Controllable Hopf bifurcations of codimension 1 and 2 in nonlinear control systems
Suppose that \(F(x)\) and \(G(x)\) are smooth vector fields with \(F(0)=0\) and \(G(0)=B \neq 0\). Assume that \(DF(0)\) has eigenvalues \(\lambda, \bar{\lambda}\), with \(\lambda = i \omega_0\), where \(\omega_0> 0\). All the other eigenvalues have nonzero real parts. Let \(q\) be an eigenvector corresponding to \(\lambda\) and \(p\) an adjoint eigenvector corresponding to \(\bar{\lambda}\). The vector \(p\) is normalized with respect to \(B\) such that \(\langle p, B \rangle = 1\); \(q\) is normalized with respect to \(p\) such that \(\langle p, q \rangle =1\). Let \( \omega= \langle p, x \rangle\), and \(\mu=(\mu_0,\mu_1,\mu_2,\mu_3)\) be the real control vector. The authors consider the system \[ \dot{x}=F(x)+G(x)\big[2 \operatorname{Re}(\omega)(\mu_0\mu_1+\mu_2\omega \bar{\omega}+\mu_3\omega^2\bar{\omega}^2)\big] . \] It is shown that there exists \(\mu_{2}^{0}\) such that if \(\mu_{2} \neq \mu_{2}^{0}\), then \((0,\mu_0,0,\mu_{2},\mu_{3})\) is a Hopf bifurcation point of codimension 1; it is a transversal point when \(\mu_0\neq 0\). Moreover, there exists \(\mu_3^0\) such that if \(\mu_0\neq 0\) and \(\mu_3\neq \mu_3^0\), then \((0,\mu_0,0,\mu_2^0,\mu_3)\) is a transversal Hopf bifurcation point of codimension 2.
scientific article; zbMATH DE number 5880260