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<p>In this paper we prove the gradient structure of solutions for a nonautonomous cascade system defined on Banach spaces, where the <i>x</i>–variable evolves independently via <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\dot{x} = Ax+f(t,x)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>x</mi> <mo>˙</mo> </mover> <mo>=</mo> <mi>A</mi> <mi>x</mi> <mo>+</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and influences the <i>y</i>–variable through <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\dot{y} = By+g(x,y)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>y</mi> <mo>˙</mo> </mover> <mo>=</mo> <mi>B</mi> <mi>y</mi> <mo>+</mo> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. By first analyzing the long–time dynamics of the nonautonomous <i>x</i>–equation and then examining the resulting <i>y</i>–dynamics for each asymptotic state of <i>x</i>, we provide a complete description of the system’s gradient structure in two levels: a more abstract and general, with less hypotheses on <i>f</i>, and a deeper level of description, when the term <i>f</i>(<i>t</i>,&#xa0;<i>x</i>) is asymptotically autonomous. Finally, we present a description when the term <i>f</i>(<i>t</i>,&#xa0;<i>x</i>) is a small nonautonomous perturbation of an autonomous term.</p>

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Gradient Structure of Dynamics for Nonautonomous Cascade Systems

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  • 发表日期：

    2026年3月2日

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  • M. C. Bortolan,
  • M. C. A. Brito,
  • A. N. Carvalho,
  • J. A. Langa

摘要

In this paper we prove the gradient structure of solutions for a nonautonomous cascade system defined on Banach spaces, where the x–variable evolves independently via \(\dot{x} = Ax+f(t,x)\) x ˙ = A x + f ( t , x ) and influences the y–variable through \(\dot{y} = By+g(x,y)\) y ˙ = B y + g ( x , y ) . By first analyzing the long–time dynamics of the nonautonomous x–equation and then examining the resulting y–dynamics for each asymptotic state of x, we provide a complete description of the system’s gradient structure in two levels: a more abstract and general, with less hypotheses on f, and a deeper level of description, when the term f(t, x) is asymptotically autonomous. Finally, we present a description when the term f(t, x) is a small nonautonomous perturbation of an autonomous term.