Inverse Jacobi multipliers: recent applications in dynamical systems. (English) Zbl 1348.37083
Ibáñez, Santiago (ed.) et al., Progress and challenges in dynamical systems. Proceedings of the international conference “Dynamical systems: 100 years after Poincaré”, Gijón, Spain, September 3–7, 2012. Berlin: Springer (ISBN 978-3-642-38829-3/hbk; 978-3-642-40137-4/ebook). Springer Proceedings in Mathematics & Statistics 54, 127-141 (2013).
Summary: In this paper we show novel applications of the inverse Jacobi multiplier focusing on questions of bifurcations and existence of periodic solutions admitted by both autonomous and non-autonomous systems of ordinary differential equations. In the autonomous case we focus on dimension \(n \geq 3\) whereas in the non-autonomous we study the cases with \(n \geq 2\). We summarize results already published and additionally we state some recent results to appear. The principal object of this research is two fold: first to prove the existence and smoothness of inverse Jacobi multiplier \(V\) in the region of interest in the phase space and second to show that the invariant set under the flow given by the zero-set of an inverse Jacobi multiplier contains under some assumptions orbits which are relevant in its phase portrait such as periodic orbits, limit cycles, stable, unstable and center manifolds and so on. In the non-autonomous \(T\)-periodic case we show some relationships between \(T\)-periodic orbits and \(T\)-periodic inverse Jacobi multipliers.
For the entire collection see [Zbl 1275.37001].
For the entire collection see [Zbl 1275.37001].
MSC:
| 37G15 | Bifurcations of limit cycles and periodic orbits in dynamical systems |
| 37C10 | Dynamics induced by flows and semiflows |
| 37C60 | Nonautonomous smooth dynamical systems |
| 37C27 | Periodic orbits of vector fields and flows |
| 34C23 | Bifurcation theory for ordinary differential equations |
| 37C80 | Symmetries, equivariant dynamical systems (MSC2010) |
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