Noose structure and bifurcations of periodic orbits in reversible three-dimensional piecewise linear differential systems
This work focuses on the periodic response of a one-parameter family of piecewise-linear ODEs given by \[ x'=y,\quad y'=z,\quad z'=1-y-\lambda(1+\lambda^2)|x|, \] with \(\lambda>0\). Special attention is given to the dynamical phenomenon referred to as noose bifurcation. It occurs when a curve of \(T\)-periodic orbits and a curve of \(2T\)-periodic orbits (connected via a period-doubling bifurcation) come together and annihilate at a fold bifurcation. The authors study in detail the curves of periodic solutions that emerge from the noose structure, which involve pitchfork bifurcations, sequences of period-doubling bifurcations, strong resonances and (crossing and non-crossing) tangential intersections of the periodic orbits with the switching boundary \(x=0\). Using the fact that the system is solvable in closed form in the regions of linearity, the authors construct a system of two equations with three unknowns describing a particular type of periodic solution, which is then solved via reduction techniques and continuation methods. Their numerical investigation also shows that in some cases the noose curve oscillates around a critical parameter value for which the period of the orbit tends to infinity, thus giving rise to a homo- or heteroclinic connection.
scientific article; zbMATH DE number 6521638
scientific article