Experimental Observation of a Chaos-to-Chaos Transition in Laser Droplet Generation

Abstract

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 perioddoubling 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.

Nonlinear analysis of laser droplet generation by means of 0–1 test for chaos

“…According to some similar results in the literature [17,27,29,30], this outcome suggests that the forced dripping regime should also be considered chaotic. These results are in agreement with our previous finding that spontaneous to forced dripping transition may in fact be characterized as chaosto-chaos transition with a non-stationary regime in between [22].…”

“…Lyapunov spectra of presented attractors consist of positive largest Lyapunov exponent, vanishing second exponent and negative divergence (sum of Lyapunov exponents). This overall indicates that both regimes are to be characterized as deterministically chaotic [22]. However, calculating Lyapunov exponents from time series may be burdened by some pitfalls inherent to the reconstruction procedure [7].…”

“…This regime is called forced dripping. In between, with medium detachment pulse power, we can observe a dynamically nonstationary intermittent regime where the two marginal regimes compete for supremacy [22]. However, in this paper we are going to focus on stationary regimes, i.e., spontaneous and forced dripping to which correspond the detachment pulse power of 0 kW and 8 kW, respectively.…”

“…The continuity of the spectra is clearly visible in both cases, meaning that the behavior is, besides being characterized by some dominant frequencies, inherently irregular, hinting toward deterministically chaotic behavior. However, periodic spikes of In this manner nonlinear time series analysis, which is based on phase space reconstruction, was performed and the dynamics of considered regimes was analyzed through Lyapunov spectra [22][23][24]. In order to determine proper embedding parameters we used mutual information [25] and false nearest neighbors methods [26].…”

Dynamics of stainless steel turning: Analysis by flicker-noise spectroscopy

“…Various characteristics/techniques of nonlinear time series analysis [42], including correlation dimension, Lyapunov exponents, Kolmogorov-Sinai entropy, surrogate data tests, singular value decomposition, and many others, have been used to examine the dynamics of steel turning [22] as well as other manufacturing processes, for instance, laser droplet formation in the welding of electrical contacts [43,44]. The estimation of correlation dimension, Lyapunov exponents, and Kolmogorov-Sinai entropy relying on the Takens' embedding theorem, which is valid only for stationary signals (when applied to real signals, it is assumed that both mean and variance are approximately constant with time at any scale) with a large number of sampling points, is complicated in this case due to the apparent nonstationarity of the signals (as clearly seen in Fig.…”