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Invariant Circles and Phase Portraits of Cubic Vector Fields on the Sphere

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Abstract

In this paper, we characterize and study dynamical properties of cubic vector fields on the sphere \(\mathbb {S}^2 = \{(x, y, z) \in \mathbb {R}^3 ~|~ x^2+y^2+z^2 = 1\}\). We start by classifying all degree three polynomial vector fields on \(\mathbb {S}^2\) and determine which of them form Kolmogorov systems. Then, we show that there exist completely integrable cubic vector fields on \(\mathbb {S}^2\) and also study the maximum number of various types of invariant great circles for homogeneous cubic vector fields on \(\mathbb {S}^2\). We find a tight bound in each case. Further, we also discuss phase portraits of certain cubic Kolmogorov vector fields on \(\mathbb {S}^2\).

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Notes

  1. SO(3) is the group of all rotations about the origin in \(\mathbb {R}^3\).

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Acknowledgements

The authors thank the anonymous reviewers for several helpful comments and suggestions. The first author was supported by a Senior Research Fellowship of the University Grants Commission of India for the duration of this work. The second author is supported by the Prime Minister’s Research Fellowship, Government of India. The third author thanks ‘ICSR office IIT Madras’ for SEED research grant.

Funding

Funding was provided by National Natural Science Foundation of China (Grant No. 42274165).

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Madras, Chennai, India

    Joji Benny, Supriyo Jana & Soumen Sarkar

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  1. Joji Benny
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  2. Supriyo Jana
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JB, SJ, and SS contributed equally to this paper.

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Correspondence to Joji Benny.

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Benny, J., Jana, S. & Sarkar, S. Invariant Circles and Phase Portraits of Cubic Vector Fields on the Sphere. Qual. Theory Dyn. Syst. 23, 121 (2024). https://doi.org/10.1007/s12346-024-00979-0

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