On piecewise linear differential systems with \(n\) limit cycles of arbitrary multiplicities in two zones. (English) Zbl 1418.34035
Summary: In this paper, we demonstrate rich dynamical phenomenon of piecewise linear differential systems having only two zones in the plane. We show that, for any given integer \(n\) and any integer tuple \(m=(m_1,m_2,\dots , m_n)\), \( m_i \ge 0 \), for \(i=1,\dots ,n\), there exists an aforementioned system which possesses exactly \(n\) limit cycles having multiplicities \(m_1\), \(m_2,\dots , m_n\), respectively. (i.e. there are totally \(m_1+m_2+\cdots +m_n\) limit cycles taking into account of multiplicities). Moreover, we can even choose the separation boundary of the zones to be an algebraic curve.
MSC:
| 34A36 | Discontinuous ordinary differential equations |
| 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 |
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