Three–step method to determine the eutectic composition of binary and ternary mixtures

A three-step method to determine the eutectic composition of a binary or ternary mixture is introduced. The method consists in creating a temperature–composition diagram, validating the predicted eutectic composition via differential scanning calorimetry and subsequent T-History measurements. To test the three-step method, we use two novel eutectic phase change materials based on \(\mathrm{Zn}(\hbox {NO}_3)_2\cdot 6\mathrm{H}_{2}{\mathrm O}\) and \(\mathrm{NH}_4\mathrm{NO}_3\)   respectively \(\mathrm{Mn}(\hbox {NO}_3)_2\cdot 6\mathrm{H}_{2}{\hbox {O}}\) and \(\mathrm{NH}_4\mathrm{NO}_3\) with equilibrium liquidus temperatures of 12.4 and 3.9  \(\,^{\circ }\mathrm {C}\) respectively with corresponding melting enthalpies of 135 J \(\mathrm{g}^{-1}\) (237 J \(\mathrm{cm}^{-3}\) ) respectively 133 J \(\mathrm{g}^{-1}\) (225 J \(\mathrm{cm}^{-3}\) ). We find eutectic compositions of 75/25 mass% for \(\mathrm{Zn}(\hbox {NO}_3)_2\cdot \mathrm{6H}_{2}{\mathrm{O}}\) and \(\mathrm{NH}_4\mathrm{NO}_3\) and 73/27 mass% for \(\mathrm{Mn}(\hbox {NO}_3)_2\cdot 6\mathrm{H}_{2}{\mathrm{O}}\) and \(\mathrm{NH}_4\mathrm{NO}_3\) . Considering a temperature range of 15 K around the phase change, a maximum storage capacity of about 172 J \(\mathrm{g}^{-1}\) (302 J \(\mathrm{cm}^{-3}\) ) respectively 162 J \(\mathrm{g}^{-1}\) (274 J \(\mathrm{cm}^{-3}\) ) was determined for \(\mathrm{Zn}(\hbox {NO}_3)_2\cdot \mathrm{6H}_{2}{\mathrm{O}}\) and \(\mathrm{NH}_4\mathrm{NO}_3\) respectively \(\mathrm{Mn}(\hbox {NO}_3)_2\cdot \mathrm{6H}_{2}{\mathrm{O}}\) and \(\mathrm{NH}_4\mathrm{NO}_3\) .     

Infrared Spectroscopy of \hbox {CH}_4/\hbox {N}_{2} and \hbox {C}_{2}\hbox {H}_{m}/\hbox {N}_{2} ({m = 2, 4, 6}) Gas Mixtures in a Dielectric Barrier Discharge

Fourier transform infrared spectroscopy of \(\hbox {CH}_{4}/\hbox {N}_{2}\) and \(\hbox {C}_{2}\hbox {H}_{m}/\hbox {N}_2\) ( \(m = 2, 4, 6\) ) gas mixtures in a medium pressure (300 mbar) dielectric barrier discharge was performed. Consumption of the initial gas and formation of other hydrocarbon and of nitrogen-containing HCN and \(\hbox {NH}_{3}\) molecules was observed. \(\hbox {NH}_{3}\) formation was further confirmed by laser absorption measurements. The experimental result for \(\hbox {NH}_{3}\) is at variance with simulation results.     

Lamellar/Disorder Phase Transition in a Mixture of Water/2,6-Dimethylpyridine/Antagonistic Salt

The effects of adding an antagonistic salt, sodium tetraphenylborate ( \(\hbox {NaBPh}_4\) ), to a binary mixture of deuterated water and 2,6-dimethylpyridine were investigated by visual inspection, optical microscopy, and small-angle neutron scattering. With increasing salt concentration, the two-phase region shrinks. When the concentration of \(\hbox {NaBPh}_4\) is \(85\hbox { mmol}{\cdot} \hbox {L}^{-1}\) , a temperature-induced lamellar/disorder phase transition is observed at 338 K. These trends are similar to those observed for a mixture of water/3-methylpyridine/ \(\hbox {NaBPh}_4\) (Sadakane et al., Phys. Rev. Lett. 103, 167803 (2009)).     

