Abstract
Dynamic states with intermittent oscillations consist of a chaotic mixture of large amplitude relaxation oscillations grouped in bursts, and between them, small-amplitude sinusoidal oscillations, or even the quiescent parts, known as gaps. In this study, intermittent dynamic states were generated in Bray–Liebhafsky (BL) oscillatory reaction in an isothermal continuously-fed, well-stirred tank reactor (CSTR) controled by changes of specific flow rate. The intermittent states were found between two regular periodic states and obtained for specific flow rate values from 0.020 to 0.082 min−1. Phenomenological analysis based on the quantitative characteristics of intermittent oscillations, as well as, the largest Lyapunov exponents calculated from experimentally obtained time series, both indicated the same type of behavior. Namely, fully developed chaos arises when approaching to the vertical asymptote which is somewhere between two bifurcations. Hence, this study proposes described route to fully developed chaos in the Bray-Liebhafsky oscillatory reaction as an explanation for experimentally observed intermittent dynamics. This is in correlation with our previously obtained results where the most chaotic intermittent chaos was achieved between the periodic oscillatory dynamic state and stable steady state, generated in BL under CSTR conditions by varying temperature and inflow potassium iodate concentration. Moreover, it was shown that, besides the largest Lyapunov exponent, analysis of chaos in experimentally obtained intermittent states can be achieved by a simpler approach which involves using the quantitative characteristics of the BL reaction evolution, that is, the number and length of gaps and bursts obtained for the various values of specific flow rates.
Highlights
Under CSTR conditions in the considered range of specific flow rate, we found the chaotic mixture of the regular sustained large-amplitude relaxation oscillations grouped in bursts and small-amplitude irregular sinusoidal ones grouped in gaps
The first chaotic emergence of low-amplitude oscillations, which occurred when the specific flow rate decreased, shifting the system from a periodic oscillatory dynamic state (Figure 2A, gray oscillogram), denotes the emerging of an initial intermittent state (Figure 2B)
This study examined a mixture of two types of dynamic state, which were chaotically combined to create a new intermittent dynamic state
Summary
The intermittent dynamic state is a chaotic state where two types of dynamic emerge in time, replacing each other stochastically. (Hilborn, 2004) The extent to which one type dominates the other can vary depending on the control parameter values. (Hilborn, 2004) In here considered case, intermittent dynamic state ( known as intermittent oscillations, intermittent chaos, or intermittency) (Pomeau and Manneville, 1980; Hilborn, 2004; Schuster and Just, 2005) represents a chaotic mixture of large amplitude relaxation oscillations grouped in bursts, and between them, there are small-amplitude sinusoidal. In the most recent papers (Bubanja et al, 2016, 2017), intermittent dynamic states were experimentally generated in the BL reaction system realized in a continuously-fed wellstirred tank reactor (CSTR) under variations of temperature and inflow concentration of potassium iodate. These two last papers, clearly show that experimentally generated intermittent chaos can be analyzed, by calculations of the largest Lyapunov exponents (λ) from experimentally obtained oscillograms and by the simple quantitative characteristics of the evolution of BL reaction, lengths of bursts, (packages of relaxing oscillations) and lengths of gaps between them (the periods of the sinusoid oscillations of relatively small amplitude when compared to relaxing oscillations). This allowed the system to stabilize in a new dynamic state with a changed control parameter
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