Nonlinear Chain Research Articles

Resilience to Supply Disruptions in a Non-Linear Two-Tier Supply Chain Model

This article describes using a nonlinear APVIOBPCS to model the resilience response to the failure of one of the suppliers of a two-tier supply chain comprising a retailer and two suppliers. The second supplier then supplies all the goods required by the retailer. The model is chosen as the simplest case to examine resilience. Time to recover from disruption to ship all goods required by customers is slightly worse when the order rate when it reaches capacity limits but is less than the delay in the system from supplier to final shipment. Just over one weeks' maximum shipments stock at each tier guarantees shipments impervious to the collapse of one supplier, controller type has little effect on the performance of the model. These results agree with other researchers in general trend but not in detail. The response does not match the ‘Resilience Triangle.'

Amplitude-induced bandgap: New type of bandgap for nonlinear elastic metamaterials

In this paper, we report a new type of bandgap phenomenon called “amplitude-induced bandgap”, which is induced by amplitude in nonlinear metamaterials. Generally, bandgap phenomena in elastic metamaterials or phononic crystals have been achieved by the artificial unit cells which prevent wave propagation with Bragg scattering or internal resonance. However, the amplitude-induced bandgap originates from the amplitude and nonlinearity unlike previously discovered bandgap phenomena. Accordingly, the amplitude-induced bandgap is shown to exhibit various peculiar characteristics that have not been previously observed. Here, an analytic investigation about the amplitude-induced bandgap is conducted based on the perturbation approach in a nonlinear softening monoatomic chain. Also, various peculiar characteristics of the amplitude-induced bandgap are studied in detail. Numerical simulations using finite element analysis are carried out to validate the amplitude induced bandgap. In addition, the transmission characteristics of nonlinear bandgap are also analytically studied based on the recursive method and numerically verified through the simulations.

A Novel Sliding Mode Control with Low-Pass Filter for Nonlinear Handling Chain System in Container Ports

Nonlinearities in a container port handling chain include mainly nonnegative arrive rate of container cargoes, limited container handling completion rate, and nonnegative unsatisfied freight requirement constraints. The nonlinearity influences the operation resources availability and consequently the planned container port handling strategies. Developments presented in this work are devoted to a novel design of sliding mode control with low-pass filter (SMC-LPF) to nonlinear handling chain system (HCS) in container ports. The SMC-LPF can effectively reduce unsatisfied freight requirement of the HCS and make chattering decrease significantly. To illustrate the effectiveness and accuracy of the proposed SMC-LPF, an application to a real container port in China is outlined. The performances of the SMC-LPF for the nonlinear HCS in container ports outperform those of the traditional method, particle swarm optimization algorithm, and slide mode control under simulations with a unit step signal and a sinusoidal signal with offset as the freight requirements. The contributions herein demonstrate the proposed control strategy in weakening chattering, reducing the unsatisfied freight requirements to 0 as close as possible in the HCS, maximizing the operation resilience and robustness of port and shipping supply chain against parametric perturbation, external disturbances, and fluctuant handling abilities.

Simultaneous energy harvesting and vibration control in a nonlinear metastructure: A spectro-spatial analysis

Considerable attention has recently been given to the study of simultaneous energy harvesting and vibration attenuation in metastructures. However, only linear metastructures were investigated although nonlinear metastructures and nonlinear electromechanical devices offer superior interesting wave propagation phenomena (e.g., birth of solitary waves, tunable bandgap, acoustic nonreciprocity) and broadband energy harvesting. In this paper, we investigate the wave propagation in a weakly nonlinear metastructure with electromechanical resonators. Explicit expressions are derived for the nonlinear dispersion relations using the method of multiple scales. These expressions are validated via direct numerical integration. We carried out parametric studies to investigate the role of different parameters of the electromechanical resonators on the linear and nonlinear band structure. To obtain further detailed information on the nonlinear wave propagation, we employ spectro-spatial analysis on the numerical simulations. This spectro-spatial analysis can reveal the output voltage distortion due to different types of nonlinearities. The results indicate that nonlinear chain can enhance energy harvesting through the birth of solitary wave and without degrading the boundary of the bandgap. The results also suggest that such a system is suitable for designing electromechanical diodes and rectifiers.

Numerical solution of time-fractional three-species food chain model arising in the realm of mathematical ecology

This research paper implements the fractional homotopy analysis transform technique to compute the approximate analytical solution of the nonlinear three-species food chain model with time-fractional derivatives. The offered technique is a fantastic blend of homotopy analysis method (HAM) and Laplace transform (LT) operator and has been used fruitfully in the numerical computation of various fractional differential equations (FDEs). This paper involves the fractional derivatives of Caputo style. The numerical solutions of this selected fractional-order food chain model are evaluated by making use of the associated initial conditions. It is revealed by the adopting procedure that the more desirable estimation of the solution can be easily acquired through the calculation of some number of iteration terms only — a fact which authenticates the easiness and soundness of the suggested hybrid scheme. The variations of fractional order of time derivative on the solutions for different specific cases have been depicted through graphical presentations. The outcomes demonstrated through the graphs expound that the adopted scheme is very fantastic and accurate.

