy per particle versus frequency ω, for a simple breather similar to that of Fig. 2(a), with “hard-spring” potential, at K=2,C(1)=−8 and 21 particles.
Grahic Jump Location
(a) Initial conditions of a simple “soft spring” breather for K=2,C(1)=8 for 21 particles (N=10). (b) Variation of the ratio of coefficients A0(3)/A0(1) in the Fourier expansion of u0(t) (position of central particle) versus the coupling parameter α, for the breather in (a).
Grahic Jump Location
For the “soft spring” breather shown in (a), with 21 particles and C(1)=8,K=2, we display in the complex plane how the distribution of the eigenvalues of the monodromy matrix changes as the coupling parameter α is increased: (b) α=0.015, (c) α=0.05, and (d) α=0.1. Note the occurrence of complex instability at α≥0.05.

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