Chaos is not an invariant of fractional-order systems: the clock shapes orbits, attractors and bifurcations

The qualitative theory of differential equations rests on a fact so elementary that it is rarely stated: multiplying a vector field by a positive function of the state, a clock, changes the speed at which the orbits are traversed but not the orbits. We show that for fractional-order systems in the Caputo sense this fact is false. The discrepancy between the system with a clock and the reference system read in the transformed time is governed, at short times, by two universal functions of the order α, which we obtain in closed form and which vanish only for α=1: c(α) measures the change in the time law, d(α) the change in the geometry of the orbits. Local stability and Matignon's critical order, on the contrary, are exactly invariant, so the effect is nonlinear and global. Numerically, a clock deforms the limit cycle of the fractional van der Pol oscillator by up to $16%$ and, in the fractional Chen system, turns a periodic regime into a chaotic one at the same order and from the same initial datum, something impossible in integer order. A general rule emerges: the local bifurcations of equilibria do not move, while the global ones shift linearly with the strength of the clock. The consequence reaches every field in which fractional models carry a state-dependent kinetic factor, from viscoelastic and glassy materials to electrochemical, biological and reacting systems: there the time in which memory accumulates is a physical hypothesis with observable consequences, and a fractional attractor is a property of the field together with its clock.