Research of nonlinear electric chains with two and more nonlinear elements in systems power supply

Work it is devoted to research of autoparametrical fluctuations at a frequency of the third subharmonica in nonlinear electric chains by two nonlinear elements. Conditions and a zone of steady excitement of a subharmonica of the third order, and also critical parameters the subharmonicas of the third providing steady excitement an order are defined.

Target triggered ultrasensitive electrochemical polychlorinated biphenyl aptasensor based on DNA microcapsules and nonlinear hybridization chain reaction.

In this work, we demonstrated an ultrasensitive detection platform for polychlorinated biphenyls (PCBs) based on DNA microcapsules and a nonlinear hybridization chain reaction (NHCR). In the process, first, electrochemical signal molecules (Methylene Blue, MB) were sealed in the prepared DNA microcapsules. In the presence of PCB-72, DNA microcapsules could be dissociated with the conjugation of the aptamer and target, and meanwhile, the released DNA strand could initiate the NHCR and trigger the chain branching growth of DNA dendrimers. Because the released MBs were intercalated into the DNA dendrimer, enhanced electrochemical responses could be detected. This method exhibited ultrahigh sensitivity to PCB-72 with a detection limit of 0.001 ng mL-1. Furthermore, the present aptasensor was also capable of discriminating different PCB congeners. Therefore, the devised label-free and enzyme-free amplification electrochemical aptasensor strategy has great potential for the detection of PCB-72 in real samples, and this strategy may also become an attractive alternative for sensitive and selective small molecule, protein, nucleic acid and nuclease activity detection.

Ergodicity Coefficients for Higher-Order Stochastic Processes

The use of higher-order stochastic processes such as nonlinear Markov chains or vertex-reinforced random walks is significantly growing in recent years as they are much better at modeling high dime...

Нелокальные солитоны в нелинейной цепочке атомов

A nonlinear 1D chain with nonlocal interaction is considered. Using the multiscale expansion method, a nonlocal equation is obtained that describes the propagation of envelope waves in a medium. The properties of the obtained equation were studied and exact soliton-like solutions were constructed by the Darboux method of transformations.

Group classification of third order nonlinear wave equations chain’s

The paper presents the results of point symmetrys Lie algebras for third-order nonlinear wave equations calculating linked into a chain by Bäcklund transformations. Calculations are carried out by using Lie-Ovsyannikov method of group analysis. The basic algebras of point symmetries of the indicated equations are found, all possible cases of their extension are revealed, and the commutator tables of algebras found are calculated.

Stability of compact breathers in translationally-invariant nonlinear chains with flat dispersion bands

The paper addresses compact oscillatory states (compact breathers) in translationally-invariant lattices with flat dispersion bands. The compact breathers appear in such systems even in the linear approximation. If the interactions are nonlinear, but comply with the flat-band symmetry, the compact breather solutions exist, but can lose their stability for certain parameter values. As benchmark nonlinear potentials, we use the $\beta$-FPU (Fermi-Pasta-Ulam) and vibro-impact models. Loss of stability is numerically observed to occur through either pitchfork or Hopf bifurcations. The loss of stability can occur through two qualitatively different mechanisms -- through internal instability in the basic lattice elements, or through interaction of the compact breather with the linear passband of the lattice. The former scenario is more typical for high-amplitude breathers, and the latter -- for low amplitudes. For the high-amplitude case, insights into the nature of compact-mode loss-of-stability are obtained by resorting to the limit of a piecewise-linear system, where interactions are represented by conservative impacts. This issue calls for detailed introspection into integrability of piecewise-linear (impacting) systems and their relation to the smooth system. An idea for a sensor based on the studied mechanisms is suggested.

Spectro-spatial analyses of a nonlinear metamaterial with multiple nonlinear local resonators

Recent focus has been given to spectro-spatial analysis of nonlinear metamaterials since they can predict interesting nonlinear phenomena not accessible by spectral analysis (i.e., dispersion relations). However, current studies are limited to a nonlinear chain with single linear resonator or linear chain with nonlinear resonator. There is no work that examines the combination of nonlinear chain with nonlinear resonators. This paper investigates the spectro-spatial properties of wave propagation through a nonlinear metamaterials consisting of nonlinear chain with multiple nonlinear local resonators. Different combinations of softening and hardening nonlinearities are examined to reveal their impact on the traveling wave features and the band structure. The method of multiple scales is used to obtain closed-form expressions for the dispersion relations. Our analytical solution is validated via the numerical simulation and results from the literature. The numerical simulation is based on spectro-spatial analysis using signal processing techniques such as spatial spectrogram, wave filtering, and contour plots of 2D Fourier transform. The spectro-spatial analysis provides a detailed information about wave distortion due to nonlinearity and classify the distortion into different features. The observations suggest that nonlinear chain with multiple nonlinear resonators can affect the waveform at all wavelength limits. Such nonlinear metamaterials are suitable for broadband vibration control and energy harvesting, as well as other applications such as acoustic switches, diodes, and rectifiers, allowing wave propagation only in a pre-defined direction.

