The combination of the averaging method with the Lyapunov-Schmidt reduction offers sufficient conditions for establishing the existence of periodic solutions in a specific class of perturbative T-periodic nonautonomous differential equations, given by the form <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$x'=F_0(t,x)+\varepsilon F(t,x,\varepsilon )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>=</mo> <msub> <mi>F</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>ε</mi> <mi>F</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>ε</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. These periodic solutions emerge from a manifold <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {Z}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">Z</mi> </math></EquationSource> </InlineEquation> of periodic solutions of the unperturbed system <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$x'=F_0(t,x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>=</mo> <msub> <mi>F</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. The stability analysis of these periodic solutions requires computing the eigenvalues of matrix-valued functions <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$M(\varepsilon )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, typically involving the theory of k-hyperbolic matrices. Traditionally, the theory demands a diagonalization process for the k-jets of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$M(\varepsilon )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, for which no general algorithm exists. In this study, we propose an alternative strategy to determine the stability of periodic solutions without relying on such diagonalization, making our approach applicable even in cases where diagonalization is not feasible. Additionally, we present applications of our results for two families of 4D vector fields: the first family corresponds to a perturbation of a two-degree-of-freedom Hamiltonian system, while the second family corresponds to a perturbation of a 4D vector field with a one-degree-of-freedom Hamiltonian sub-system.

On the Stability of Smooth Branches of Periodic Solutions for Higher Order Perturbed Differential Equations

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2026年3月7日

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Murilo R. Cândido,
Douglas D. Novaes

摘要

The combination of the averaging method with the Lyapunov-Schmidt reduction offers sufficient conditions for establishing the existence of periodic solutions in a specific class of perturbative T-periodic nonautonomous differential equations, given by the form $x'=F_0(t,x)+\varepsilon F(t,x,\varepsilon )$ $x^{'} = F_{0} (t, x) + ε F (t, x, ε)$ . These periodic solutions emerge from a manifold $\mathcal {Z}$ $Z$ of periodic solutions of the unperturbed system $x'=F_0(t,x)$ $x^{'} = F_{0} (t, x)$ . The stability analysis of these periodic solutions requires computing the eigenvalues of matrix-valued functions $M(\varepsilon )$ $M (ε)$ , typically involving the theory of k-hyperbolic matrices. Traditionally, the theory demands a diagonalization process for the k-jets of $M(\varepsilon )$ $M (ε)$ , for which no general algorithm exists. In this study, we propose an alternative strategy to determine the stability of periodic solutions without relying on such diagonalization, making our approach applicable even in cases where diagonalization is not feasible. Additionally, we present applications of our results for two families of 4D vector fields: the first family corresponds to a perturbation of a two-degree-of-freedom Hamiltonian system, while the second family corresponds to a perturbation of a 4D vector field with a one-degree-of-freedom Hamiltonian sub-system.