On boundedness of solutions of quadratic differential equation sets

V.Ye. Belozerov (Dnipropetrovsk National University), https://orcid.org/0000-0003-2888-8876
V.S. Chernyshenko (Dnipropetrovsk National University)
S.V. Chernyshenko (Dnipropetrovsk National University), https://orcid.org/0000-0002-2852-6878


We obtain new sufficient conditions of asymptotic stability on solution cone of the set of usual uniform quadratic differential equations. For non-uniform quadratic sets (with non-zero linear part) we give the conditions of boundedness of solutions. We determine quadratic sets whose solution is bounded for any linear part and for any initial conditions. We provide examples of applications of the method in mathematical ecology.


Belozerov V.Ye., Kapustian V.Ye. Geometric methods of modal control, Kyiv, 1999. (in Russian)

Volterra V. Mathematical theory of struggle for existence, 1976. (in Russian)

Kuzenkov A.A., Chernyshenko S.V. "Dynamics of intraspecies interrelation of races: description by means of Volterra systems", Proc. Conf. MPZIS-2006, Dnipropetrovsk, 2006; pp. 90-91. (in Russian)

Chernyshenko V.S. "Complete topological classification of one model of ecologic concurrency in regular case", Dynamic Systems, 2006; 21: pp. 21-42. (in Russian)

Chernyshenko S.V. "Principal mathematical models of optimal usage of nature", Problems of steppe forestology and forest recultiv. of lands, Dnipropetrovsk, 1998; pp. 78-82. (in Ukrainian)

Belozyorov V.Ye. "On an Invariant Design of Feedback for Bilinear Control Systems of Second Order", Int. J. App. Math. and Comp. Sci., 2001; 11(2): pp. 377-389.

Belozyorov V.Ye. "Design of Linear Feedback for Bilinear Control Systems", Int. J. App. Math. and Comp. Sci., 2002; 12(4): pp. 493-511.

Khalil H.K. Nonlinear Systems, New-Jersey: Prentice Hall, 1996.

Lotka A.G. Elements of physical biology, Baltimore, 1925.

Sachkov Yu.L. "On Invariant Orthants of Bilinear Systems", J. Dynamical and Control Systems, 1998; 4(1): pp. 137-147. doi:10.1023/A:1022829201840

Wang H. "Feedback Stabilization of Bilinear Control Systems", SIAM J. Control and Optimization, 1998; 36(5): pp. 1669-1684. doi:10.1137/S0363012996305498

DOI: https://doi.org/10.15421/240706



  • There are currently no refbacks.

Copyright (c) 2007 V.Ye. Belozerov, V.S. Chernyshenko, S.V. Chernyshenko

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

Registered in


ISSN (Online): 2664-5009
ISSN (Print): 2664-4991