A note on Li–Yorke chaos in a coupled lattice system related with Belusov–Zhabotinskii reaction

This paper is concerned with the following system which comes from a lattice dynamical system stated by Kaneko in (Phys Rev Lett 65:1391–1394, 1990) and is related to the Belusov–Zhabotinskii reaction: $$\begin{aligned} x_{n}^{m+1}=(1-\varepsilon )f(x_{n}^{m})+\frac{1}{2}\varepsilon \left[ f(x_{n-1}^{m})+f(x_{n+1}^{m})\right] , \end{aligned}$$ x n m + 1 = ( 1 ? ε ) f ( x n m ) + 1 2 ε [ f ( x n ? 1 m ) + f ( x n + 1 m ) ] , where $m$ m is discrete time index, $n$ n is lattice side index with system size $L$ L (i.e., $n=1, 2, \ldots , L$ n = 1 , 2 , … , L ), $\varepsilon $ ε is coupling constant, and $f(x)$ f ( x ) is the unimodal map on $I$ I (i.e., $f(0)=f(1)=0$ f ( 0 ) = f ( 1 ) = 0 and $f$ f has unique critical point $c$ c with $0<c<1$ 0 < c < 1 and $f(c)=1$ f ( c ) = 1 ). It is proved that for coupling constant $\varepsilon =1$ ε = 1 , this CML (Coupled Map Lattice) system is chaotic in the sense of Li–Yorke for each unimodal selfmap on the interval $I=[0, 1]$ I = [ 0 , 1 ] .     

Chaos of a coupled lattice system related with the Belusov–Zhabotinskii reaction

In this paper we present a lattice dynamical system stated by Kaneko in (Phys Rev Lett, 65: 1391–1394, 1990) which is related to the Belusov-Zhabotinskii reaction. We prove that this CML (Coupled Map Lattice) system has positive topological entropy for zero coupling constant.     

The principal measure and distributional (p, q)-chaos of a coupled lattice system related with Belusov–Zhabotinskii reaction

García Guirao and Lampart (J Math Chem 48:66–71, 2010; J Math Chem 2 48:159–164, 2010) said that for non-zero couplings constant, the lattice dynamical system is more complicated. Motivated by this, in this paper, we prove that this coupled lattice system is distributionally (p, q)-chaotic for any pair 0?≤ p?≤ q?≤ 1 and its principal measure is not less than ${\frac{2}{3} + \sum_{n=2}^{\infty} \frac{1}{n} \frac{2^{n-1}}{(2^{n}+1)(2^{n-1}+1)}}$ for coupling constant ${0 < \epsilon < 1}$ .     

A note on the principal measure and distributional ({\varvec{p, q}})-chaos of a coupled lattice system related with Belusov–Zhabotinskii reaction

García Guirao and Lampart in (J Math Chem 48:159–164, 2010) presented a lattice dynamical system stated by Kaneko in (Phys Rev Lett 65:1391–1394, 1990) which is related to the Belusov–Zhabotinskii reaction. In this paper, we prove that for any non-zero coupling constant $\varepsilon \in (0, 1)$ , this coupled map lattice system is distributionally $(p, q)$ -chaotic for any pair $0\le p\le q\le 1$ , and that its principal measure is not less than $(1-\varepsilon )\mu _{p}(f)$ . Consequently, the principal measure of this system is not less than $$\begin{aligned} (1-\varepsilon )\left( \frac{2}{3}+\sum \limits _{n=2}^{\infty }\frac{1}{n}\frac{2^{n-1}}{(2^{n}+1) (2^{n-1}+1)}\right) \end{aligned}$$ for any non-zero coupling constant $\varepsilon \in (0, 1)$ and the tent map $\Lambda $ defined by $$\begin{aligned} \Lambda (x)=1-|1-2x|,\quad x\in [0, 1]. \end{aligned}$$      

The principal measure and distributional (p, q)-chaos of a coupled lattice system with coupling constant \varepsilon =1 related with Belusov–Zhabotinskii reaction

We consider the following system coming from a lattice dynamical system stated by Kaneko (Phys Rev Lett, 65:1391–1394, 1990) which is related to the Belusov–Zhabotinskii reaction: $$\begin{aligned} x_{n}^{m+1}=(1-\varepsilon )f\left( x_{n}^{m}\right) +\frac{1}{2}\varepsilon \left[ f(x_{n-1}^{m})+f\left( x_{n+1}^{m}\right) \right] , \end{aligned}$$ where $m$ is discrete time index, $n$ is lattice side index with system size $L$ (i.e., $n=1, 2, \ldots , L$ ), $\varepsilon \ge 0$ is coupling constant, and $f(x)$ is the unimodal map on $I$ (i.e., $f(0)=f(1)=0$ , and $f$ has unique critical point $c$ with $0<c<1$ and $f(c)=1$ ). In this paper, we prove that for coupling constant $\varepsilon =1$ , this CML (Coupled Map Lattice) system is distributionally $(p, q)$ -chaotic for any $p, q\in [0, 1]$ with $p\le q$ , and that its principal measure is not less than $\mu _{p}(f)$ . Consequently, the principal measure of this system is not less than $\frac{2}{3}+\sum _{n=2}^{\infty }\frac{1}{n}\frac{2^{n-1}}{(2^{n}+1) (2^{n-1}+1)}$ for coupling constant $\varepsilon =1$ and the tent map $\Lambda $ defined by $\Lambda (x)=1-|1-2x|, x\in [0, 1]$ . So, our results complement the results of Wu and Zhu (J Math Chem, 50:2439–2445, 2012).     

