Families of vector fields with many numerical invariants

Nataliya Goncharuk; Yury Kudryashov

doi:10.3934/dcds.2021114

Article Contents

2022, Volume 42, Issue 1: 239-259. Doi: 10.3934/dcds.2021114

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Families of vector fields with many numerical invariants

Nataliya Goncharuk and
Yury Kudryashov

University of Toronto Mississauga, 3359 Mississauga Road, Mississauga, ON, L5L 1C6, Canada

Received: March 2021

Revised: May 2021

Early access: August 2021

Published: January 2022

Both authors are partially supported by RFBR grant No. 20-01-00420

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Abstract

We study bifurcations in finite-parameter families of vector fields on $S^2$. Recently, Yu. Ilyashenko, Yu. Kudryashov, and I. Schurov provided examples of (locally generic) structurally unstable $3$-parameter families of vector fields: topological classification of these families admits at least one numerical invariant. They also provided examples of $(2D+1)$-parameter families such that the topological classification of these families has at least $D$ numerical invariants and used those examples to construct families with functional invariants of topological classification.

In this paper, we construct locally generic $4$-parameter families with any prescribed number of numerical invariants and use them to construct $5$-parameter families with functional invariants. We also describe a locally generic class of $3$-parameter families with a tail of an infinite number sequence as an invariant of topological classification.

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Mathematics Subject Classification: Primary: 34C23, 37G99; Secondary: 37E35.

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Figure 1. A vector field with a polycycle "tears of the heart". This figure was first published in Nonlinearity [14]. A similar figure was earlier published in Invent. Math. [17]

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Figure 2. Vector fields with structurally unstable unfoldings from [14]. This figure was first published in Nonlinearity [14]

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Figure 3. Degenerate vector fields with structurally unstable unfoldings

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Figure 4. A vector field of class $ \mathbf{WG}_{1, 1}$

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Figure 5. Ensemble "lips"

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Figure 6. A vector field of class $ \mathbf{LEG}_{2}$

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Figure 7. An unfolding of a vector field $v_{0}\in \mathbf{LEG}_{2}$ satisfying assertions of Lemma 4.10 for $k = 2$

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