<p>We investigate periodic orbits of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$C^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation> autonomous vector fields in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> using inverse Jacobi multipliers that may depend explicitly on time. We establish a localization principle for <i>T</i>-periodic orbits in arbitrary dimension, extending known planar results and deriving nonexistence conditions through the relation between the time-slices <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$V(0, \cdot )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$V(T, \cdot )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We further characterize hyperbolicity and orbital stability, including a decomposition of characteristic multipliers along invariant surfaces associated with autonomous inverse Jacobi multipliers. A test for the algebraicity of periodic orbits in 3-dimensional vector fields is given based on non-autonomous inverse Jacobi multipliers. The interplay between normalizers, inverse Jacobi multipliers and invariants is analyzed, with applications to the Lorenz and Rössler systems.</p>

On Periodic Orbits of Vector Fields in Arbitrary Dimension Via Autonomous and Nonautonomous Inverse Jacobi Multipliers

期刊论文
出版模式：

开放获取
发表日期：

2026年3月30日

查看全文

Isaac A. García,
Ernest Latorre,
Susanna Maza

摘要

We investigate periodic orbits of $C^1$ $C^{1}$ autonomous vector fields in $\mathbb {R}^n$ $R^{n}$ using inverse Jacobi multipliers that may depend explicitly on time. We establish a localization principle for T-periodic orbits in arbitrary dimension, extending known planar results and deriving nonexistence conditions through the relation between the time-slices $V(0, \cdot )$ $V (0, \cdot)$ and $V(T, \cdot )$ $V (T, \cdot)$ . We further characterize hyperbolicity and orbital stability, including a decomposition of characteristic multipliers along invariant surfaces associated with autonomous inverse Jacobi multipliers. A test for the algebraicity of periodic orbits in 3-dimensional vector fields is given based on non-autonomous inverse Jacobi multipliers. The interplay between normalizers, inverse Jacobi multipliers and invariants is analyzed, with applications to the Lorenz and Rössler systems.