Periodic solutions of travelling-wave type in circular gene networks (Q2821884)

Consider the system NEWLINE\[NEWLINE{du_j\over dt}=-u_j+{1+\beta\over 1+\sum^s_{k=1} \delta_k u^{\gamma_k}_{j-k}},\quad j= 1,2,\dots, m\tag{\(*\)}NEWLINE\]NEWLINE under the conditions \(m>s+1\), \(\beta>0\), \(\delta_k>0\), \(\gamma_k= {\gamma^0_t\over\varepsilon}\) for \(k=1,\dots, s\), \(0<\varepsilon<<1\), \(u_{-l}= u_{m-l}\) for \(l= 1,\dots, m\).NEWLINENEWLINE The authors look for periodic solutions of \((*)\) of the form NEWLINE\[NEWLINEu_j(t)= u(t+(j-1)\Delta),\quad j= 1,\dots, m,\;\Delta>0.\tag{\(**\)}NEWLINE\]NEWLINE They prove that to any \(p\) satisfying NEWLINE\[NEWLINE{m\over s+1+\beta+ s/\beta}< p<{m\over s+1}NEWLINE\]NEWLINE there is a periodic solution of \((*)\) of type \((**)\). This solution is stable in case \(m= p(s+1)+1\) and unstable otherwise. It is obvious that the number of unstable solutions grows with increasing \(m\).