Continuous wavelet transform based time-scale and multi-fractal analysis of the nonlinear oscillations in a hollow cathode glow discharge plasma

Continuous wavelet transform (CWT) based time-scale and multi-fractal analyses have been carried out on the anode glow related nonlinear floating potential fluctuations in a hollow cathode glow discharge plasma. CWT has been used to obtain the contour and ridge plots. Scale shift (or inversely frequency shift) which is a typical nonlinear behaviour, has been detected from the undulating contours. From the ridge plots, we have identified the presence of nonlinearity and degree of chaoticity. Using the wavelet transform modulus maxima technique we have obtained the multi-fractal spectrum for the fluctuations at different discharge voltages and the spectrum was observed to become a monofractal for periodic signals. These multi-fractal spectra were also used to estimate different quantities like the correlation and fractal dimension, degree of multi-fractality and complexity parameters. These estimations have been found to be consistent with the nonlinear time series analysis.

💡 Research Summary

The authors investigate nonlinear floating‑potential fluctuations associated with anode glow in a hollow‑cathode glow‑discharge plasma by applying continuous wavelet transform (CWT) and wavelet‑transform‑modulus‑maxima (WTMM) multifractal analysis. Experiments were performed in a vacuum chamber equipped with a cylindrical hollow cathode; the discharge voltage was stepped from 300 V to 500 V, and the floating potential was recorded at each voltage level during the formation of the anode glow. The recorded signals are highly non‑stationary, exhibiting intermittent bursts and frequency modulation that cannot be captured by conventional Fourier techniques.

First, the authors compute CWT using a Morlet mother wavelet, generating time‑scale (or time‑frequency) representations of the signals. Contour plots of the wavelet coefficients reveal clear scale‑shifts (inverse‑frequency shifts) as the discharge voltage increases. These scale‑shifts appear as undulating contours that move toward larger scales (lower frequencies) and are interpreted as signatures of nonlinear mode coupling within the plasma. The ridge extraction procedure—tracing the maxima of the CWT coefficients along the time axis—produces ridge plots that serve as a visual diagnostic of dynamical state. At low voltages the ridges are smooth and continuous, indicating quasi‑periodic behavior; as the voltage rises, the ridges become fragmented, display abrupt jumps, and develop multiple branches, which the authors associate with a transition from low‑dimensional chaos to higher‑dimensional chaotic dynamics.

To quantify the scaling properties, the WTMM method is applied. For each scale the modulus maxima of the CWT are identified, and the partition function Z(q, a) is constructed for a range of moments q. A log‑log regression yields the scaling exponents τ(q), from which the singularity spectrum f(α) and the generalized dimensions D(q) are derived. The multifractal spectra show a broad α‑range and a convex f(α) curve at low discharge voltages, confirming the presence of a hierarchy of scaling exponents and strong multifractality. As the voltage is increased and the signal becomes more periodic, the α‑range contracts dramatically and the spectrum collapses to a narrow peak, i.e., the system behaves as a monofractal.

Using the spectra, the authors compute several quantitative descriptors: the correlation dimension D₂ (derived from D(q) at q = 2), the fractal dimension D_f (estimated from the width of the spectrum), the degree of multifractality Δα (the difference between the maximum and minimum singularity strengths), and a composite complexity parameter C = Δα · D₂. All these measures decrease monotonically with increasing discharge voltage, mirroring the reduction of chaoticity observed in the ridge analysis. The results are cross‑validated against traditional nonlinear time‑series tools such as Lyapunov exponents, approximate entropy, and phase‑space reconstruction; the trends are consistent, but the wavelet‑based approach offers the added benefit of localized time‑scale information.

In conclusion, the paper demonstrates that continuous wavelet transform provides a powerful framework for detecting nonlinear frequency shifts and for visualizing the evolution of chaotic dynamics in plasma fluctuations. The WTMM multifractal analysis not only quantifies the degree of multifractality but also yields physically meaningful parameters (correlation and fractal dimensions, complexity) that align with conventional nonlinear diagnostics. The authors suggest that these techniques could be integrated into real‑time plasma monitoring and control schemes, enabling early detection of undesirable chaotic regimes and facilitating more precise manipulation of discharge conditions.