Pitchfork and transcritical bifurcations in systems with homogeneous nonlinearities and an almost periodic time coefficient
Consider the vector differential equation \[ {dx\over dt}= A(\varepsilon)x+ b(t) H_m(x,t), \] where \(A(0)\) has the simple eigenvalue \(\mu(0)= 0\) and all other eigenvalues have negative real parts, \(b\) is an almost periodic function which is strictly positive or strictly negative, \(H_m\) is homogeneous in \(x\) of order \(m\geq 2\), \(\varepsilon\) is the bifurcation parameter. The author shows that \(x\equiv 0\) undergoes a transcritical or pitchfork bifurcation to a small nontrivial almost periodic solution when the eigenvalue \(\mu(\varepsilon)\) crosses the imaginary axis, he also determines its stability.
scientific article; zbMATH DE number 2141499