Non-linear normal modes for systems with discrete symmetry

Chechin, G. M.; Sakhnenko, V. P.; Stokes, Harold T.; Smith, Anthony D.; Hatch, Dorian M.

doi:10.1016/s0020-7462(99)00037-2

Cited by 28 publications

(64 citation statements)

References 23 publications

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“…Taking into account the above results (see (116), (117), (121), (122)), we can conclude that in the case N = 12, the linearized systemδ = J(t) · δ for studying the stability of the bush B[â 4 ,î] splits into two 2 × 2 and two 4 × 4 independent systems of differential equations of the second order. In the real form, these systems can be written as follows:…”

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

confidence: 82%

“…2 we reproduce the stability diagram for the two-dimensional bush B[â 4 ,î] from our paper [9]. This diagram corresponds to the FPU-α chain with N = 12 particles.…”

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

confidence: 99%

“…Now, let us consider the stability of the two-dimensional bush B[â 4 ,î] that can be determined by the displacement pattern [9] …”

Section: Consideringmentioning

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

confidence: 82%

“…2 we reproduce the stability diagram for the two-dimensional bush B[â 4 ,î] from our paper [9]. This diagram corresponds to the FPU-α chain with N = 12 particles.…”

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

confidence: 99%

“…Now, let us consider the stability of the two-dimensional bush B[â 4 ,î] that can be determined by the displacement pattern [9] …”

Section: Consideringmentioning

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%