Publication detail

Instable trivial solution of autonomous differential systems with quadratic right-hand sides in a cone

DIBLÍK, J. ŠMARDA, Z. SVOBODA, Z. KHUSAINOV, D.

Original Title

Instable trivial solution of autonomous differential systems with quadratic right-hand sides in a cone

English Title

Instable trivial solution of autonomous differential systems with quadratic right-hand sides in a cone

Type

journal article - other

Language

en

Original Abstract

The present investigation deals with global instability of a general n-dimensional system of ordinary differential equations with quadratic right-hand sides. The global instability of the zero solution in a given cone is proved by Chetaevs method, assuming that the matrix of linear terms has a simple positive eigenvalue and the remaining eigenvalues have negative real parts. The sufficient conditions for global instability obtained are formulated by inequalities involving norms and eigenvalues of auxiliary matrices. In the proof, a result is used on the positivity of a general third-degree polynomial in two variables to estimate the sign of the full derivative of an appropriate function in a cone.

English abstract

The present investigation deals with global instability of a general n-dimensional system of ordinary differential equations with quadratic right-hand sides. The global instability of the zero solution in a given cone is proved by Chetaevs method, assuming that the matrix of linear terms has a simple positive eigenvalue and the remaining eigenvalues have negative real parts. The sufficient conditions for global instability obtained are formulated by inequalities involving norms and eigenvalues of auxiliary matrices. In the proof, a result is used on the positivity of a general third-degree polynomial in two variables to estimate the sign of the full derivative of an appropriate function in a cone.

Keywords

instability, general n-dimensional system of ordinary differential equations with quadratic right-hand sides, the zero solution, cone, Chetaevs method

RIV year

2011

Released

15.03.2011

Pages from

1

Pages to

23

Pages count

23

BibTex


@article{BUT49861,
  author="Denys {Khusainov} and Josef {Diblík} and Zdeněk {Svoboda} and Zdeněk {Šmarda}",
  title="Instable trivial solution of autonomous differential systems with quadratic right-hand sides in a cone",
  annote="The present investigation deals with global instability of a general n-dimensional system of ordinary differential equations with quadratic right-hand sides. The global instability of the zero solution in a given cone is proved by Chetaevs method, assuming that the matrix of linear
terms has a simple positive eigenvalue and the remaining eigenvalues have negative real parts. The sufficient conditions for global instability obtained are formulated by inequalities involving norms and eigenvalues of auxiliary matrices. In the proof, a result is used on the positivity of a general third-degree polynomial in two variables to estimate the sign of the full derivative of an appropriate function in a cone.",
  chapter="49861",
  journal="Abstract and Applied Analysis",
  number="Article ID 15491",
  volume="2011",
  year="2011",
  month="march",
  pages="1--23",
  type="journal article - other"
}