Non-algebraic limit cycles for parametrized planar polynomial systems. (English) Zbl 1121.34036
It is well known that planar systems of the form
\[ x'=P_n(x,y)+x R_m(x,y),\quad y'=Q_n(x,y)+y R_m(x,y), \]
where \(P_n, Q_n\) and \(R_m\) are homogeneous polynomials of degrees \(n,n\) and \(m,\) respectively, have at most one limit cycle. This assertion becomes clear by writing them in polar coordinates. In this paper the author develops the tools introduced in [A. Gasull, H. Giacomini and J. Torregrosa, “Explicit non-algebraic limit cycles for polynomial systems.” J. Comput. Appl. Math. 200, No. 1, 448–457 (2007; Zbl 1171.34021)] to study whether this limit cycle is algebraic or not. The particular case \(P_3(x,y)=-(x-y)(x^2-xy+y^2),\) \(Q_3(x,y)=-(x+y)(2x^2-xy+2y^2),\) \(R_4(x,y)=ax^4+bx^2y^2+cy^4\) is studied with detail. Among other things it is proved that if \((a-b+c)(a-2b+4c)\neq 0\), then the only algebraic solutions of the planar system are \(x^2+y^2=0\) and \(2x^2+y^2=0\) and so, the limit cycle, whenever it exists, is non-algebraic.
\[ x'=P_n(x,y)+x R_m(x,y),\quad y'=Q_n(x,y)+y R_m(x,y), \]
where \(P_n, Q_n\) and \(R_m\) are homogeneous polynomials of degrees \(n,n\) and \(m,\) respectively, have at most one limit cycle. This assertion becomes clear by writing them in polar coordinates. In this paper the author develops the tools introduced in [A. Gasull, H. Giacomini and J. Torregrosa, “Explicit non-algebraic limit cycles for polynomial systems.” J. Comput. Appl. Math. 200, No. 1, 448–457 (2007; Zbl 1171.34021)] to study whether this limit cycle is algebraic or not. The particular case \(P_3(x,y)=-(x-y)(x^2-xy+y^2),\) \(Q_3(x,y)=-(x+y)(2x^2-xy+2y^2),\) \(R_4(x,y)=ax^4+bx^2y^2+cy^4\) is studied with detail. Among other things it is proved that if \((a-b+c)(a-2b+4c)\neq 0\), then the only algebraic solutions of the planar system are \(x^2+y^2=0\) and \(2x^2+y^2=0\) and so, the limit cycle, whenever it exists, is non-algebraic.
Reviewer: Armengol Gasull (Barcelona)
MSC:
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 34C07 | Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations |
Citations:
Zbl 1171.34021References:
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