Normal form construction for nearly-integrable systems with dissipation (Q691238)

Nearly integrable Hamiltonian systems can be modeled by a Hamiltonian function as \(H(y,x,t)=h_{00}(y)+\epsilon h_{10}(y,x,t)\) where \(y\in R^\ell\), and \( (x,t)\in T^{\ell+1}\), \(T\) is the torus and \(\epsilon\) is a small parameter. If \(\epsilon=0\) then the system is integrable with frequency \(\omega(y)\equiv \frac{\partial h_{00}(y)}{\partial y}\). For \(\epsilon\neq 0\), the Hamilton equations are \({\dot x}=\omega(y)+\epsilon h_{10,y}(y,x,t),\;{\dot y}=-\epsilon h_{10,x}(y,x,t)\). For a nearly integrable dissipative system, the authors' equations are \({\dot x}=\omega(y)+\epsilon h_{10,y}(y,x,t)+\mu f_{01}(y,x,t),\;{\dot y}=-\epsilon h_{10,x}(y,x,t)+\mu(g_{01}(y,x,t)-\eta)\) where \(\mu\in R_+\) is the dissipative parameter and \(\eta \in R^{\ell}\) is called the drift vector. Here the dissipative term \(\mu\) is assumed to be small. The authors show how to achieve a normal form transformation \((Y,X,t)=\Xi^{(N)}(y,x,t)\) to a suitable order \(N\) such that the system becomes \({\dot X}=\Omega_d (Y)+ h.o.t.,\;{\dot Y}=h.o.t\), where \(\Xi\) is of order not greater than \(N\) as a polynomial in \(\epsilon,\mu\) and \(X=x\), \(Y=y\) when \(\epsilon=\mu=0\). This transformation is possible for \(\omega(y)\) irrational and if \(\eta\) satisfies some compatibility properties. The construction is complicated, but the authors provide good motivation for their work and the exposition is very clear. They illustrate their normal form calculation with the example \({\dot x}=y-\mu(\sin(x-t)+\sin(x)),\;{\dot y}=-\epsilon(\sin(x-t)+\sin(x))-\mu(y-\eta)\). They construct the normal form for this example, make a comparison with numerical integration, and determine the region of validity for the normal form.