Hydrogen Peroxide Production in an Atmospheric Pressure RF Glow Discharge: Comparison of Models and Experiments

The production of \(\hbox {H}_2\hbox {O}_2\) in an atmospheric pressure RF glow discharge in helium-water vapor mixtures has been investigated as a function of plasma dissipated power, water concentration, gas flow (residence time) and power modulation of the plasma. \(\hbox {H}_2\hbox {O}_2\) concentrations up to 8 ppm in the gas phase and a maximum energy efficiency of 0.12 g/kWh are found. The experimental results are compared with a previously reported global chemical kinetics model and a one dimensional (1D) fluid model to investigate the chemical processes involved in \(\hbox {H}_2\hbox {O}_2\) production. An analytical balance of the main production and destruction mechanisms of \(\hbox {H}_2\hbox {O}_2\) is made which is refined by a comparison of the experimental data with a previously published global kinetic model and a 1D fluid model. In addition, the experiments are used to validate and refine the computational models. Accuracies of both model and experiment are discussed.     

Symmetry-itemized enumeration of RS-stereoisomers of allenes. I. The fixed-point matrix method of the USCI approach combined with the stereoisogram approach

After the RS-stereoisomeric group \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}\) of order 16 has been defined by starting point group \(\mathbf{D}_{2d}\) of order 8, the isomorphism between \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}\) and the point group \(\mathbf{D}_{4h}\) of order 16 is thoroughly discussed. The non-redundant set of subgroups (SSG) of \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}\) is obtained by referring to the non-redundant set of subgroups of \(\mathbf{D}_{4h}\) . The coset representation for characterizing the orbit of the four positions of an allene skeleton is clarified to be \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}(/\mathbf{C}_{s\widetilde{\sigma }\widehat{I}})\) , which is closely related to the \(\mathbf{D}_{4h}(/\mathbf{C}_{2v}^{\prime \prime \prime })\) . According to the unit-subduced-cycle-index (USCI) approach (Fujita, Symmetry and combinatorial enumeration of chemistry. Springer, Berlin 1991), the subduction of \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}(/\mathbf{C}_{s\widetilde{\sigma }\widehat{I}})\) is examined so as to generate unit subduced cycle indices with chirality fittingness (USCI-CFs). Then, the fixed-point matrix method of the USCI approach is applied to the USCI-CFs. Thereby, the numbers of quadruplets are calculated in an itemized fashion with respect to the subgroups of \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}\) . After the subgroups of \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}\) are categorized into types I–V, type-itemized enumeration of quadruplets is conducted to illustrate the versatility of the stereoisogram approach.     

Heat capacity and density of potassium iodide solutions in mixed N-methylpyrrolidone-water solvent at 298.15 K

The heat capacity and density of potassium iodide solutions in a mixed N-methylpyrrolidone (MP)-water solvent with a low content of the organic component are measured via calorimetry and densimetry at 298.15 K. Standard partial molal heat capacities \(\bar C_{p,2}^ \circ \) and volumes \(\bar V_2^ \circ \) of potassium iodide in MP-water mixtures are calculated. Standard heat capacities \(\bar C_{p,i}^ \circ \) and volumes \(\bar V_i^ \circ \) of potassium and iodide ions are determined. The character of the changes in heat capacity and volume are discussed on the basis of calculating additivity coefficients δ _c and δ _v upon the mixing of isomolal binary solutions KI-MP and KI-water, depending on the composition of the MP-H₂O mixture and the concentration of the electrolyte.     

