Volume 7, no. 3Pages 23 - 32
The Form of Key Function in the Problem of Branching of Periodic Extremals with Resonance 1:1:1E.V. Bukhonova
This article contains a method for calculating approximately the standardized key functions in the problem of branching of periodic extremals of a continuously differentiable action functional near its minimum. The periodic extremals of such functionals are used as a prototype for periodic oscillations of dynamical systems, ferroelectric crystal phases, nonlinear periodic waves, as so on. Recently Karpova, Kopytin, Derunova, and Sapronov studied cycle bifurcations in dynamical systems using key equations and key functions in the cases of double resonances $p_1:p_2:p_3$ with $p_1<p_2<p_3$. This article deals with the poorly understood case $p_1=p_2=p_3 = 1$. As a demonstration model, we consider an order six ODE. We use the Lyapunov-Schmidt method. Full text
- continuously differentiable functional; extremal; circular symmetry; resonance; bifurcation; Lyapunov-Schmidt method.
- 1. Darinskii B.M., Sapronov Yu. I., Zarev S.L. Bifurcations of Extremals of Fredholm Functionals. Journal of Mathematical Sciences, 2007, vol. 145, issue 6, pp. 5311-5453. DOI: 10.1007/s10958-007-0356-2
2. Karpova A.P., Kopytin N.A., Sapronov Yu.I. [The Key Equation in Dynamical Systems with 2-tuple Resonances]. Matematicheskie modeli i operatornye uravneniya [Symbolic Model and Operator Equations], 2009, vol. 6, pp. 51-58
3. Derunova E. V., Sapronov Yu. I. [Three-Mode Extremals' Bifurcations from the Minimum Point of Fredholm Functional in Terms of Circular Symmetric]. Vestnik VGU. Seriya: Fizika. Matematika [Proceedings of Voronezh State University. Series: Physics. Mathematics], 2014, no. 1, pp. 64-77.
4. Darinskiy B.M., Kolesnikova I.V., Sapronov Yu.I. Branching phases crystal determines the thermodynamic potential of the sixth order. Sistemy upravleniya i informatsionnye tekhnologii [Control Systems and Information Technology], 2009, no. 1 (35), pp. 72-76.
5. Zachepa A.V., Sapronov Yu. I. The Bifurcation of Extremals Fredholm Functional from a Degenerate Minimum Point with Feature 3-dimensional Assembly. Sbornik trudov matematicheskogo fakul'teta VGU [Proceedings of the Math. Faculty of VSU], 2005, issue 9, pp. 57-71.
6. Strygin V.V., Severin G.Yu. Bifurcation of Self-Oscillations of Small Synchronous Two Dynamical Systems with Similar Frequencies. Vestnik VGU. Seriya: Sistemnyy Analiz i Informatsionnye Tekhnologii [Proceedings of Voronezh State University. Series: System Analysis and Information Technologies], 2006. no. 2. pp. 36-45.
7. Arnold V.I., Varchenko A.N., Husein-Zade S.M. Features Differentiable Maps. Classification of Critical Points Caustics and Wave Fronts. Moskow, 1982.
8. Siersma D. Singularities of Functions on Boundaries, Corners, etc. Quart. J. Oxford Ser., 1981, vol. 32, no. 125, pp. 119-127. DOI: 10.1093/qmath/32.1.119
9. Gnezdilov A.V. [Bifurcation of Critical Tori for Functionals with 3-circular Symmetry]. Funktsional'nyy analiz [Function Analysis], 2000, vol. 34, no. 1, pp. 83-86.
10. Borisovich Yu.G., Bliznaykov N.M., Izrelevich Ya.A., Fomenko T.N. Vvedenie v topologiyu [Introduction to Topology]. Moskow, Nauka, 1995. 416 p.