A new class of quadratic systems. (English) Zbl 0733.58037

The qualitative features of the phase portraits of the class of quadratic systems given by \[ \dot x=ax+by+(Dx+Ey)(Ax+By),\quad \dot y=cx+dy+(Dx+Ey)(Cx+Ay), \] is studied. For example, it is shown that if such a system has at most two critical points at infinity, then either it has a center or it has at most one periodic orbit. If there is a unique periodic orbit it is a hyperbolic limit cycle. A nice application of the theory developed in the paper is made to a quadratic system which arises in the study of shear flow of a non-Newtonian fluid.


37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
Full Text: DOI


[1] Coppel, W. A., A survey of quadratic systems, J. Differential Equations, 2, 293-304 (1966) · Zbl 0143.11903
[2] Coppel, W. A., The limit cycle configurations of quadratic systems, (Pitman Research Notes in Mathematics, Vol. 157 (1987)), 52-65 · Zbl 0328.34007
[3] Coppel, W. A., Some quadratic systems with at most one limit cycle, Dynamics Reported, 2, 61-88 (1989) · Zbl 0674.34026
[4] Coppel, W. A., Quadratic systems with a degenerate critical point, Bull. Austral. Math. Soc., 38, 1-10 (1988) · Zbl 0634.34013
[6] Ryčkov, G. S., Some criteria for the presence and absence of limit cycles in a second order dynamical system, Sibirsk. Mat. Zh., 7, 1425-1431 (1966), [Russian] · Zbl 0148.33502
[7] Sibirskii, K. S., Introduction to the Algebraic Theory of Invariants of Differential Equations (1982), Shtiintsa, [Russian] · Zbl 0559.34046
[9] Yanqian, Ye, Theory of Limit Cycles, (Translations of Mathematical Monographs, Vol. 66 (1986), Amer. Math. Soc: Amer. Math. Soc Providence, RI) · Zbl 0969.34026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.