On k-resonance of grid graphs on the plane,torus and cylinder

Grid graphs on the plane, torus and cylinder are finite 2-connected bipartite graphs embedded on the plane, torus and cylinder, respectively, whose every interior face is bounded by a quadrangle. Let \(k\) be a positive integer, a grid graph is \(k\) -resonant if the deletion of any \(i \le k\) vertex-disjoint quadrangles from \(G\) results in a graph either having a perfect matching or being empty. If \(G\) is \(k\) -resonant for any integer \(k \ge 1\) , then it is called maximally resonant. In this study, we provide a complete characterization for the \(k\) -resonance of grid graphs \(P_m\times P_n\) on plane, \(C_m\times C_n\) on torus and \(P_m\times C_n\) on cylinder.     

Densities,Refractive Indices,Ultrasonic Speeds and Excess Properties of Acetonitrile + Poly(ethylene glycol) Binary Mixtures at Temperatures from 298.15 to 313.15 K

The densities, ρ, refractive indices, n _D, and ultrasonic speeds, u, of binary mixtures of acetonitrile (AN) with poly(ethylene glycol) 200 (PEG200), poly(ethylene glycol) 300 (PEG300) and poly(ethylene glycol) 400 (PEG400) were measured over the entire composition range at temperatures (298.15, 303.15, 308.15 and 313.15) K and at atmospheric pressure. From the experimental data, the excess molar volumes, \( V_{\text{m}}^{\text{E}} \) , deviations in refractive indices, \( \Delta n_{\text{D}} \) , excess molar isentropic compressibility, \( K_{{s , {\text{m}}}}^{\text{E}} \) , excess intermolecular free length, \( L_{\text{f}}^{\text{E}} \) , and excess acoustic impedance, Z ^E, have been evaluated. The partial molar volumes, \( \overline{V}_{\text{m,1}} \) and \( \overline{V}_{\text{m,2}} \) , partial molar isentropic compressibilities, \( \overline{K}_{{s , {\text{m,1}}}} \) and \( \overline{K}_{{s , {\text{m,2}}}} \) , and their excess values over whole composition range and at infinite dilution have also been calculated. The variations of these properties with composition and temperature are discussed in terms of intermolecular interactions in these mixtures. The results indicate the presence of specific interactions among the AN and PEG molecules, which follow the order PEG200 < PEG300 < PEG400.     

Spin correlation in Pr IV and Tm IV free ions

The coefficients \(c_{k}\) (k = 2, 4, 6) that pertain to spin-correlated matrix elements of the tensor operator \({{\varvec{U}}}^{{\varvec{(k)}}}\) have been evaluated by means of the differences \({{\varvec{U}}}^{{\varvec{(k)}}}\) (intermediate) \(-\) \({{\varvec{U}}}^{{\varvec{(k)}}}\) (LS) and the reduced matrix elements of the operator \({{\varvec{V}}}^{{\varvec{(1k)}}}\) . Only spin-allowed transitions have been considered from each ground level to the excited energy levels within the \(4\hbox {f}^{2}\) and \(4\hbox {f}^{12}\) configurations of the free ions Pr (3+) and Tm (3+), respectively. The values of the coefficients \(c_{k}\) thus found correspond in most cases by sign and order of magnitude to those determined in other sources as corrections to lanthanide (3+) crystal-field parameters.     

A preliminary study of gross alpha/beta activity concentrations in drinking waters from Albania

A pilot study involving screening measurements of gross alpha/beta activity concentration in drinking water was performed in 12 cities with the highest population density in Albania. The aim of this study was to develop a first insight regarding the radiological quality of drinking and bottled water supplies. The tap and bottled water samples investigated are acceptable for consumption, complying with the WHO recommendations for drinking water. The average gross alpha and beta activity concentrations were \( 36_{ - 18}^{ + 37} \) and \( 269_{ - 150}^{ + 337} \)  mBq/L, respectively in tap waters. While for bottled water the gross alpha and beta activities were respectively \( 39_{ - 23}^{ + 55} \) and \( 220_{ - 132}^{ + 336} \)  mBq/L. The data obtained can provide information for authorities regarding the quality of drinking water and a baseline for future contaminations.     

