Invariants for dissipative nonlinear systems by using rescaling

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Abstract
A rescaling transformation of space and time is introduced in the study of nonlinear dissipative systems that are described by a second-order differential equation with a friction term proportional to the velocity, β(t)v. The transformation is of the form (x,t)→(ξ,θ), where x=ξC(t)+α(t), dθ=dt/A2(t). This rescaling is used to find each potential for which there exists an exact invariant quadratic in the velocity and to find the invariant. The invariants are found explicitly for a power-law potential, γ(t)xm+1/(m+1), and an arbitrary coefficient of friction β(t). We show in an example how the rescaling transformation can be chosen to give an asymptotic solution of the equation in cases where the exact invariant does not exist. For certain parameters, the asymptotic solution is a self-similar solution that is an attractor for all initial conditions. The technique of applying a rescaling transformation has been useful in other problems and may have additional practical applications.