Gradient Structure of Dynamics for Nonautonomous Cascade Systems

Abstract 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 $$\dot{x} = Ax+f(t,x)$$ <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>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> and influences the y –variable through $$\dot{y} = By+g(x,y)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover> <mml:mi>y</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> <mml:mo>=</mml:mo> <mml:mi>B</mml:mi> <mml:mi>y</mml:mi> <mml:mo>+</mml:mo> <mml:mi>g</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . 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.