Bifurcations in spatially distributed chains of two-dimensional systems of equations

Kashchenko, S. A.

doi:10.1070/rm9986

Cited by 6 publications

(5 citation statements)

References 6 publications

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“…Thus, we can say that critical cases of infinite dimension are realised. Critical cases of this type have been considered in [18][19][20]. The methodology of these works will be essentially used here.…”

Section: Functionmentioning

confidence: 99%

“…Functions u j are 2π γ(δ) + εθγ (δ) −1 periodic in t and 2π periodic in x. The expression for u 1 is the same as (20), with the only difference that now ξ(τ, x) and η(τ, x) are unknown complex amplitudes. We substitute ( 21) into ( 5) and successively collect coefficients of the same powers of ε.…”

Section: Asymptotic Behaviour Of Irregular Solutionsmentioning

confidence: 99%

“…We substitute ( 21) into ( 5) and successively collect coefficients of the same powers of ε. In the first step, we arrive at an equation for u 1 , whose solution is presented in (20). In the second step, we obtain the same equation for u 2 :…”

Section: Asymptotic Behaviour Of Irregular Solutionsmentioning

confidence: 99%

“…We use the technique for constructing quasi-normal forms developed in [20,38]. We seek the asymptotes of the solutions to the boundary value problem (A20)-(A22) based on solution (A26).…”

Section: Appendix Amentioning

confidence: 99%

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Asymptotics of Regular and Irregular Solutions in Chains of Coupled van der Pol Equations

Kashchenko

2023

Mathematics

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Chains of coupled van der Pol equations are considered. The main assumption that motivates the use of special asymptotic methods is that the number of elements in the chain is sufficiently large. This allows moving from a discrete system of equations to the use of a continuity argument and obtaining an integro-differential boundary value problem as the initial model. In the study of the behaviour of all its solutions in a neighbourhood of the equilibrium state, infinite-dimensional critical cases arise in the problem of the stability of solutions. The main results include the construction of special families of quasi-normal forms, namely non-linear boundary value problems of either Schrödinger or Ginzburg–Landau type. Their solutions make it possible to determine the main terms of the asymptotic expansion of both regular and irregular solutions to the original system. The main goal is the study of chains with diffusion- and advective-type couplings, as well as fully connected chains.

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Section: Functionmentioning

confidence: 99%

Section: Asymptotic Behaviour Of Irregular Solutionsmentioning

confidence: 99%

Section: Asymptotic Behaviour Of Irregular Solutionsmentioning

confidence: 99%

“…We use the technique for constructing quasi-normal forms developed in [20,38]. We seek the asymptotes of the solutions to the boundary value problem (A20)-(A22) based on solution (A26).…”

Section: Appendix Amentioning

confidence: 99%

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Asymptotics of Regular and Irregular Solutions in Chains of Coupled van der Pol Equations

Kashchenko

2023

Mathematics

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“…We use the technique developed in [11][12][13][14][15]. It is based on the assumption that a certain set of solutions to the boundary value problem ( 4) and ( 6) can be represented as an asymptotic series…”

Section: Justification Of Theoremmentioning

confidence: 99%

New Irregular Solutions in the Spatially Distributed Fermi–Pasta–Ulam Problem

Kashchenko

Tolbey

2021

Mathematics

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For the spatially-distributed Fermi–Pasta–Ulam (FPU) equation, irregular solutions are studied that contain components rapidly oscillating in the spatial variable, with different asymptotically large modes. The main result of this paper is the construction of families of special nonlinear systems of the Schrödinger type—quasinormal forms—whose nonlocal dynamics determines the local behavior of solutions to the original problem, as t→∞. On their basis, results are obtained on the asymptotics in the main solution of the FPU equation and on the interaction of waves moving in opposite directions. The problem of “perturbing” the number of N elements of a chain is considered. In this case, instead of the differential operator, with respect to one spatial variable, a special differential operator, with respect to two spatial variables appears. This leads to a complication of the structure of an irregular solution.

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Dynamics of chains of a large number of fully coupled oscillators, as well as oscillators with one-way and two-way delay couplings

Kashchenko

2024

Partial Differential Equations in Applied Mathematics

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Bifurcations in spatially distributed chains of two-dimensional systems of equations

Cited by 6 publications

References 6 publications

Asymptotics of Regular and Irregular Solutions in Chains of Coupled van der Pol Equations

Asymptotics of Regular and Irregular Solutions in Chains of Coupled van der Pol Equations

New Irregular Solutions in the Spatially Distributed Fermi–Pasta–Ulam Problem

Dynamics of chains of a large number of fully coupled oscillators, as well as oscillators with one-way and two-way delay couplings

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