Bifurcation equations of continuous piecewise-linear vector fields (Q1199759)

The author studies the continuous 3-dimensional 2-region piecewise-linear vector fields, that are vector fields defined by an ordinary differential equation: \[ {dx\over dt}=f(x)=\begin{cases} Ax & (x\in\mathbb{R}_ -) \\ Bx-p & (x\in\mathbb{R}_ +) \end{cases} \] where \(\mathbb{R}_ \pm=\{x\in\mathbb{R}^ 3:\pm(\langle\alpha,x\rangle-1)>0,\;\alpha\in\mathbb{R}^ 3,\;\alpha\neq 0\}\), \(\langle , \rangle\) denotes a usual inner product, \(A\) and \(B\) are \(3\times 3\) matrices, \(B=A+p^ T\alpha\). Namely, the homoclinic and heteroclinic bifurcations for singular points, and saddle-node, period-doubling and Hopf bifurcations for periodic orbits are studied. The bifurcation equations are numerically solved by the Newtonian method in order to describe the structure of bifurcation sets. Finally, the discussion is extended to the case of general \(n\)- dimensional multi-region systems.