The problem of estimating the maximal number H(m) of limit cycles that planar polynomial vector fields of degree m can exhibit has long been a central question in the qualitative theory of planar dynamical systems. A natural extension to the three-dimensional space is to study the maximum number N(m) of limit tori that can occur in spatial polynomial vector fields of degree m. In this work, we focus on normally hyperbolic limit tori and show that the corresponding maximum number <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$N_h(m)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>N</mi> <mi>h</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, if finite, increases strictly with m. More precisely, we prove that <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$N_h(m+1) \geqslant N_h(m) + 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>N</mi> <mi>h</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>⩾</mo> <msub> <mi>N</mi> <mi>h</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Our proof relies on two central results established in this paper. The first is that the normal hyperbolicity of compact invariant manifolds is preserved under time reparametrizations. Despite the fundamental nature of this statement, a complete proof has, surprisingly, not previously appeared in the literature, except under rather restrictive assumptions on the flow restricted to the invariant manifold. The second result concerns the torus bifurcation phenomenon near Hopf–Zero equilibria in spatial vector fields. While the conditions for such bifurcations are typically expressed in terms of higher-order normal form coefficients, we derive explicit and verifiable criteria for the occurrence of torus bifurcation, assuming only that the linear part of the unperturbed vector field is in Jordan normal form. This approach not only circumvents intricate computations involving higher-order normal forms but also ensures the normal hyperbolicity of the bifurcated torus.

Variation in the Number of Normally Hyperbolic Limit Tori in 3D Polynomial Vector Fields via Time Reparametrization and Hopf-Zero Bifurcation

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2026年4月28日

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Lucas Q. Arakaki,
Douglas D. Novaes,
Pedro C. C. R. Pereira

摘要

The problem of estimating the maximal number H(m) of limit cycles that planar polynomial vector fields of degree m can exhibit has long been a central question in the qualitative theory of planar dynamical systems. A natural extension to the three-dimensional space is to study the maximum number N(m) of limit tori that can occur in spatial polynomial vector fields of degree m. In this work, we focus on normally hyperbolic limit tori and show that the corresponding maximum number $N_h(m)$ $N_{h} (m)$ , if finite, increases strictly with m. More precisely, we prove that $N_h(m+1) \geqslant N_h(m) + 1$ $N_{h} (m + 1) ⩾ N_{h} (m) + 1$ . Our proof relies on two central results established in this paper. The first is that the normal hyperbolicity of compact invariant manifolds is preserved under time reparametrizations. Despite the fundamental nature of this statement, a complete proof has, surprisingly, not previously appeared in the literature, except under rather restrictive assumptions on the flow restricted to the invariant manifold. The second result concerns the torus bifurcation phenomenon near Hopf–Zero equilibria in spatial vector fields. While the conditions for such bifurcations are typically expressed in terms of higher-order normal form coefficients, we derive explicit and verifiable criteria for the occurrence of torus bifurcation, assuming only that the linear part of the unperturbed vector field is in Jordan normal form. This approach not only circumvents intricate computations involving higher-order normal forms but also ensures the normal hyperbolicity of the bifurcated torus.