Interaction of phonons with discrete breathers in one-dimensional chain with tunable type of anharmonicity

In this paper, we consider the interaction of small amplitude waves (phonons) with standing discrete breather (DB) in the one-dimensional chain of harmonically coupled particles interacting with the anharmonic one-site potential, which can be of hard-type or soft-type anharmonicity. The coefficients of phonon reflection and transmission are calculated numerically. It is found that for the case of hard-type anharmonicity (soft-type anharmonicity) DBs are more transparent for short-wavelength (long-wavelength) phonon waves, while they efficiently reflect long-wavelength (short-wavelength) phonons. In thermal equilibrium, when all phonons have equal energy density, it is found that for the same width of the transparency window, DB transmits less energy in the case of the hard-type anharmonicity. This is so because, in this case, DB reflects long-wavelength phonons, which have larger group velocity and hence greater contribution to the net energy flux through the DB. In this sense, DBs more efficiently suppress thermal conductivity in the chain with hard-type anharmonicity. Our results contribute to a better understanding of the role of discrete breathers in the heat flow in nonlinear chains.

Impact of non-linear resonators in periodic structures using a perturbation approach

Abstract The work describes the wave propagation in a periodic structure formed by a linear spring-mass chain with local Duffing non-linear resonators. The wave propagation of the system is studied using the Floquet-Bloch theorem combined with a perturbation approach to identify the dispersion relations in the nonlinear periodic structure. The theoretical model is benchmarked by a numerical approach based on imposing a wave number on a finite resonant spring-mass system as an initial condition, and then obtaining the corresponding frequency from the chain amplitude in the time domain. The perturbation and the numerical methods are compared to discuss the behaviour of the wave propagation in the nonlinear resonators periodic chain considered in this work. The results from the perturbation techniques are also compared with the ones generated through an Harmonic Balance Method previously used in open literature.

Stabilization of bianisotropic fluid models for time domain simulation

Willis fluids are characterized by constitutive relations that couple the pressure and momentum density to both the particle velocity and the volume strain. This effective dynamic response coupling may arise due to microstructural asymmetry, long range order, or time-varying material properties and has been shown to be analogous to electromagnetic bianisotropy [Phys. Rev. B 96, 104303 (2017)]. Recent work has shown that, when wavevectors exceed a critical magnitude, the currently employed model for passive Willis fluids possesses a linear instability stemming from the truncation of higher-order coupling terms in the derivation of the constitutive relationships [J. Acoust. Soc. Am. 144(3), 1832 (2018)]. While this instability can be avoided when using frequency domain methods, it poses significant problems for time domain simulations, as short-wavelength numerical noise may grow exponentially. In the present work, we report on methods of stabilizing the Willis fluid model. Specifically, we consider mathematical manipulations of the wave equation derived from the lowest-order constitutive equations, similar to methods that have been explored in continuum approximations of nonlinear spring-mass chains, as well as higher-order coupling terms. Physical interpretations and implications for time domain finite difference and finite element simulations are discussed. [Work supported by the postdoctoral program at ARL:UT.]Willis fluids are characterized by constitutive relations that couple the pressure and momentum density to both the particle velocity and the volume strain. This effective dynamic response coupling may arise due to microstructural asymmetry, long range order, or time-varying material properties and has been shown to be analogous to electromagnetic bianisotropy [Phys. Rev. B 96, 104303 (2017)]. Recent work has shown that, when wavevectors exceed a critical magnitude, the currently employed model for passive Willis fluids possesses a linear instability stemming from the truncation of higher-order coupling terms in the derivation of the constitutive relationships [J. Acoust. Soc. Am. 144(3), 1832 (2018)]. While this instability can be avoided when using frequency domain methods, it poses significant problems for time domain simulations, as short-wavelength numerical noise may grow exponentially. In the present work, we report on methods of stabilizing the Willis fluid model. Specifically, we consider mathema...

