Periodic-chaotic alternating regimes during a fast solar metric-radio pulsation event

In the magnetically dominated corona, loop-like structures are expected to undergo pulsating events which are highly structured in time and classi�ed into different types from strictly periodic to irregular. In the case of periodic pulsations, they differ in their characteristic period: from tens of minutes (associated with global large-scale structure oscillations) to milliseconds (fast oscillations associated with a small local scale). These pulsations are strongly related to the basic physical properties of the solar corona and the evolution of magnetic loop-like structures that make up the entire solar atmosphere. This contribution describes a real-world scenario where a route to chaos and reverse processes take place. We examine the dynamic characteristics of a group of a train of pulsations at metric waveband solar radio emission as it evolves in time. The associated time series was recorded with the radio polarimeter of the INAF Trieste Astronomical Observatory, Basovizza Station on April 17, 2002, with a resolution time of 10 ms. Possible scenarios and mechanisms for the observed deterministic-chaos alternating regimes are discussed.


Introduction
The elastic nature of the magnetic eld and the inhomogeneous coronal plasma environment suggest that the magnetic solar corona structures should support MDH oscillations and pulsating events.Such kinds of events can serve as a diagnostic tool for the main variables governing the plasma environment of the solar corona like magnetic eld strength, plasma magnetic pressure, plasma density, and temperature.There is a signi cant number of papers in which different models are proposed that describe different scenarios, as well as generation mechanisms, for the evolution of such oscillations and pulsations (e.g., Roberts et al., 1984;Aschwanden, 1987Aschwanden, , 2006;;Aschwanden andBenz, 1988a, 1988b; Nakariakov and Kolotkov, 2020).These pulsations include sinusoidal-like periodic or quasi-periodic patterns and irregular non-periodic structures, all observed in a wide range from metric to microwave frequencies.
Pulsations at the radio wave range are observed recurrently showing different time structures from rigorously periodic to non-periodic.In the last-mentioned case of a non-periodic time structure, studies are pointing to deterministic chaotic behavior (e.g., Kurths and Herzel, 1986;Kurths and Karlicky, 1989;Mendez, et al., 2015).In this paper, we examine the time series structure associated with the dynamic characteristics of a group of a train of pulsations at 237 MHz solar radio emission of a Type IV radio burst on April 17, 2002.The train of pulsations can be described as periodic-deterministic chaos alternating regimes as they evolve in time.The evolution of the observed alternating sequence periodicchaotic-periodic is presented, and the route to chaos and the restoration to periodic behavior are discussed.Different techniques for examining time series, including nonlinear dynamics analysis, are applied.

Data description
Figure 1 shows the solar polarimeter radio emission data on April 17, 2002, at 235 MHz recorded at the Trieste Solar Radio System (INAF-Trieste Astronomical Observatory) in the time interval from 08:54:00 to 08:57:00 UT.The data were recorded with very high time resolution with a sampling rate of 100 Hz (10 ms).The polarization degree of these trains of pulsations is in average about 73% left-handed, which means a signi cantly high polarization degree (80% or more is considered a strong polarization degree).This radio emission is associated with a 2N solar are/M x-ray class in the NOAA/USAF active region 9906 (heliographic position S14W34).This activity has a broad-band Type IV burst as a radio counterpart.Three well-de ned, consecutive trains of pulsating structures were detected in this interval.
The rst train of pulsations (starting at 8.9049 hours and lasting 24.99 s) shows very regular sine-shaped pulsations followed by a second train but of irregular pulsations (lasting 37.96 s) which is followed again by other very regular sine-shaped pulsations similar to the former (starting at 8.9223 hours and lasting 23.25 s), indicating an alternating regime as the pulsations evolve.These trains are superimposed above a smooth variable base level of radio emission.To remark on the pulsating structures and refer all pulsation trains to the same background level, this smooth variable base level was removed using a median lter of 0.1 s (Fig. 2).After removing the base level and remarking the pulsating structures of the three trains, it is noticeable the remarkable sinusoidal behavior for the rst and third trains, interrupted by a second train with an irregular pro le.

