On the number of hyperelliptic limit cycles of Liénard systems. (English) Zbl 1453.34045
This paper is devoted to a study of the maximum number \(H(m, n)\) of hyperelliptic limit cycles for the Liénard system
\[\dot{x}=y, \quad \dot{y}=-f_m(x)y-g_n(x),\]
where \(f_m(x)\) and \(g_n(x)\) are real polynomials of degree \(m\) and \(n\) respectively.
A limit cycle is called an algebraic limit cycle if it is contained in an invariant algebraic curve defined by equation \(F(x, y) = 0\). If \(F(x, y)\) takes the form \(F(x, y) = (y + P(x))^2 - Q(x)\), where \(P\) and \(Q\) are polynomials, then the invariant curve is called as hyperelliptic. Correspondingly, a limit cycle is said to be hyperelliptic if it is contained in an invariant hyperelliptic curve.
The main results of the paper represent the upper as well as the lower bounds for \(H(m,n)\). The authors claim that in most cases these bounds are sharp.
A limit cycle is called an algebraic limit cycle if it is contained in an invariant algebraic curve defined by equation \(F(x, y) = 0\). If \(F(x, y)\) takes the form \(F(x, y) = (y + P(x))^2 - Q(x)\), where \(P\) and \(Q\) are polynomials, then the invariant curve is called as hyperelliptic. Correspondingly, a limit cycle is said to be hyperelliptic if it is contained in an invariant hyperelliptic curve.
The main results of the paper represent the upper as well as the lower bounds for \(H(m,n)\). The authors claim that in most cases these bounds are sharp.
Reviewer: Alexander Grin (Grodno)
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 |
| 34A34 | Nonlinear ordinary differential equations and systems |
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