LIE SYMMETRIES AND THE CENTER PROBLEM

Jaume Giné

doi:10.11948/2011033

2011 Volume 1 Issue 4

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Jaume Giné. LIE SYMMETRIES AND THE CENTER PROBLEM[J]. Journal of Applied Analysis & Computation, 2011, 1(4): 487-496. doi: 10.11948/2011033

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Jaume Giné. LIE SYMMETRIES AND THE CENTER PROBLEM[J]. Journal of Applied Analysis & Computation, 2011, 1(4): 487-496. doi: 10.11948/2011033

LIE SYMMETRIES AND THE CENTER PROBLEM

Jaume Giné

Departament de Matemàtica. Universitat de Lleida. Avda. Jaume Ⅱ, 69. 25001. Lleida. Spain

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Abstract

Abstract

In this short survey we discuss the narrow relation between the center problem and the Lie symmetries. It is well known that an analytic vector field X having a non-degenerate center has a non-trivial analytic Lie symmetry in a neighborhood of it, i.e. there exists an analytic vector field Y such that[X, Y]=µX. The same happens for a nilpotent center with an analytic first integral as can be seen from the recent results about nilpotent centers. From the recent results for nilpotent and degenerate centers it also can be proved that any nilpotent or degenerate center has a trivial smooth (of class C^∞) Lie symmetry. It remains as open problem if there always exists also a non-trivial Lie symmetry for any nilpotent and degenerate center.
- Lie symmetries /
- center problem /
- vector fields
MSC: 34C14

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References

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