Stability Theory of Stochastic Models in Opinion Dynamics

We consider a certain class of nonlinear maps that preserve the probability simplex, i.e., stochastic maps, which are inspired by the DeGroot–Friedkin model of belief/opinion propagation over influence networks. The corresponding dynamical models describe the evolution of the probability distribution of interacting species. Such models where the probability transition mechanism depends nonlinearly on the current state are often referred to as nonlinear Markov chains . In this paper, we develop stability results and study the behavior of representative opinion models. The stability certificates are based on the contractivity of the nonlinear evolution in the $\ell _1$ -metric. We apply the theory to two types of opinion models where the adaptation of the transition probabilities to the current state is exponential and linear—both of these can display a wide range of behaviors. We discuss continuous-time and other generalizations.

Multi-polaron solutions, nonlocal effects and internal modes in a nonlinear chain

Multipolaron solutions were studied in the framework of the Holstein one-dimensional molecular crystal model. The study was performed in the continuous limit where the crystal model maps into the nonlinear Schrödinger equation for which a new periodic dnoidal solution was found for the multipolaron system. In addition, the stability of the multi-polaron solutions was examined, and it was found that cnoidal and dnoidal solutions stabilize in different ranges of the parameter space. Moreover, the model was studied under the influence of nonlocal effects and the polaronic dynamics was described in terms of internal solitonic modes.

Construction of an Autonomous Nonlinear Hybridization Chain Reaction for Extracellular Vesicles-Associated MicroRNAs Discrimination.

Extracellular vesicles (EVs) have emerged as promising tumor biomarkers for early cancer diagnosis, as primary tumor-secreted EVs carry characteristic molecular information on parent cells. It is thus desirable to realize the efficient discrimination of the signatured EVs-associated microRNAs (miRNAs) with low expression and subtle variation. Here, we introduce an autonomous nonlinear enzyme-free signal amplification paradigm for EVs discrimination through a highly sensitive and selective detection of their inherent miRNAs in situ. Our proposed amplifier consists of a modularized DNAzyme-amplified two-stage cascaded hybridization chain reaction (CHCR-DNAzyme) circuit, where the analyte-generated output of the preceding hybridization chain reaction (HCR1) stage serves as input to motivate the following hybridization chain reaction (HCR2) stage and the concomitant assembly of numerous DNAzyme biocatalysts. By incorporating a flexibly configurable sensing module, this modular CHCR-DNAzyme circuit can further extend to "plug-and-play" sensing mode that enables the miRNA assay with high specificity. The sophisticated design and the detecting performance of our CHCR-DNAzyme scheme were systematically investigated in vitro. The optimized CHCR-DNAzyme system was further applied for distinguishing EVs derived from different cells through the amplified detection of a putative miRNA biomarker in EVs. This compact CHCR-DNAzyme amplifier provides a universal and facile toolbox for highly efficient identification of multiple miRNAs-involved EVs and thus holds great potential for early cancer diagnosis.

From modulational instability to self-trapping in nonlinear chains with power-law hopping amplitudes

The modulational instability problem in tight-binding nonlinear chains with long-ranged hopping amplitudes decaying as 1∕rα is investigated. Exploring that the spectral dimension of the corresponding linear chain deff=2∕(α−1) for α≥3, we demonstrate that the nonlinear strength χMI above which a uniform solution becomes unstable scales with the chain size N as χMI∝N2−α for α≤3 while χMI∝1∕N for α≥3. We also present numerical data unveiling that the intermediate regimes of breathing and chaotic-like solutions appearing during the crossover from the stable uniform solution to self-trapping are fully suppressed when the hopping amplitudes decay slower than 1∕r2.

Rheology of Concentrated Polymer/Ionic Liquid Solutions: An Anomalous Plasticizing Effect and a Universality in Nonlinear Shear Rheology.

An anomalous plasticizing effect was observed in polymer/ionic liquid (IL) solutions by applying broad range of rheological techniques. Poly(ethylene oxide)(PEO)/IL solutions exhibit stronger dynamic temperature dependence than pure PEO, which is in conflict with the knowledge that lower-Tg solvent increases the fractional free volume. For poly(methy methacrylate)(PMMA)/IL solutions, the subtle anomaly was detected from the fact that the effective glass transition temperature Tg,eff of PMMA in IL is higher than the prediction of the self-concentration model, while in conventional polymer solutions, Tg,eff follows the original Fox equation. Observations in both solutions reveal retarded segmental dynamics, consistent with a recent simulation result (Macromolecules, 2018, 51, 5336) that polymer chains wrap the IL cations by hydrogen bonding interactions and the segmental unwrapping delays their relaxation. Start-up shear and nonlinear stress relaxation tests of polymer/IL solutions follow a universal nonlinear rheological behavior as polymer melts and solutions, indicating that the segment-cation interaction is not strong enough to influence the nonlinear chain orientation and stretch. The present work may arouse the further theoretical, experimental, and simulation interests in interpreting the effect of complex polymer-IL interaction on the dynamics of polymer/IL solutions.