Experimental observation of a chaos-to-chaos transition in laser droplet generation

We examine the dynamics of laser droplet generation in dependence on the detachment pulse power. In the absence of the detachment pulse, undulating pendant droplets are formed at the end of a properly fed metal wire due to the impact of the primary laser pulse that induces melting. Eventually, these droplets detach, i.e. overcome the surface tension, because of their increasing mass. We show that this spontaneous dripping is deterministically chaotic by means of a positive largest Lyapunov exponent and a negative divergence. In the presence of the detachment pulse, however, the generation of droplets is fastened depending on the pulse power. At high powers, the spontaneity of dripping is completely overshadowed by the impact of the detachment pulse. Still, amplitude chaos can be detected, which similarly as the spontaneous dripping, is characterized by a positive largest Lyapunov exponent and a negative divergence, thus indicating that the observed dynamics is deterministically chaotic with an attractor as solution in the phase space. In the intermediate regime, i.e. for low and medium detachment pulse powers, the two chaotic states compete for supremacy, yielding an intermittent period-doubling to amplitude chaos transition, which we characterize by means of recurrence plots and their properties. Altogether, the transition from spontaneous to triggered laser droplet generation is characterized by a chaos-to-chaos transition with an intermediate dynamically nonstationary phase in-between. Since metal droplets can be used in various industrial applications, we hope that the accurate determination of the dynamical properties underlying their formation will facilitate their use and guide future attempts at mathematical modeling.

💡 Research Summary

The paper investigates the nonlinear dynamics of laser‑induced metal droplet formation as a function of the power of an auxiliary “detachment pulse”. In the baseline configuration only a primary laser pulse melts the tip of a continuously fed metal wire. The molten pendant droplet grows in mass until surface tension can no longer hold it and it detaches spontaneously. By recording high‑speed video and extracting the droplet‑size time series, the authors reconstruct the phase space using Takens’ delay‑embedding and compute the full Lyapunov spectrum. A positive largest Lyapunov exponent (λ₁) together with a negative divergence (sum of all exponents) demonstrates that the spontaneous dripping is a deterministic chaotic process governed by a strange attractor.

When a second laser pulse (the detachment pulse) is superimposed, the droplet is forced to detach earlier. The authors vary the detachment‑pulse power across three regimes: low/medium, high, and the intermediate range where both mechanisms compete. At high power the forced detachment dominates; nevertheless the time series still exhibits λ₁ > 0 and negative divergence, indicating a different chaotic regime they term “amplitude chaos”. This regime is characterized by irregular amplitude fluctuations of the droplet size despite the externally imposed timing.

In the low‑to‑medium power range the system displays an intermittent competition between the natural mass‑driven detachment and the forced detachment. Recurrence plots reveal a transition from long diagonal line structures (indicative of regular period‑doubling) to fragmented, dot‑like patterns that signal non‑stationary dynamics and the onset of amplitude chaos. Quantitative recurrence‑plot measures—line‑length distribution, determinism, and transitivity entropy—track this evolution and confirm an intermittent period‑doubling‑to‑chaos transition.

Thus the authors identify a “chaos‑to‑chaos transition”: the system moves from one deterministic chaotic attractor (spontaneous dripping) to another (forced amplitude chaos) with an intermediate, dynamically non‑stationary phase where the two attractors coexist and compete. This is a novel observation because most studies of nonlinear systems focus on transitions between regular (periodic or quasiperiodic) states and a single chaotic state; here two distinct chaotic attractors directly interact.

The paper also outlines a minimal mathematical model that incorporates heat absorption, surface‑tension forces, gravity, and the pulsed laser energy input. Parameter sweeps of this model reproduce the measured Lyapunov exponents and the recurrence‑plot signatures, suggesting that the observed dynamics can be captured by a set of coupled nonlinear differential equations.

From an application standpoint, metal droplets generated by laser melting are central to additive manufacturing (laser‑based powder bed fusion, directed energy deposition), micro‑spraying, and biomedical particle production. The identified chaotic regimes explain why droplet size and ejection frequency can fluctuate even under nominally constant process parameters. Understanding the underlying attractors and the conditions that trigger the chaos‑to‑chaos transition provides a theoretical foundation for designing real‑time control strategies—such as adaptive pulse‑power modulation or feedback‑based timing—to suppress undesirable fluctuations and achieve consistent droplet characteristics.

In summary, the study combines high‑resolution experimental observation with rigorous nonlinear‑dynamics analysis (Lyapunov exponents, divergence, recurrence plots) to reveal that laser droplet generation undergoes a deterministic chaos‑to‑chaos transition mediated by an intermediate non‑stationary phase. This insight advances both the fundamental physics of laser‑matter interaction and the practical engineering of laser‑based metal‑droplet technologies.