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Journal of Applied Nonlinear Dynamics 13(3) (2024) 431--438 | DOI:10.5890/JAND.2024.09.001

Abdigappar Narmanov, Eldor Rajabov

Department of Mathematics, National University of Uzbekistan, Tashkent, 100174, Uzbekistan

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Abstract

The geometry of orbits of families of smooth vector fields was studied by many mathematicians due to its importance in applications, in the theory of optimal control of dynamic systems, in geometry, and in the theory of foliations. In this paper it is studied geometry of orbits of vector fields in four dimensional Euclidean space. It is shown that orbits generate singular foliation ever regular leaf of which is a surface of negative Gauss curvature and zero normal torsion. In addition, the invariant functions of the considered vector fields are used to find solutions of the two-dimensional heat equation that are invariant under the groups of transformations generated by these vector fields. In the present paper, smoothness is understood as smoothness of the class $C^{\infty } $.

References

1. [1]	Azamov A. and Narmanov A. (2004), On the limit sets of orbits of systems of vector fields, Differential Equations, 40(2), 257-260.

2. [2]	Narmanov A. and Saitova S. (2014), On the geometry of orbits of killing vector fields, Differential Equations, 50(12), 1584-1591.

3. [3]	Narmanov A. and Saitova S. (2017), On the geometry of the reachability set of vector fields, Differential Equations, 53, 311-316.

4. [4]	Narmanov A. and Rajabov E. (2019), On the geometry of orbits of conformal vector fields, Journal of Geometry and Symmetry in Physics, 51, 29-39.

5. [5]	Sussman H. (1973), Orbits of families of vector fields and integrability of distributions, Transactions of the AMS, 180, 171-188.

6. [6]	Olver P. (1993), Applications of Lie Groups to Differential Equations, Springer.

7. [7]	Fomenko, V. (2004), Classification of two-dimensional surfaces with zero normal torsion in four-dimensional spaces of constant curvature, Mathematical Notes, 75(5), 690-701.

8. [8]	Fomenko, V. (1979), Some properties of two-dimensional surfaces with zero normal torsion in $E^{4}$, Matematicheskii Sbornik, 35(2), 251-265.

9. [9]	Kadomcev, S. (1975), A study of certain properties of normal torsion of a two-dimensional surface in four-dimensional space, (Russian), Problems in geometry, 7, 267-278.

10. [10]	Ramazanova, K. (1966), The theory of curvature of $X^{2} $ in $E^{4} $, Izvestiya Vysshikh Uchebnykh Zavedenii Matematika, 6, 137-143

11. [11]	Dorodnitsyn, V.A.E., Knyazeva, I.V., and Svirshchevskii, S.R. (1983), Group properties of the heat equation with source in the two-dimensional and three-dimensional cases, Differential Equations, 19(7), 1215-1223.

12. [12]	Narmanov, O. (2019), Invariant solutions of two dimensional heat equation, Journal of Applied Mathematics and Physics, 7, 1488-1497

13. [13]	Narmanov, O. (2019), Invariant solutions of the two-dimensional heat equation, Bulletin of Udmurt University. Mathe-matics, Mechanics, Computer Science, 29(1), 52-60