Comment on “A note on the principal measure and distributional (p,q)-chaos of a coupled lattice system related with Belusov–Zhabotinskii reaction”

In García Guirao and Lampart (J Math Chem 48:159–164, 2010) presented a lattice dynamical system stated by Kaneko (Phys Rev Lett 65:1391–1394, 1990) which is related to the Belusov–Zhabotinskii reaction. In this note, we give an example which shows that the proofs of Theorems 3.1 and 3.2 in [J Math Chem 51:1410–1417, 2013] are incorrect, and two open problems.     

Eliminating chaos in the Belousov–Zhabotinsky reaction by no-delay feedback and delayed feedback

We present numerical simulations of the three-variable model of the Belousov–Zhabotinsky reaction developed by Gyorgyi and Field, where chaos can occur in the free-running system. As the rate of in-flow in the continuous-flow stirred-tank reactor is controlled by the concentration of one of the reactor species through a feedback loop, we find two simple ways to bring the chaotic behavior to periodic behavior. One is to modify the feedback strength without time delay in the feedback loop, and the other is to modify the delay time of the feedback at constant feedback strength. The possible mechanisms for the two ways are discussed.     

Direct determination of copper in urine using a sol–gel optical sensor coupled to a multicommutated flow system

In this work, a multicommutated flow system incorporating a sol–gel optical sensor is proposed for direct spectrophotometric determination of Cu(II) in urine. The optical sensor was developed by physical entrapment of 4-(2-pyridylazo)resorcinol (PAR) in sol–gel thin films by means of a base-catalysed process. The immobilised PAR formed a red 2:1 complex with Cu(II) with maximum absorbance at 500 nm. Optical transduction was based on a dual-colour light-emitting diode (LED) (green/red) light source and a photodiode detector. The sensor had optimum response and good selectivity towards Cu(II) at pH 7.0 and its regeneration was accomplished with picolinic acid. Linear response was obtained for Cu(II) concentrations between 5.0 and 80.0 g L^–1, with a detection limit of 3.0 g L^–1 and sampling frequency of 14 samples h^–1. Interference from foreign ions was studied at a 10:1 (w/w) ion:Cu(II) ratio. Results obtained from analysis of urine samples were in very good agreement with those obtained by inductively coupled plasma mass spectrometry (ICP–MS); there was no significant differences at a confidence level of 95%.     

On the kinetics of pozzolanic reaction in metakaolin–lime–water system

The kinetics of pozzolanic reaction metakaolin–lime is studied in the present work. Metakaolin is prepared by calcination of enriched kaolin (deposit “Senovo”, Bulgaria) at temperature of 830 ± 10 °C in a labscale muffle oven. The reaction is performed in intensively stirred water suspension at different temperatures in the range 20–100 °C. The kinetics is analyzed by comparing the experimental data with theoretical curves, derived according to appropriate kinetic and diffusion models taking into account the grain size distribution of metakaolin. The macroscopic mechanism and activation energy of the reaction are determined. It is found, that the activation energy decreases gradually from 71 to 45 kJ/mol[Ca(OH)₂] with the increase of the reaction degree from 0.2 up to 0.6, respectively, which is a characteristic for transition regime reactions.     

Phase equilibria involving solid solutions in the Li–Mn–O system

Phase equilibria involving LiMn₂O_4-, Li₂MnO_3-, LiMnO_2-, Mn₃O_4-, and MnO-base solid solutions were studied with varied temperature and partial oxygen pressure. The \({P_{{o_2}}}\)–T and x–y projections of the P–T–x–y phase diagram of the Li–Mn?O system were constructed, as well as the key x–y isotherms of the Li₂O–MnO–MnO₂ quasi-ternary system. In some experiments, the authors’ hydride lithiation method was employed to prepare lithium-rich homogeneous three-component nonstoichiometric phases.     

Phase equilibria in a polyethylene–polystyrene system

The method of optical interferometry is used to study the interaction of PE with PS in situ. On the basis of the obtained data, phase diagrams of the PE–PS system are constructed for a number of molecular masses of the components. For PE and PS oligomers, the UCMT values are determined. Pair parameters for the interaction of homopolymers are calculated, and their dependences on temperature and molecular mass are considered. The quantitative analysis of the behavior of high-molecular-mass fractions of PE and PS at high temperatures is carried out, and the regions of a partial compatibility of the components are predicted.     

