ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

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Tsegelnik V. V. Some analytical properties of solutions of the 3D-dynamical system with square nonlinearity. Izvestiya VUZ. Applied Nonlinear Dynamics, 1998, vol. 6, iss. 5, pp. 84-88. DOI: 10.18500/0869-6632-1998-6-5-84-88

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Language: 
Russian
Heading: 
Deterministic Chaos
Article type: 
Article
UDC: 
517.9
DOI: 
10.18500/0869-6632-1998-6-5-84-88

Some analytical properties of solutions of the 3D-dynamical system with square nonlinearity

Autors: 
Tsegelnik Vladimir Vladimirovich, Belarusian State University of Informatics and Radioelectronics
Abstract: 

We consider the analytical properties of solutions of the 3D—dynamical system, related with а model of turbulence with spectrally narrow range and model а neuron with complex oscillatory activity. The algorithm of forming of е system exact solutions, expressed in terms of the solutions of the third Painleve’s equation, is offered.

Key words: 
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Acknowledgments: 
The work was supported by the Ministry of Education of the Republic of Belarus (project 97-3020).
Reference: 
  1. Vyshkind SYa, Rabinovich MI. The phase stochastization mechanism and the structure of wave turbulence in dissipative media. J. Exp. Theor. Phys. 1976;44(2):292-299.
  2. Bazhenov MV, Rabinovich MI, Rubchinskii LA. A simple model of a neuron with complex oscillatory activity. Izvestiya VUZ. Applied Nonlinear Dinamics. 1996;4(1):33-39. (in Russian).
  3. Bountis TS, Ramani А, Grammaticos В, Dorizzi В. On the complete and partial integrability of non—Hamiltonian systems. Physica A. 1984;128(1-2):268-288. DOI: 10.1016/0378-4371(84)90091-8.
  4. Gromak VI, Tsegelnik VV. On solutions of a system of three differential equations with quadratic nonlinearities. Differential Equations. 1991;27(3):396-402.
  5. Ablowitz MJ, Segur H. Solitons and the Inverse Scattering Transform. Philadelfia: SIAM; 1981. 424 p. DOI: 10.1137/1.9781611970883.
  6. Gromak VI, Lukashevich NA. Analytical Properties of Solutions to the Painlevé Equations. Minsk: Universitetskoe; 1990. 156 p. (in Russian).
  7. Manakov SV. Pulse propagation in a long laser amplifier. JETP Lett. 1982;35(5):238-240.
  8. Zakharov VE, Kuznetsov ЕА, Musher SL. Semiclassical regime of a three-dimensional wave collapse. JETP Lett. 1985;41(3):154-156.
  9. Its AR, Kapaev АА. Method of isomonodrome deformations and coupling formulas for the second Painleve transcendent. Izvestiya: Mathematics. 1987;51(4):878-892.
  10. Ince EL. Ordinary Differential Equations. North Chelmsford: Courier Corporation; 1956. 558 p.
  11. Tsegelnik VV. On a relationship between solutions of the third Painlevé equation. Theor. Math. Phys. 1995;102(3):265-266.
  12. Gromak VI On the transcendence of Painlevé equation. Differential Equations. 1996;32(2):154. DOI: 10.1007/BF01017877.
  13. Giacomini HJ, Repetto СЕ, Zandron ОР. Integrals оf motion for three-dimensional non-Hamiltonian dynamical systems. J. Phys. A: Math. Gen. 1991;24(19):4567-4574. DOI: 10.1088/0305-4470/24/19/020.
Received: 
19.04.1998
Accepted: 
05.10.1998
Published: 
25.02.1999
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