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

Abstract 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 \emph{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 $\alpha$, which we obtain in closed form and which vanish only for $\alpha=1$: $c(\alpha)$ measures the change in the time law, $d(\alpha)$ 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.