On the kinetics of pozzolanic reaction in the system kaolin–lime–water

The kinetics of the pozzolanic reaction of enriched kaolin from the “Senovo” deposit (Bulgaria) with lime is the object of this article. The kaolin contains kaolinite as a major clay mineral as well as admixtures of quartz and illite. The experimental data of pozzolanic activity at temperatures of 100 and 23 °C are obtained for different reaction times. The reaction degrees of kaolinite and lime at 100 °C are determined from the pozzolanic activity data using a powder X-ray diffraction analysis. The kinetic analysis is performed by joint presentation of theoretical and experimental data in dimensionless coordinates having in mind the influence of particle size distribution on the reaction rate. It is found by the kinetic analysis that the rate of entire reaction is limited by the rate of chemical reaction on the reaction surface up to degree of reaction near to 0.4. The rate of penetration of the chemical reaction into the kaolinite particles for this area—from the beginning to degree of reaction 0.4, is determined to be equal to 2.10⁻¹¹ m/s.     

Underpinning energetics of lithium bonding and stability in the Li–Pt–Sn system

Within the Li–Pt–Sn system, we examine the electronic structures and Li-binding of LiPtSn₂, Li₂PtSn and Li₃Pt₂Sn₃ with fluorite-related crystal structures. The structures with totally de-intercalated lithium keep the characteristics of the pristine ternary compound with a reduction of the volume. In Li₃Pt₂Sn₃ the binding energies of lithium belonging to three crystallographically inequivalent Wyckoff sites are different and point to distinct activities of de-intercalation concomitant with site-selective bonding magnitudes. The derived potentials are within the range of non-oxide binary and ternary lithium based compounds and indicate the possibility of at least partial delithiation.     

Suzuki reaction with aryl bromides at room temperature in the presence of a simple “ligand-free” catalytic system

Possibility was examined of using the simplest catalytic system (palladium salt and NaOH) in ethanol or in a mixture of ethanol and water as a catalyst precursor in order to obtain high product yields in the Suzuki reaction with nonactivated aryl bromides at room temperature without an inert atmosphere or ligand additives.     

Porous carbon coupled with an interlaced MoP–MoS2 heterojunction hybrid for efficient hydrogen evolution reaction

The design and development of electrocatalysts composed of non-noble-metal catalysts with both large surface area and high electrical conductivities are crucial for the hydrogen evolution reaction(HER).Here,a xylose-based porous carbon is coupled with a MoS2-Mo P heterojunction(MoS2-Mo P/FPC)hybrid and used as a promising catalyst for HER.The hybrid is prepared by immobilizing petal-like MoS2 nanosheets on porous carbon(MoS2/FPC),followed by controlling the phosphidation in Ar/H2 to form MoS2-Mo P/FPC.Red phosphorus provides the P species that can induce the construction of the heterojunction under the reducing atmosphere,along with the generation of a Mo P phase and the splitting of the MoS2 phase.The as-prepared MoS2-Mo P/FPC catalyst offers a low overpotential of 144 mV at a current density of 10 m A cm^-2 and a small Tafel slope of 41 m V dec^-1 for the HER in acidic media,as well as remarkable stability.Apart from the active nature of the hybrid,its outstanding activity is attributed to the MoS2-Mo P heterojunction,and the good charge/mass-transfer ability of porous carbon.This strategy provides a new method to develop and design low-cost and high-performance catalysts for the HER.     

Accelerating S↔Li2S Reactions in Li–S Batteries through Activation of S/Li2S with a Bifunctional Semiquinone Catalyst

The reaction rate bottleneck during interconversion between insulating S₈ (S) and Li₂S fundamentally leads to incomplete conversion and restricted lifespan of Li−S battery, especially under high S loading and lean electrolyte conditions. Herein, we demonstrate a new catalytic chemistry: soluble semiquinone, 2-tertbutyl-semianthraquinone lithium (Li⁺TBAQ⋅⁻), as both e^-/Li⁺ donor and acceptor for simultaneous S reduction and Li₂S oxidation. The efficient activation of S and Li₂S by Li⁺TBAQ⋅⁻ in the initial discharging/charging state maximizes the amount of soluble lithium polysulfide, thereby substantially improve the rate of solid–liquid-solid reaction by promoting long-range electron transfer. With in situ Raman spectra and theoretical calculations, we reveal that the activation of S/Li₂S is the rate-limiting step for effective S utilization under high S loading and low E/S ratio. Beyond that, the S activation ratio is firstly proposed as an accurate indicator to quantitatively evaluate the reaction rate. As a result, the Li−S batteries with Li⁺TBAQ⋅⁻ deliver superior cycling performance and over 5 times higher S utilization ratio at high S loading of 7.0 mg cm⁻² and a current rate of 1 C compared to those without Li⁺TBAQ⋅⁻. We hope this study contributes to the fundamental understanding of S redox chemical and inspires the design of efficient catalysis for advanced Li−S batteries.