Densities,viscosities, and excess properties for binary mixtures of ethylene glycol with amides at 308.15 K

The densities, ρ, and viscosities, η, of binary mixtures of ethylene glycol with formamide, N,N-dimethyl formamide and N,N-dimethyl acetamide, have been measured over the entire composition range at 308.15 K. From this experimental data, excess molar volume, \( V_{\text{m}}^{\text{E}} \) , deviation in viscosity, Δη, and excess Gibbs free energy of activation of viscous flow, \( \Delta G^{{ * {\text{E}}}}, \) have been determined. Negative values of \( V_{\text{m}}^{\text{E}} \) , Δη, and \( \Delta G^{{ * {\text{E}}}} \) are observed over the entire composition range in the mixtures studied. The observed negative values of various excess and deviation parameters are attributed to the existence of strong interactions, like dipole–dipole interactions, H-bonding between the carbonyl group of amide molecules, and hydroxyl group of glycol molecules, geometrical fitting of smaller molecules into the voids created by larger molecules in the liquid mixtures. The excess properties have been fitted to Redlich–Kister-type polynomial, and the corresponding standard deviations have been calculated. The derived partial molar volumes and excess partial molar volumes also support the \( V_{\text{m}}^{\text{E}} \) results. The experimental viscosity data of all of these liquid mixtures have been correlated with four viscosity models.     

Chaos in oscillatory sorption of hydrogen in palladium

Aperiodic oscillations in the sorption of hydrogen and deuterium in palladium have been observed. An expression relating the square of a function, with the derivative and integral with variable upper limit of the same function has been proved and proposed to be used as a base for a chaos-vs.-random test. The result of one “branch” of the test is a real number \(D \in [0,2]\) ; close to zero for the deterministic and smooth datasets, and approaching two for the random or discrete datasets. Another “branch” of the test, based on the same mathematical relation, produces two functions that appear to be convergent for deterministic and smooth datasets, but run totally divergent for random or discrete ones. The \(D\) -values yielded by deterministic time series, recorded in the periodic and quasiperiodic sorptions of H \(_{2}\) or D \(_{2}\) in Pd, are around 0.001. On the other hand, the databases that were presumably random or non-smooth yielded the test results from \(D= 0.2\) to \(D= 1.9\) . Against these benchmarks, the experimental, aperiodic oscillations scored around 0.003 in \(D\) , which is much closer to the deterministic than to a random manner.     

Thermodynamic investigations of 1-ethyl-3-methylimidazolium tetrafluoroborate and cycloalkanone mixtures

The densities, ρ, speeds of sound, u, and heat capacities, (C _P)_mix, for binary 1-ethyl-3-methylimidazolium tetrafluoroborate (1) + cyclopentanone or cyclohexanone (2) mixtures within temperature range (293.15–308.15 K) and excess molar enthalpies, H ^E, at 298.15 K have been measured over the entire composition range. The excess molar volumes, V ^E, excess isentropic compressibilities, \( \kappa_{\text{S}}^{\text{E}}, \) and excess heat capacities, \( C_{\text{P}}^{\text{E}}, \) have been computed from the experimental results. The V ^E, \( \kappa_{\text{S}}^{\text{E}} \) , H ^E, and \( C_{\text{P}}^{\text{E}} \) values have been calculated and compared with calculated values from Graph theory. It has been observed that V ^E, \( \kappa_{\text{S}}^{\text{E}} \) , H ^E, and \( C_{\text{P}}^{\text{E}} \) values were predicted by Graph theory compare well with their experimental values. The V ^E, \( \kappa_{\text{S}}^{\text{E}}, \) and H ^E thermodynamic properties have also been analyzed in terms of Prigogine–Flory–Patterson theory.     

