## Abstract

A geometric generalization of contraction theory called <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-contraction was recently developed using <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-compound matrices. In this note, we focus on the relations between <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-contraction and two other generalized contraction frameworks: partial contraction (also known as virtual contraction) and horizontal contraction. We show that in general these three notions of contraction are different. We here provide new sufficient conditions guaranteeing that partial contraction implies horizontal contraction, and that horizontal contraction implies <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-contraction. We use the Andronov-Hopf oscillator to demonstrate some of the theoretical results.

Original language | English |
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Pages (from-to) | 1-8 |

Number of pages | 8 |

Journal | IEEE Transactions on Automatic Control |

DOIs | |

State | Accepted/In press - 2023 |

Externally published | Yes |

## Keywords

- <inline-formula xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"> <tex-math notation="LaTeX">$k$</tex-math> </inline-formula>-contraction
- Andronov-Hopf oscillator
- Behavioral sciences
- compound matrix
- Compounds
- horizontal contraction
- Manifolds
- Oscillators
- partial contraction
- Sufficient conditions
- Synchronization
- Trajectory
- virtual contraction