<p>We study the existence of almost periodic solutions to dynamical systems of the form <Equation ID="Equ11"> <EquationSource Format="TEX">$\begin{aligned} \ddot{u}(t) + a u(t) = \nabla _{u}H(t, u(t)), \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mover accent="true"> <mi>u</mi> <mo>¨</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>a</mi> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi mathvariant="normal">∇</mi> <mi>u</mi> </msub> <mi>H</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ u(t) = (u_1(t), \ldots , u_N(t)) \in \mathbb {R}^N $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>u</mi> <mi>N</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$a \ge 0 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> is a constant. The system models the interaction between acceleration, position, and time through a gradient-type forcing term. By employing variational methods and the min-max principle, and under suitable conditions on the potential function <i>H</i>, we prove the existence of quasi-periodic solutions to the system.</p>

Almost periodic solutions for second-order nonlinear time-dependent dynamical systems via variational methods

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发表日期：

2025年8月22日

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Najoua Barhoumi,
Sameh Benatti,
Mohsen Miraoui

摘要

We study the existence of almost periodic solutions to dynamical systems of the form $\begin{aligned} \ddot{u}(t) + a u(t) = \nabla _{u}H(t, u(t)), \end{aligned}$ $\begin{matrix} \ddot{u} (t) + a u (t) = \nabla_{u} H (t, u (t)), \end{matrix}$ where $ u(t) = (u_1(t), \ldots , u_N(t)) \in \mathbb {R}^N $ $u (t) = (u_{1} (t), \dots, u_{N} (t)) \in R^{N}$ , and $a \ge 0 $ $a \geq 0$ is a constant. The system models the interaction between acceleration, position, and time through a gradient-type forcing term. By employing variational methods and the min-max principle, and under suitable conditions on the potential function H, we prove the existence of quasi-periodic solutions to the system.