Interactions between normal modes in nonlinear dynamical systems with discrete symmetry. Exact results

Chechin, G. M.; Sakhnenko, V. P.

doi:10.1016/s0167-2789(98)80012-2

Cited by 93 publications

(251 citation statements)

References 16 publications

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“…These 'bushes' are simply invariant manifolds of a certain type. Their definition and how to find them are discussed more elaborately in [6]. The basic idea is the well-known physical principle that the fixed point set of a symmetry forms an invariant manifold for the equations of motion.…”

Section: In Whichmentioning

confidence: 99%

Symmetric invariant manifolds in the Fermi–Pasta–Ulam lattice

Rink

2003

Physica D: Nonlinear Phenomena

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The Fermi-Pasta-Ulam (FPU) lattice with periodic boundary conditions and n particles admits a large group of discrete symmetries. The fixed point sets of these symmetries naturally form invariant symplectic manifolds that are investigated in this short note. For each k dividing n we find k degree of freedom invariant manifolds. They represent short wavelength solutions composed of k Fourier-modes and can be interpreted as embedded lattices with periodic boundary conditions and only k particles. Inside these invariant manifolds other invariant structures and exact solutions are found which represent for instance periodic and quasi-periodic solutions and standing and traveling waves. Some of these results have been found previously by other authors via a study of mode coupling coefficients and recently also by investigating 'bushes of normal modes'. The method of this paper is similar to the latter method and much more systematic than the former. We arrive at previously unknown results without any difficult computations. It is shown moreover that similar invariant manifolds exist also in the Klein-Gordon lattice and in the thermodynamic and continuum limits.

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Section: In Whichmentioning

confidence: 99%

Symmetric invariant manifolds in the Fermi–Pasta–Ulam lattice

Rink

2003

Physica D: Nonlinear Phenomena

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“…In square brackets, the group of the bush symmetry is indicated by listing its generators (â 2 andî, in our case), while the characteristic fragment of the bush displacement pattern is presented next to the colon. The bush symmetry group G fully determines the form (displacement pattern) of the bush B[G] (see, for example, [3,9]). Indeed, in the case of the bush B[â 2 ,î], it is easy to show that this form, X = {A(t), −A(t), A(t), −A(t)}, can be obtained as the general solution to the following linear algebraic equation representing the invariance of the configuration vector X:ĝ 1 X = X,ĝ 2 X = X, whereĝ 1 =â 2 andĝ 2 =î are the generators of the group G. In our previous papers we often write these invariance conditions for the bush B[G] in the form…”

Section: X(t) = {0 A(t) B(t) 0 −B(t) −A(t) | }mentioning

confidence: 99%

“…Substituting the vector X(t) in the form (61) into the equationîX(t) = X(t), we obtain z(t) ≡ −x(t), y(t) ≡ −y(t) and, therefore, y(t) ≡ 0. The displacement pattern for the bush B[â 3 ,î] then can be written as follows:…”

Section: E Further Decomposition Of Linearized Systems Based On Highmentioning

confidence: 99%

“…Low-dimensional bushes in mechanical systems with various kinds of symmetry and structures were studied in [1, 2,3,4,5,6,7,8,9]. The problem of bush stability was discussed in [3,7,8,9].…”

Section: Introductionmentioning

confidence: 99%

“…Let us emphasize that group-theoretical methods developed in our papers [1, 2,3] can be applied efficiently not only to the monoatomic chains (as was illustrated in [8,9]), but to all other physical systems with discrete symmetry groups (see, [1,2,3,4,5,6,7]). …”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Stability analysis of dynamical regimes in nonlinear systems with discrete symmetries

Chechin

Zhukov

2006

Phys. Rev. E

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We present a theorem that allows to simplify linear stability analysis of periodic and quasiperiodic nonlinear regimes in N -particle mechanical systems (both conservative and dissipative) with different kinds of discrete symmetry. This theorem suggests a decomposition of the linearized system arising in the standard stability analysis into a number of subsystems whose dimensions can be considerably less than that of the full system. As an example of such simplification, we discuss the stability of bushes of modes (invariant manifolds) for the Fermi-Pasta-Ulam chains and prove another theorem about the maximal dimension of the above mentioned subsystems.

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Delocalized Nonlinear Vibrational Modes in Graphene: Second Harmonic Generation and Negative Pressure

Korznikova

Shcherbinin

Ryabov

et al. 2018

Physica Status Solidi (b)

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With the help of molecular dynamics simulations, delocalized nonlinear vibrational modes (DNVM) in graphene are analyzed. Such modes are dictated by the lattice symmetry, they are exact solutions to the atomic equations of motion, regardless the employed interatomic potential and for any mode amplitude (though for large amplitudes they are typically unstable). In this study, only one-and two-component DNVM are analyzed, they are reducible to the dynamical systems with one and two degrees of freedom, respectively. There exist 4 one-component and 12 two-component DNVM with in-plane atomic displacements. Any two-component mode includes one of the one-component modes. If the amplitudes of the modes constituting a two-component mode are properly chosen, periodic in time vibrations are observed for the two degrees of freedom at frequencies ω and 2ω, that is, second harmonic generation takes place. For particular DNVM, the higher harmonic can have frequency nearly two times larger than the maximal frequency of the phonon spectrum of graphene. Excitation of some of DNVM results in the appearance of negative in-plane pressure in graphene. This counterintuitive result is explained by the rotational motion of carbon hexagons. Our results contribute to the understanding of nonlinear dynamics of the graphene lattice.

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Interactions between normal modes in nonlinear dynamical systems with discrete symmetry. Exact results

Cited by 93 publications

References 16 publications

Symmetric invariant manifolds in the Fermi–Pasta–Ulam lattice

Symmetric invariant manifolds in the Fermi–Pasta–Ulam lattice

Stability analysis of dynamical regimes in nonlinear systems with discrete symmetries

Delocalized Nonlinear Vibrational Modes in Graphene: Second Harmonic Generation and Negative Pressure

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