Glasnik Matematicki, Vol. 55, No. 2 (2020), 351-366.

PARTIAL QUALITATIVE ANALYSIS OF PLANAR 𝓐Q-RICCATI EQUATIONS

Borut Zalar, Brigita Ferčec, Yilei Tang and Matej Mencinger

University of Maribor, Faculty of civil engineering, Transportation engineering and architecture, Smetanova 17, 2000 Maribor, Slovenia
e-mail: borut.zalar@um.si

University of Maribor, Faculty of energy technology, Hočevarjev trg 1, 8270 Krško, Slovenia
and
Center for applied mathematics and theoretical physics, University of Maribor, Mladinska 3, 2000 Maribor, Slovenia
e-mail: brigita.fercec@um.si

School of mathematical sciences, Shanghai Jiao Tong University, 800 Dongchuan Road, Minhang District Shanghai, 200240, China
e-mail: mathtyl@sjtu.edu.cn

University of Maribor, Faculty of civil engineering, transportation engineering and architecture, Smetanova 17, 2000 Maribor, Slovenia
and
Institute of mathematics, physics and mechanics, Jadranska 19, 1000 Ljubljana, Slovenia
and
Center for applied mathematics and theoretical physics, University of Maribor, Mladinska 3, 2000 Maribor, Slovenia
e-mail: matej.mencinger@um.si

Abstract.   If we view the field of complex numbers as a 2-dimensional commutative real algebra, we can consider the differential equation z'=az2+bz+c as a particular case of 𝓐- Riccati equations z'=a · (z · z)+b · z+c where 𝓐=( ℝn,·) is a commutative, possibly nonassociative algebra, a,b,c∈𝓐 and z:I → 𝓐 is defined on some nontrivial real interval. In the case 𝓐=ℂ, the nature of (at most two) critical points can be described using purely algebraic conditions involving involution * of ℂ. In the present paper we study the critical points of 𝓛(π)- Riccati equations, where 𝓛(π) is the limit case of the so-called family of planar Lyapunov algebras, which characterize 2-dimensional homogeneous systems of quadratic ODEs with stable origin. The number of possible critical points is 1, 3 or ∞, depending on coefficients. The nature of critical points is also completely described. Finally, simultaneous stability of the origin is considered for homogeneous quadratic part corresponding to algebras 𝓛(θ).

2010 Mathematics Subject Classification.   34A34, 34C60, 17A99

Key words and phrases.   Differential systems, Riccati equation, commutative algebra, singular points, stability, center problem

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https://doi.org/10.3336/gm.55.2.11

References:

  1. J.C. Artés, J. Llibre, D. Schlomiuk and N. Vulpe, Global topological configurations of singularities for the whole family of quadratic differential systems, Qual. Theory Din. Syst. 19 (2020), no. 51, 32 pp.
    MathSciNet     CrossRef

  2. H. Boujemaa and S. El Qotbi, On unbounded polynomial dynamical systems, Glas. Mat. Ser. III 53(73) (2018), 343-357.
    MathSciNet     CrossRef

  3. H. Boujemaa, S. El Qotbi and H. Rouiouih, Stability of critical points of quadratic homogeneous dynamical systems, Glas. Mat. Ser. III 51(71) (2016), 165-173.
    MathSciNet     CrossRef

  4. H. Boujemaa, M. Rachidi and A. Micali, On a class of nonassociative algebras: a reduction theorem for their associated quadratic systems, Algebras Groups Geom. 19 (2002), 73-83.
    MathSciNet    

  5. J. Chavarriga and J. Giné, Integrability of a linear center perturbed by a fifth degree homogeneous polynomial, Publ. Mat. 41 (1997) 335-356.
    MathSciNet     CrossRef

  6. H. Dulac, Détermination et intégration d'une certaine classe d'équations différentielles ayant pour point singulier un centre, Bull. Sci. Math. (2) 32 (1908), 230-252.

