Necessary conditions for similarity between two lattice differential equations (Q6642455)

In this paper the authors study similarity between two lattice differential equations:\N\[\N\dot{u}_i = \nu (u_{i+1}-2u_i+u_{i-1}) - \lambda u_i + f_i(u) + \mathfrak{f}_i(t),\quad i\in\mathbb{Z},\ t\geq 0,\N\]\Nand\N\[\N\dot{v}_i = \nu (v_{i+1}-2v_i+v_{i-1}) - \lambda v_i + g_i(v) + \mathfrak{g}_i(t),\quad i\in\mathbb{Z},\ t\geq 0,\N\]\Nwith \(f,g:\ell^2\rightarrow\ell^2\) being sublinear and lipschitz continuous.\N\NThe systems are conjugate if there exists a homeomorphism \(K\) between the solution sets of the two equations. If the systems are not similar, the cost functional\N\[\NJ[K] := \sup\limits_{u^0\in D} \frac{1}{T}\int\limits_0^T \|K\left( u(t,u^0) \right) -v(t,K(u^0))\|_2^2 \mathrm{d} t\N\]\Nis applied to measure dissimilarity. If the cost functional vanishes \(J[K]=0\), the systems are conjugate. On the other hand if \(J[K]=\infty\), the systems are completely dissimilar. The authors introduce the similarity function \(\rho(x)\) as a decreasing function with \(\rho(0)=1\) and \(\rho(\infty)=0\) to quantify intermediate cases.