Effect of activation volume on the defect-induced anomalous electronic transport in Rb_{0.8}Fe_2Se_2

We adopted an absolute-reaction model which is considering the hole(defect)-induced charged frictionless transport to explain the unusual experiment: Two single crystalline samples of the same nominal composition Rb \(_{0.8}\) Fe \(_2\) Se \(_2\) were prepared using the self-flux technique via two different precursor routes. Although the difference in the final chemical composition falls within a narrow range, one was superconducting with a \(T_c \sim 31\) K, while the other behaves like a narrow gap semiconductor.     

Aspects of size extensivity in unitary group adapted multi-reference coupled cluster theories: the role of cumulant decomposition of spin-free reduced density matrices

We present in this paper a comprehensive study of the various aspects of size extensivity of a set of unitary group adapted multi-reference coupled cluster (UGA-MRCC) theories recently developed by us. All these theories utilize a Jeziorski–Monkhorst (JM) inspired spin-free cluster Ansatz of the form \(|\varPsi \rangle = \sum\nolimits_\mu \varOmega _\mu |\phi _\mu \rangle c_\mu\) with \(\varOmega _\mu =\{\exp ({T_\mu })\}\) , where \(T_\mu\) is expressed in terms of spin-free generators of the unitary group \(U(n)\) for n-orbitals with the associated cluster amplitudes. \(\{...\}\) indicates normal ordering with respect to the common closed shell \(core\) part, \(|0\rangle\) , of the model functions, \(\{\phi _\mu \}\) which is taken as the vacuum. We argue and emphasize in the paper that maintaining size extensivity of the associated theories is consequent upon (a) connectivity of the composites, \(G_\mu\) , containing the Hamiltonian \(H\) and the various powers of \(T\) connected to it, (b) proving the connectivity of the MRCC equations which involve not only \(G_\mu\) s but also the associated connected components of the spin-free reduced density matrices (RDMs) obtained via their cumulant decomposition and (c) showing the extensivity of the cluster amplitudes for non-interacting groups of orbitals and eventually of the size-consistency of the theories in the fragmentation limits. While we will discuss the aspect (a) above rather briefly, since this was amply covered in our earlier papers, the aspect (b) and (c), not covered in detail hitherto, will be covered extensively in this paper. The UGA-MRCC theories dealt with in this paper are the spin-free analogs of the state-specific and state-universal MRCC developed and applied by us recently.We will explain the unfolding of the proof of extensivity by analyzing the algebraic structure of the working equations, decomposed into two factors, one containing the composite \(G_\mu\) that is connected with the products of cumulants arising out of the cumulant decomposition of the RDMs and the second term containing some RDMs which is disconnected from the first and can be factored out and removed. This factorization ultimately leads to a set of connected MRCC equations. Establishing the extensivity and size-consistency of the theories requires careful separation of truly extensive cumulants from the ones which are a measure of spin correlation and are thus connected but not extensive. We have discussed in detail, using diagrams, the factorization procedure and have used suitable example diagrams to amplify the meanings of the various algebraic quantities of any diagram. We conclude the paper by summarizing our findings and commenting on further developments in the future.     

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).     

Volumetric and Viscometric Studies in Binary Liquid Mixtures of 2-Methyl-2-propanol with Some Ketones at Different Temperatures

Densities, ??, and viscosities, ??, of binary mixtures of 2-methyl-2-propanol with acetone (AC), ethyl methyl ketone (EMK) and acetophenone (AP), including those of the pure liquids, were measured over the entire composition range at 298.15, 303.15 and 308.15?K. From these experimental data, the excess molar volume $V_{\mathrm{m}}^{\mathrm{E}}$ , deviation in viscosity ????, partial and apparent molar volumes ( $\overline{V}_{\mathrm{m},1}^{\,\circ }$ , $\overline{V}_{\mathrm{m},2}^{\,\circ }$ , $\overline{V}_{\phi ,1}^{\,\circ}$ and $\overline{V}_{\phi,2}^{\,\circ} $ ), and their excess values ( $\overline{V}_{\mathrm{m},1}^{\,\circ \mathrm{E}}$ , $\overline{V}_{\mathrm{m,2}}^{\,\circ \mathrm{ E}}$ , $\overline {V}_{\phi \mathrm{,1}}^{\,\circ \mathrm{ E}}$ and $\overline{V}_{\phi \mathrm{,2}}^{\,\circ \mathrm{ E}}$ ) of the components at infinite dilution were calculated. The interaction between the component molecules follows the order of AP > AC > EMK.     

