Chaotic systems are sensitive to initial conditions. A small perturbation grows exponentially — the Lyapunov exponent measures the growth rate. The perturbation also spreads spatially — the butterfly velocity measures how fast the disturbance propagates through an extended system. Together, these two quantities characterize how chaos eats information: how fast locally, and how fast it spreads.
Aron and Kulkarni (arXiv:2412.21043) show that stochastically resetting a chaotic system to its initial conditions controls both. At a critical resetting rate, both the Lyapunov exponent and the butterfly velocity vanish simultaneously. Below the critical rate, the resets are too infrequent to arrest the chaos — perturbations still grow and spread between resets. Above the critical rate, the resets overwhelm the chaos — the system never has time to diverge before being pulled back.
The critical rate defines a dynamical phase transition. Below it, the system is chaotic. Above it, the system is controlled. The transition is sharp — not a gradual decrease in chaos but a sudden arrest. The Lyapunov exponent doesn't approach zero asymptotically as the reset rate increases. It hits zero at a finite rate and stays there.
The mechanism is a competition between two timescales. The Lyapunov time — the time for perturbations to grow by a factor of e — sets the chaos timescale. The mean time between resets sets the control timescale. When the control timescale is shorter than the Lyapunov time, the system cannot accumulate enough divergence between resets to sustain chaos. The resets clip the exponential growth before it becomes significant.
The result holds for discrete maps and their lattice extensions — the logistic map and its spatiotemporal generalization. But the framework applies to any chaotic extended system. The only requirement is that the reset genuinely returns the system to its initial state. Partial resets, approximate returns, or state-dependent resetting would produce different critical rates and possibly different phase transition structures.
The general principle: chaos is not an inherent property of a system. It is a property of a system left to itself. Interrupt the system often enough, and chaos cannot establish itself. The information that would have spread is reset before it propagates. The butterfly never flaps long enough for the storm to form.