Nonautonomous Dispersiveness in Linear Differential Systems

Abstract This manuscript studies dispersive concepts of a nonautonomous dynamical system associated with a linear differential equation $$\dot{x}=\textbf{A}x+ \textbf{g}\left( t\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover> <mml:mi>x</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> <mml:mo>=</mml:mo> <mml:mi>A</mml:mi> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:mi>g</mml:mi> <mml:mfenced> <mml:mi>t</mml:mi> </mml:mfenced> </mml:mrow> </mml:math> . 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 ( $$\det \textbf{A}=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>det</mml:mo> <mml:mi>A</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> ), the mean limit criterion is formulated, based on the computation of the mean limit set of the adding function $$\textbf{g}\left( t\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>g</mml:mi> <mml:mfenced> <mml:mi>t</mml:mi> </mml:mfenced> </mml:mrow> </mml:math> . The results of the paper are applied to various special cases, including nilpotent systems, polynomial systems, higher order differential equations, and mechanical systems.