Invariants and chaos in the Volterra gyrostat without energy conservation

The model of the Volterra gyrostat (VG) has not only played an important role in rigid body dynamics but also served as foundation low-order models many naturally occurring systems. It is well known that VG possesses two invariants, or constants motion, corresponding to kinetic energy and squared angular momentum, giving oscillatory solutions its equations motion. Nine distinct subclasses have been identified, which Euler gyroscope Lorenz are each constants. This paper characterizes quadratic invariants subclasses, showing how these enjoy even when rendered terms a non-invertible transformation parameters, leading transformed (TVG). If coefficients TVG sum zero, they do for VG, system conserves energy. In all cases, flows preserve volume. However, physical where zero ubiquitous, characterization resulting this more general class with volume conservation without lacking. provides first such absence conservation, number depends on linear feedback terms. shown three no invariants. circumscribes possible three-dimensional flows, those any admit rich including chaos. gives rise broad conserving chaotic arising from reduction techniques.