A mathematical approach to chemical equilibrium theory for gaseous systems—III: \hbox {Q}_\mathrm{p}, \hbox {Q}_\mathrm{c}, and \hbox {Q}_{\mathrm{x}}

The reaction quotient Q can be expressed in partial pressures as $\hbox {Q}_\mathrm{P}$ or in mole fractions as $\hbox {Q}_{\mathrm{x}}$ . $\hbox {Q}_\mathrm{P}$ is ostensibly more useful than $\hbox {Q}_{\mathrm{x}}$ because the related $\hbox {K}_{\mathrm{x}}$ is a constant for a chemical equilibrium in which T and P are kept constant while $\hbox {K}_{\mathrm{P}}$ is an equilibrium constant under more general conditions in which only T is constant. However, as demonstrated in this work, $\hbox {Q}_{\mathrm{x}}$ is in fact more important both theoretically and technically. The relationships between $\hbox {Q}_{\mathrm{x}}$ , $\hbox {Q}_\mathrm{P}$ , and $\hbox {Q}_{\mathrm{C}}$ are discussed. Four examples of applications are given in detail.     

Acid–Base Properties of the \mathrm{Fe}(\mathrm{CN})_{6}^{3-}/\mathrm{Fe}(\mathrm{CN})_{6}^{4-} Redox Couple in the Presence of Various Background Mineral Acids and Salts

The acid?Cbase behavior of $\mathrm{Fe}(\mathrm{CN})_{6}^{4-}$ was investigated by measuring the formal potentials of the $\mathrm{Fe}(\mathrm{CN})_{6}^{3-}$ / $\mathrm{Fe}(\mathrm{CN})_{6}^{4-}$ couple over a wide range of acidic and neutral solution compositions. The experimental data were fitted to a model taking into account the protonated forms of $\mathrm{Fe}(\mathrm{CN})_{6}^{4-}$ and using values of the activities of species in solution, calculated with a simple solution model and a series of binary data available in the literature. The fitting needed to take account of the protonated species $\mathrm{HFe}(\mathrm{CN})_{6}^{3-}$ and $\mathrm{H}_{2}\mathrm{Fe}(\mathrm{CN})_{6}^{2-}$ , already described in the literature, but also the species $\mathrm{H}_{3}\mathrm{Fe}(\mathrm{CN})_{6}^{-}$ (associated with the acid?Cbase equilibrium $\mathrm{H}_{3}\mathrm{Fe}(\mathrm{CN})_{6}^{-}\rightleftharpoons \mathrm{H}_{2}\mathrm{Fe}(\mathrm{CN})_{6}^{2-} + \mathrm{H}^{+}$ ). The acidic dissociation constants of $\mathrm{HFe}(\mathrm{CN})_{6}^{3-}$ , $\mathrm{H}_{2}\mathrm{Fe}(\mathrm{CN})_{6}^{2-}$ and $\mathrm{H}_{3}\mathrm{Fe}(\mathrm{CN})_{6}^{-}$ were found to be $\mathrm{p}K^{\mathrm{II}}_{1}= 3.9\pm0.1$ , $\mathrm{p}K^{\mathrm{II}}_{2} = 2.0\pm0.1$ , and $\mathrm{p}K^{\mathrm{II}}_{3} = 0.0\pm0.1$ , respectively. These constants were determined by taking into account that the activities of the species are independent of the ionic strength.     

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 ] .