The buffer phenomenon in ring-like chains of unidirectionally connected generators (Q2925760)

Consider the ring-like system NEWLINE\[NEWLINE{d^2 x_j\over dt^2}+ 2\mu {dx_j\over dt}+ x_j= F\Biggl({x_j-1\over\varepsilon}\Biggr),\leqno{(*)}NEWLINE\]NEWLINE \(j= 1,\dots, m\), \(x_0= x_m\), \(m\geq 3\), \(x_j(t)\in\mathbb{R}\). The function \(F:\mathbb{R}\to\mathbb{R}\) is sufficiently smooth, satisfies \(F(0)= 0\), \(F'(x)<0\) for \(x\in\mathbb{R}\) and has some asymptotic behavior as \(x\to\pm\infty\). Typical examples of \(F\) are: \(-{2\over\pi}\arctan x\), \(- {x\over\sqrt{x^2+1}}\), \(-\tanh x\). The authors look for periodic solutions of \((*)\) of the form NEWLINE\[NEWLINEx_j= x(t+ (j-1)\Delta),\quad \Delta>0,\quad j=1,\dots, m.\leqno{(**)}NEWLINE\]NEWLINE This problem is reduced to the problem of the existence of a periodic solution of the auxiliary delay-differential equation NEWLINE\[NEWLINE{d^2x\over dt^2}+ 2\mu{dx\over dt}+ x= F\Biggl({x(t-\Delta)\over\varepsilon}\Biggr).\leqno{(***)}NEWLINE\]NEWLINE It is shown that under the condition \(0<\Delta<\pi\) there are sufficiently small positive numbers \(\varepsilon_0(\Delta)\), \(\mu_0(\Delta)\) such that for any \(\varepsilon\), \(\mu\) satisfying \(0<\varepsilon\leq\varepsilon_0(\Delta)\), \(0<\mu\leq \mu_0(\Delta)\), \((***)\) has a periodic solution \(x=x(t,\varepsilon,\mu)\) with period \(T(\varepsilon,\mu)\). Using this result and corresponding investigations on the stability, the authors finally prove that \((*)\) has exponentially orbitally stable periodic solutions the number of which increases unboundedly as \(m\to\infty\). That means that the buffer phenomen can be realized.