Analysis of the data
We applied well-known time series analysis techniques to characterize the dynamics of these alternating trains of pulsations.

FFT spectral analysis
The rst examination carried out was a frequency spectral analysis by applying the fast Fourier transform (FFT) to each of the trains to nd out possible candidates for being classi ed as regular periodic behavior or signs for a chaotic-deterministic process.A strongly single dominant spectral component with a period of 50 ms is clearly noticeable for the rst and third trains of pulsations, indicating a candidate for a sinusoidal-like periodic process.On the other hand, the second train of pulsations exhibits a broadband spectrum with several important components of comparable amplitudes.This broadband spectrum could be due to a random or nonlinear process.Figure 3 shows the FFT spectral analysis for the rst and second pulsation trains, and Table 1 summarizes the general characteristics of the studied trains of pulsations.

Determinism test
The above FFT spectral analysis is not enough to judge the underlying mechanism generating these pulsations, in particular those related to irregular pulsations.That is, it does not allow us to conclude the deterministic or stochastic nature of the processes generating these pulsations.Therefore, what follows is the application of nonlinear analysis tools to examine whether these pulsations are due to periodic, deterministic, or stochastic processes.
The determinism test is a tool for guessing if the dynamic of a process is generated by a deterministic, rather than a stochastic, process.For this purpose, we used the reliable determinism test (Kaplan and Glass, 1992) based on a proper reconstruction of the system's state space (Takens, 1981).By using the Kaplan-Glass deterministic test, if the process is fully deterministic, the index of determinism is one; on the contrary, for a stochastic process, the index tends to zero.For the rst and third sinusoidal-like pulsation trains, a deterministic factor very close to one was found, in agreement with a deterministic periodic process.On the contrary, for the second train of irregular pulsations, the deterministic factor is 0.63, pointing to a non-periodic but deterministic nature of the process generating these irregular pulsations (Table 2).

Lyapunov exponents and correlation dimension
The Lyapunov exponent (λ) is a measure of sensitivity to initial conditions of a dynamical system, providing strong evidence of the predictability of the underlying mechanism generating such a dynamical system.The Lyapunov exponents measure the exponential divergence of nearby trajectories in the statespace, an abstract multidimensional space that plots all possible states of a dynamical system.A zero value for the Lyapunov exponents (λ = 0) means all orbits or trajectories connecting all possible states converge in a stable limit cycle in the state-space which is proper for a fully deterministic or periodic process.On the other hand, if the maximal Lyapunov exponent is larger than zero (λmax > 0) the orbits in the state-space diverge exponentially fast, which indicates a process that is sensitive to initial conditions, that is, chaotic.For the three trains of pulsations, the maximal Lyapunov exponents were calculated (Wolf et al. 1985) and we found values very close to zero (10 − 3 ) of the maximal Lyapunov exponent for the rst and third trains, while the value corresponding to the second train of pulsations is larger than zero (Table 2).These values are in line with the values found in the previous determinism test, reinforcing the idea of a periodic nature of the rst and third trains of pulsations and the chaotic-deterministic nature of the second train of pulsations.
Although the Lyapunov exponent is a valuable index for deciding whether the system is completely deterministic (λ ≤ 0), chaotic-deterministic (0 < λ < ∞), or stochastic (λ → ∞), it is not a quantitative parameter for measuring how deterministic a system is.To quantify how chaotic a system is, the correlation dimension is an extended-used parameter.The correlation dimension is related to the scaling properties of a dynamical system and its value is an index of how many degrees of freedom play a role, that is, how much sensitivity to the initial conditions is present (Eckmann and Ruelle 1985) giving a lower limit for the number of independent variables or degree of freedom involved in the system.Table 2 shows the calculated correlation dimension for the second train of pulsations with a maximal Lyapunov exponent larger than one indicating a low dimensional chaotic-deterministic process that can be characterized by a relatively low number of exited degrees of freedom.

