Multiplicity of limit cycles and analytic \(m\)-solutions for planar differential systems. (English) Zbl 1345.34037
Summary: This work deals with limit cycles of real planar analytic vector fields. It is well-known that given any limit cycle \(\Gamma\) of an analytic vector field it always exists a real analytic function \(f_0(x,y)\), defined in a neighborhood of \(\Gamma\), and such that \(\Gamma\) is contained in its zero level set.
In this work we introduce the notion of \(f_0(x,y)\) being an \(m\)-solution, which is a merely analytic concept. Our main result is that a limit cycle \(\Gamma\) is of multiplicity \(m\) if and only if \(f_0(x,y)\) is an \(m\)-solution of the vector field. We apply it to study in some examples the stability and the bifurcation of periodic orbits from some non-hyperbolic limit cycles.
In this work we introduce the notion of \(f_0(x,y)\) being an \(m\)-solution, which is a merely analytic concept. Our main result is that a limit cycle \(\Gamma\) is of multiplicity \(m\) if and only if \(f_0(x,y)\) is an \(m\)-solution of the vector field. We apply it to study in some examples the stability and the bifurcation of periodic orbits from some non-hyperbolic limit cycles.
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 |
| 34C23 | Bifurcation theory for ordinary differential equations |
| 37G15 | Bifurcations of limit cycles and periodic orbits in dynamical systems |
Keywords:
multiple limit cycle; planar vector field; stability; bifurcation; invariant algebraic curve; \(m\)-solutionReferences:
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