For a linear damped system; xdot=-k*x, the solution is x=x0*e^(-kt), hence the system is of exponential dissipation rate.

But what can we say about some other systems, for example; xdot=-k*x^3? Should we solve the equation first, and then judge its dissipation rate based on that solution. What if the typical equation has no analytic solution. Lyapunov function method, Lyapunov exponent, Sontag lemmas, or so on...? which one to undertake to quantitatively or qualitatively characterize the dissipation rate?

Consider v=v(t), and set dv/dt=-v*e^t,

it means vdot=-v*exp(t), then how can we judge and discuss the dissipation rate of v, without solving the preceding differential equation (ode)?.

In some text, I've seen vdot= -ρ(v), then upon some conditions of the function

ρ (.), they discuss stability and dissipation rate. I want to know how it is undertaken for the function v, where we know; vdot=-v*exp(t)= ρ(v,t). In some other texts ρ is replaced by β as they say if x(t)= β(|x0|,t) is class KL, then some conclusions could be drawn for the stability of x(t) trajectory, and these are mainly due to Eduardo Sontag. I am interested to qualitatively get some insight on dissipation rate of the damped system xdot=-x*e^t, without solving the equation, while it is clear that the system is globally asymptotically stable, but what can we say about its dissipation rate through a rigorous mathematical proof?.

Assume no connection between v and x, they are the same, only denoting system variable as x or v. Do not assume any connection between v and x (they are the same) and only a formulation notation to elaborate this question on ResearchGate.