Bifurcation of cycles of automatic control systems with ideal relay (Q544718)

Let \(U(t) = e^{At}\) be the fundamental matrix of the system \(\dot{h}= Ah\) whose solutions \(h(t)=U(t-t_{s-1})h(t_{s-1}+0)\), \(t_{0}=0\) and \(h(+0)\) represent the initial conditions, satisfy the renormalization conditions \[ h(t_{s}+0)=h(t_{s}-0)+(\dot{x}_{0}(t_{s}+0)-\dot{x}_{0}(t_{s}-0))\frac{(b,h(t_{s}-0))}{\dot{\sigma}_{0}(t_{s}-0)}, s=1,\cdots,p. \] The monodromy matrix to a periodic solution \(x_0^{(t)}\) with period \(T_0\) is introduced in this paper as the matrix generated by the map \(h=h(+0)\to h(T_{0}-0)\). A result on the stability of the cycle \(x_0^{(t)}\) is given in terms of the monodromy matrix. An analog of the Andronov-Hopf bifurcation for systems with ideal relay is also proved.