Interrelation between the stability of extended normal modes and the existence of intrinsic localized modes in nonlinear lattices with realistic potentials

1994 ◽  
Vol 50 (2) ◽  
pp. 866-887 ◽  
Author(s):  
K. W. Sandusky ◽  
J. B. Page
Keyword(s):  
Normal Modes ◽  
Localized Modes ◽  
Intrinsic Localized Modes ◽  
Nonlinear Lattices ◽  
The Stability ◽  
Realistic Potentials

Intrinsic Localized Modes and Nonlinear Normal Modes in Micro-Resonator Arrays

2005 ◽  
Author(s):  
A. J. Dick ◽  
B. Balachandran ◽  
C. D. Mote
Keyword(s):  
Normal Modes ◽  
Nonlinear Phenomena ◽  
Strong Nonlinearity ◽  
Nonlinear Phenomenon ◽  
Solid State Physics ◽  
Localized Modes ◽  
Intrinsic Localized Modes ◽  
Micro Scale ◽  
Resonator Arrays ◽  
Nonlinear Normal

In the solid-state physics literature, nonlinear phenomena such as localizations have been studied for a number of decades in lattice structures. These localizations, which occur in periodic and strongly nonlinear discrete systems, have been found to result from a combination of the discreteness and the strong nonlinearity rather than the defects or impurities in the system. Intrinsic Localized Modes (ILMs), which are defined as spatial localization due to strong nonlinearity within arrays of oscillators, have been studied recently in the context of coupled micro-scale cantilevers. Within this paper, the hypothesis that an intrinsic localized mode may be realized as a nonlinear normal mode is explored in order to gain a better understanding of this nonlinear phenomenon. It is believed that an understanding of this phenomenon would be valuable for the design of piezoelectric micro-scale resonator arrays that are being developed for signal processing, communication, and sensor applications.

q-BREATHERS IN FPU-LATTICES — SCALING AND PROPERTIES FOR LARGE SYSTEMS

2007 ◽  
Vol 21 (23n24) ◽  
pp. 3925-3932 ◽  
Author(s):  
SERGEJ FLACH ◽  
OLEG I. KANAKOV ◽  
KONSTANTIN G. MISHAGIN ◽  
MIKHAIL V. IVANCHENKO
Keyword(s):  
Energy Density ◽  
Normal Modes ◽  
Localization Length ◽  
Nonlinearity Parameter ◽  
First Finding ◽  
Nonlinear Lattices ◽  
Intensive Quantities ◽  
Time Periodic Solutions ◽  
Large Systems ◽  
Time Periodic

Recently q-breathers - time-periodic solutions which localize in the space of normal modes and maximize the energy density for some mode vector q0 - were obtained for finite nonlinear lattices. We scale these solutions to arbitrarily large lattices in various lattice dimensions. We study the scaling consequence for previously obtained analytical estimates of the localization length of q-breathers for β-FPU and α-FPU lattices. The first finding is that the degree of localization depends only on intensive quantities and is size independent. Secondly, a critical wave vector km is identified, which depends on one effective nonlinearity parameter, q-breathers minimize the localization length at k0 = km and completely delocalize in the limit k0 → 0, π.

An Application of the Poincare´ Map to the Stability of Nonlinear Normal Modes

1980 ◽  
Vol 47 (3) ◽  
pp. 645-651 ◽  
Author(s):  
L. A. Month ◽  
R. H. Rand
Keyword(s):  
Floquet Theory ◽  
Poincaré Map ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Theory Approach ◽  
Poincare Map ◽  
Linearized Stability ◽  
Nonlinear Normal ◽  
The Stability ◽  
Two Degree Of Freedom

The stability of periodic motions (nonlinear normal modes) in a nonlinear two-degree-of-freedom Hamiltonian system is studied by deriving an approximation for the Poincare´ map via the Birkhoff-Gustavson canonical transofrmation. This method is presented as an alternative to the usual linearized stability analysis based on Floquet theory. An example is given for which the Floquet theory approach fails to predict stability but for which the Poincare´ map approach succeeds.

scholarly journals Selective excitations of intrinsic localized modes of atomic scales in carbon nanotubes

2008 ◽  
Vol 77 (2) ◽  
Author(s):  
Y. Kinoshita ◽  
Y. Yamayose ◽  
Y. Doi ◽  
A. Nakatani ◽  
T. Kitamura
Keyword(s):  
Carbon Nanotubes ◽  
Localized Modes ◽  
Intrinsic Localized Modes

Stability of the Base Flow to Axisymmetric and Plane-Polar Disturbances in an Electrically Driven Flow Between Infinitely-Long, Concentric Cylinders

2000 ◽  
Vol 123 (1) ◽  
pp. 31-42
Author(s):  
J. Liu ◽  
G. Talmage ◽  
J. S. Walker
Keyword(s):  
Magnetic Field ◽  
Electric Current ◽  
Axial Magnetic Field ◽  
Normal Modes ◽  
Azimuthal Velocity ◽  
Base Flow ◽  
Transition To Turbulence ◽  
Electrically Conducting ◽  
Concentric Cylinders ◽  
The Stability

The method of normal modes is used to examine the stability of an azimuthal base flow to both axisymmetric and plane-polar disturbances for an electrically conducting fluid confined between stationary, concentric, infinitely-long cylinders. An electric potential difference exists between the two cylinder walls and drives a radial electric current. Without a magnetic field, this flow remains stationary. However, if an axial magnetic field is applied, then the interaction between the radial electric current and the magnetic field gives rise to an azimuthal electromagnetic body force which drives an azimuthal velocity. Infinitesimal axisymmetric disturbances lead to an instability in the base flow. Infinitesimal plane-polar disturbances do not appear to destabilize the base flow until shear-flow transition to turbulence.

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