<p>This manuscript studies dispersive concepts of a nonautonomous dynamical system associated with a linear differential equation <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\dot{x}=\textbf{A}x+ \textbf{g}\left( t\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>x</mi> <mo>˙</mo> </mover> <mo>=</mo> <mi mathvariant="bold">A</mi> <mi>x</mi> <mo>+</mo> <mi mathvariant="bold">g</mi> <mfenced close=")" open="("> <mi>t</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation>. The dispersiveness is a property characterized by the absence of recursiveness, which is determined by the computation of the nonautonomous prolongational limit set. Differently from the autonomous case, due to the time initial dependence, the nonautonomous dispersiveness has distinct meanings for positive and negative times. Innovative analytical techniques to treat the dispersive concepts are employed, by using the matrix convolution product and the Laplace multi-transform. For non-invertible systems (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\det \textbf{A}=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo movablelimits="true">det</mo> <mi mathvariant="bold">A</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>), the mean limit criterion is formulated, based on the computation of the mean limit set of the adding function <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\textbf{g}\left( t\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">g</mi> <mfenced close=")" open="("> <mi>t</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation>. The results of the paper are applied to various special cases, including nilpotent systems, polynomial systems, higher order differential equations, and mechanical systems.</p>

Nonautonomous Dispersiveness in Linear Differential Systems

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2026年3月10日

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Jean G. Silva,
Josiney A. Souza

摘要

This manuscript studies dispersive concepts of a nonautonomous dynamical system associated with a linear differential equation $\dot{x}=\textbf{A}x+ \textbf{g}\left( t\right) $ $\dot{x} = A x + g (t)$ . The dispersiveness is a property characterized by the absence of recursiveness, which is determined by the computation of the nonautonomous prolongational limit set. Differently from the autonomous case, due to the time initial dependence, the nonautonomous dispersiveness has distinct meanings for positive and negative times. Innovative analytical techniques to treat the dispersive concepts are employed, by using the matrix convolution product and the Laplace multi-transform. For non-invertible systems ( $\det \textbf{A}=0$ $det A = 0$ ), the mean limit criterion is formulated, based on the computation of the mean limit set of the adding function $\textbf{g}\left( t\right) $ $g (t)$ . The results of the paper are applied to various special cases, including nilpotent systems, polynomial systems, higher order differential equations, and mechanical systems.