A three-dimensional autonomous system with unbounded ‘bending’ solutions

We consider the system ẋ=ayz+bz+cy, ẏ=dzx+ex+fz, ż=gxy+hy+kxfor real functions x(t), y(t) and z(t), where the overdot denotes differentiation with respect to a time-like independent variable t, and the coefficients a to k are real constants. Such equations arise in mechanical and fluid-dynamical contexts. Depending on parameter values, solutions may exhibit blowup in finite time; or they may be bounded oscillatory, or unbounded, as time t→∞. The local shape of the latter unbounded solutions is typically helical, sometimes with and sometimes without a 90° bend in the axis of the helix. Complete solutions are obtained in cases where certain coefficients are zero. Other cases are investigated numerically and asymptotically. The numerical solutions reveal an interesting “four-leaf” structure connected to the helical trajectories: this structure largely determines whether these trajectories bend through 90° or not. A fluid-dynamical application is discussed in Appendix A.