  7. F. Dumortier, J. Llibre and J.C. Artés, Qualitative theory of planar differential systems, Springer, Berlin, 2006.
    MathSciNet    

  8. M. Han, T. Petek and V. Romanovski. Reversibility in polynomial systems of ODE's, Appl. Math. Comput. 338 (2018) 55-71.
    MathSciNet     CrossRef

  9. N.C. Hopkins, Quadratic differential equations in the complex domain I, Trans. Amer. Math. Soc. 367 (2015), 6771-6782.
    MathSciNet     CrossRef

  10. Y. Krasnov and I. Messika, Differential and integral equations in algebra, Funct. Differ. Equ. 21 (2014), 137-146.
    MathSciNet    

  11. Y. Krasnov, Properties of ODEs and PDEs in algebras, Complex Anal. Oper. Theory 7 (2013), 623-634.
    MathSciNet     CrossRef

  12. Y. Krasnov, Differential equations in algebras, in: Hypercomplex analysis, Birkhauser Verlag, Basel, 2009, 187-205.
    MathSciNet    

  13. A. M. Liapunov, Stability of motion. With a contribution by V. Pliss and an introduction by V. P. Basov. Translated from Russian by F. Abramovici and M. Shimshoni, Academic Press, New York, 1966.
    MathSciNet    

  14. N.G. Lloyd and J.M. Pearson, Computing centre condition for certain cubic systems, J. Comput. Appl. Math. 40 (1992), 323-336.
    MathSciNet     CrossRef

  15. K.E. Malkin, Criteria for the center for a certain differential equation, Volž. Mat. Sb. Vyp. 2 (1964), 87-91.
    MathSciNet    

  16. L. Markus, Quadratic differential equations and non-associative algebras, in: 1960 Contributions to the theory of nonlinear oscillations, Vol. V, Princeton Univ. Press, Princeton, 1960, 185-213.

  17. M. Mencinger and B. Zalar, A class of nonassociative algebras arising from quadratic ODEs, Comm. Algebra. 33 (2005), 807-828.
    MathSciNet     CrossRef

  18. M. Mencinger, On stability of the origin in quadratic systems of ODEs via Markus approach, Nonlinearity 16 (2003), 201-218.
    MathSciNet     CrossRef

  19. M. Mencinger, On the stability of Riccati diferential equation X'=TX+Q(X) in ℝn, Proc. Edinb. Math. Soc. (2) 45 (2002), 601-615.
    MathSciNet     CrossRef

  20. M. Mencinger and B. Zalar, On stability of critical points of quadratic differential equations in nonassociative algebras, Glas. Mat. Ser. III 38(58) (2003), 19-27.
    MathSciNet     CrossRef

  21. M. Mencinger and B. Zalar, Planar Lyapunov algebras, Algebra Colloq. 27 (2020), 433-446.
    MathSciNet     CrossRef

  22. H. Poincaré, Mémoire sur les courbes définies par une équation différentielle. J. Math. Pures et. Appl. (Sér. 3) 7 (1881), 375-422; (Sér. 3) 8 (1882), 251-296; (Sér. 4) 1 (1885), 167-244; (Sér. 4) 2 (1886), 151-217.

  23. V.G. Romanovski and D.S. Shafer, The center and cyclicity problems: a computational algebra approach, Birkhäuser, Boston, 2009.
    MathSciNet     CrossRef

  24. A. P. Sadovskii, Solution of the center and focus problem for a cubic system of nonlinear oscillations, Differential Equations 33 (1997), 236-244.
    MathSciNet    

  25. A. Sagle and K. Schmitt, On second-order quadratic systems and algebras, Differential Integral Equations 24 (2011), 877-894.
    MathSciNet    

  26. A. Sagle and K. Schmitt, Remarks on second-order quadratic systems in algebras, Electron. J. Differential Equations (2017), no. 248, 9 pp.
    MathSciNet    

  27. F. Takens, Singularities of vector fields, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 47-100.
    MathSciNet     CrossRef

  28. S. Walcher, Algebras and differential equations, Hadronic Press, Inc., Palm Harbor, 1991.
    MathSciNet    

  29. Y.Q. Ye et al, Theory of limit cycles, AMS, Providence, 1986.
    MathSciNet    
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