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Rigid centres on the center manifold of tridimensional differential systems
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 152 / Issue 4 / August 2022
- Published online by Cambridge University Press:
- 07 September 2021, pp. 1058-1080
- Print publication:
- August 2022
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- Article
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Motivated by the definition of rigid centres for planar differential systems, we introduce the study of rigid centres on the center manifolds of differential systems on
$\mathbb {R}^{3}$. On the plane, these centres have been extensively studied and several interesting results have been obtained. We present results that characterize the rigid systems on 
$\mathbb {R}^{3}$ and solve the centre-focus problem for several families of rigid systems.
CENTER MANIFOLDS FOR PARTIALLY HYPERBOLIC SETS WITHOUT STRONG UNSTABLE CONNECTIONS
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 15 / Issue 4 / October 2016
- Published online by Cambridge University Press:
- 11 March 2015, pp. 785-828
- Print publication:
- October 2016
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- Article
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We consider compact sets which are invariant and partially hyperbolic under the dynamics of a diffeomorphism of a manifold. We prove that such a set
$K$ is contained in a locally invariant center submanifold if and only if each strong stable and strong unstable leaf intersects 
$K$ at exactly one point.
