Buffer phenomenon and chaos in circular chains of unidirectionally coupled oscillators
Consider the chain of weakly unidirectionally coupled oscillators \[ {du_j\over dt}= F(u_j,\varepsilon)+\varepsilon G(u_{j-1},\varepsilon),\quad j= 1,2,\dots, m,\qquad u_0= u_m,\leqno{(\ast)} \] with \(m\geq 3\), \(u_j\in \mathbb R^n\), \(n\geq 2\), \(\varepsilon\) is a small positive parameter, \(F\), \(G\in C^\infty(\mathbb R^n\times [0,\varepsilon_0],\mathbb R^n)\), \(F(0,\varepsilon)\equiv G(0,\varepsilon)\equiv 0\). It is assumed that the system \[ {du\over dt}= F(u,\varepsilon) \] has a classical Hopf bifurcation from the origin for increasing \(\varepsilon\). The authors derive conditions implying the bifurcation of a stable periodic solution of \((*)\) and construct its asymptotic representation in the form of a power series in \(\sqrt{\varepsilon}\): \[ u_j(t,\varepsilon)= \sqrt{\varepsilon}\,u_{j,0}(t,\tau)+\varepsilon u_{j,1}(t,\tau)+ \varepsilon^{3/2} u_{j,2}(t,\tau)+\cdots, \] where \(\tau=\varepsilon t\), \(u_{j,0}= z_j(\tau) a \exp(i\varepsilon_0t)+\text{c.c., }j= 1,\dots, m\).
scientific article; zbMATH DE number 6381295