Discussion and Conclusions
In this contribution, we describe a real-world scenario where a route to chaos and reverse processes take place.The results shown in Table 1 indicate a rst train of pulsations with a periodic regime where a clear single activated component of 50 ms (Fig. 3a), followed by a second train with a broadband spectrum with several signi cant components (Fig. 3b) and a third train that returns to a periodic regime with the same general characteristics of the rst train but with the only remarkable difference in the amplitude of the pulsations reduced by half (Table 1).The second train of pulsations points to a clear irregular nonperiodic behavior, accentuated by the fact that it shows a variable amplitude of the pulsations.This nonperiodic behavior could take place both from stochastic or from chaotic-deterministic processes.The results shown in Table 2 better suggest a chaotic-deterministic process as the driven agent causing the observed non-periodic behavior, rather than a stochastic one, characterized by a low-dimension attractor as the value for the correlation dimension indicates.

Route to chaos scenarios
There are several ways in which regular dynamics transform into chaos as some system parameter is changed.The route to chaos, the process by which a periodic dynamical system becomes chaoticdeterministic, has been su ciently studied from decades ago with ample literature on the matter (e.g., Eckmann, 1981, Swinney, 1983, and Argyris, 1993).The route to chaos consists of the activation of a determined number of grades of freedom as a parameter of the system reaches some critical value.
Feigenbaum, Ruelle-Takens-Newhouse, and Pomeau-Manneville are the most frequently cited scenarios for a route to chaos.Feigenbaum transition scenario, also known as the period doubling (Feigenbaum, 1978 and1979) is a transition to chaos in which the periodicity loses stability via period doublings, a cascade of halving frequencies (or doubling periods) from the fundamental f in the form of f/2, f/4, f/8… emerges.The process continues inde nitely and a chaotic behavior takes place.Ruelle-Takens-Newhouse scenario (Newhouse, Ruelle, and Takens, 1978) also named periodic-quasiperiodic-chaotic sequence, is a transition sequence described rst in Ruelle and Takens (1971), where the transition takes place from a periodic oscillation with a fundamental frequency f 1 followed by a quasiperiodic with two fundamental frequencies f 1 and f 2 , and then a broadband spectrum indicating the emergence of a chaotic motion.On the other hand, in the Pomeau-Manneville scenario, also known as intermittency (Pomeau and Manneville, 1980) the route to chaos via intermittency consists of several turbulent outbreaks of occasional bursts of noise must be observed at irregular intervals until the complete transition occurs when the regular or periodic behavior becomes a chaotic one as the intervals between the bursts decrease and it becomes impossible to recognize a periodic state.
In both Feigenbaum and Ruelle-Takens-Newhouse routes to chaos scenarios, since the chaotic regime takes place after the activation of a cascade or sequence of new components in the power spectrum, it is expected a transition time su ciently long as new components appear.In the same sense, in the Pomeau-Manneville route to chaos, outbreaks of irregular series that interrupt a regular pro le should be observed.However, the observed transition time between the periodic-chaotic deterministic-periodic phases is signi cantly shorter than the duration of each of the three phases.We understood this fact to mean that the transitions do not occur gradually as would be expected in the previous three mentioned routes to chaos scenarios.In addition, from the FFT analysis, no sign of any sequence of activation of period doubling or harmonics is evident during the transition from the periodic regime to the chaoticdeterministic one.
Furthermore, the reverse process remains to be explained, that is, the transition or return to periodic behavior so that the entire process can be explained as a whole.Tomita and Tsuda (1980) and D'Humiers et al. (1982) describe an attractive scenario for an alternating periodic-chaotic sequence occurring in the well-known Belousov-Zhabotinsky reaction and in a forced pendulum respectively.

remarks
above theoretical models for a route to chaos, which have been mathematically well described, the discussion of a real-world scenario event where the transition in both directions takes place, is the main goal of this contribution.The results discussed in this contribution show a change in the structure of a

Table 1
General characteristics of the studied trains of pulsations.

Table 2
Calculated parameters characterizing the studied trains of pulsations.