Search results for: Relaxation dynamics.

Authors: Yudha Prambudia, Masaru Nakano

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Due to rapid economic growth, Indonesia's energy needs is rapidly increasing. Indonesia-s primary energy consumption has doubled in 2007 compared to 2003. Indonesia's status change from oil net-exporter to oil net-importer country recently has increased Indonesia's concern over energy security. Due to this, oil import becomes center of attention in the dynamics of Indonesia's energy security. Conventional studies addressing Indonesia's energy security have focused on energy production sector. This study explores Indonesia-s energy security considering energy import sector by modeling and simulating Indonesia-s energy-related policies using system dynamics. Simulation result of Indonesia's energy security in 2020 in Business-As-Usual scenario shows that in term of supply demand ratio, energy security will be very high, but also it poses high dependence on energy import. The Alternative scenario result shows lower energy security in term of supply demand ratio and much lower dependence on energy import. It is also found that the Alternative scenario produce lower GDP growth.

Keywords: Energy security, modeling, simulation, system dynamics.

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Authors: Javad Khazaei, Rick S. Blum

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Distributed control is an efficient and flexible approach for coordination of multi-agent systems. One of the main challenges in designing a distributed controller is identifying the governing dynamics of the dynamical systems. Data-driven system identification is currently undergoing a revolution. With the availability of high-fidelity measurements and historical data, model-free identification of dynamical systems can facilitate the control design without tedious modeling of high-dimensional and/or nonlinear systems. This paper develops a distributed control design using consensus theory for linear and nonlinear dynamical systems using sparse identification of system dynamics. Compared with existing consensus designs that heavily rely on knowing the detailed system dynamics, the proposed model-free design can accurately capture the dynamics of the system with available measurements and input data and provide guaranteed performance in consensus and tracking problems. Heterogeneous damped oscillators are chosen as examples of dynamical system for validation purposes.

Keywords: Consensus tracking, distributed control, model-free control, sparse identification of dynamical systems.

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Authors: Bo Liu, Liangtian Gao, Idrees Qasim

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The impact of the storm leads to accidents even in the case of vessels that meet the computed safety criteria for stability. That is why, in order to clarify the causes of the accident and shipwreck, it is necessary to study the dynamics of the ship under the complex sudden impact of external forces. The task is to determine the movement and landing of the ship in the complex and sudden impact of external forces, i.e. when the ship's load changes over a relatively short period of time. For the solution, a technique was used to study the ship's dynamics, which is based on the compilation of a system of differential equations of motion. A coordinate system was adopted for the equation of motion of the hull and the determination of external forces. As a numerical method of integration, the 4^th order Runge-Kutta method was chosen. The results of the calculation show that dynamic deviations were lower for high-altitude vessels. The study of the movement of the hull under a difficult situation is performed: receiving of cargo, impact of a flurry of wind and subsequent displacement of the cargo. The risk of overturning and flooding was assessed.

Keywords: Dynamics, statics, roll, trim, dynamic load, tilt, vertical displacement.

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Authors: Shumin Hou, Yourong Li, Sanxing Zhao

Abstract:

Much time series data is generally from continuous dynamic system. Firstly, this paper studies the detection of the nonlinearity of time series from continuous dynamics systems by applying the Phase-randomized surrogate algorithm. Then, the Delay Vector Variance (DVV) method is introduced into nonlinearity test. The results show that under the different sampling conditions, the opposite detection of nonlinearity is obtained via using traditional test statistics methods, which include the third-order autocovariance and the asymmetry due to time reversal. Whereas the DVV method can perform well on determining nonlinear of Lorenz signal. It indicates that the proposed method can describe the continuous dynamics signal effectively.

Keywords: Nonlinearity, Time series, continuous dynamics system, DVV method

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Authors: R. O. Ocaya, J. J. Terblans

Abstract:

The laws of Newtonian mechanics allow ab-initio molecular dynamics to model and simulate particle trajectories in material science by defining a differentiable potential function. This paper discusses some considerations for the coding of ab-initio programs for simulation on a standalone computer and illustrates the approach by C language codes in the context of embedded metallic atoms in the face-centred cubic structure. The algorithms use velocity-time integration to determine particle parameter evolution for up to several thousands of particles in a thermodynamical ensemble. Such functions are reusable and can be placed in a redistributable header library file. While there are both commercial and free packages available, their heuristic nature prevents dissection. In addition, developing own codes has the obvious advantage of teaching techniques applicable to new problems.

Keywords: C-language, molecular dynamics, simulation, embedded atom method.

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Authors: Miriam Sosa-Díaz, Faustino Sánchez-Garduño

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In population dynamics the study of both, the abundance and the spatial distribution of the populations in a given habitat, is a fundamental issue a From ecological point of view, the determination of the factors influencing such changes involves important problems. In this paper a mathematical model to describe the temporal dynamic and the spatiotemporal dynamic of the interaction of three populations (pollinators, plants and herbivores) is presented. The study we present is carried out by stages: 1. The temporal dynamics and 2. The spatio-temporal dynamics. In turn, each of these stages is developed by considering three cases which correspond to the dynamics of each type of interaction. For instance, for stage 1, we consider three ODE nonlinear systems describing the pollinator-plant, plant-herbivore and plant-pollinator-herbivore, interactions, respectively. In each of these systems different types of dynamical behaviors are reported. Namely, transcritical and pitchfork bifurcations, existence of a limit cycle, existence of a heteroclinic orbit, etc. For the spatiotemporal dynamics of the two mathematical models a novel factor are introduced. This consists in considering that both, the pollinators and the herbivores, move towards those places of the habitat where the plant population density is high. In mathematical terms, this means that the diffusive part of the pollinators and herbivores equations depend on the plant population density. The analysis of this part is presented by considering pairs of populations, i. e., the pollinator-plant and plant-herbivore interactions and at the end the two mathematical model is presented, these models consist of two coupled nonlinear partial differential equations of reaction-diffusion type. These are defined on a rectangular domain with the homogeneous Neumann boundary conditions. We focused in the role played by the density dependent diffusion term into the coexistence of the populations. For both, the temporal and spatio-temporal dynamics, a several of numerical simulations are included.

Keywords: Bifurcation, heteroclinic orbits, steady state, traveling wave.

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Authors: Chanchal Saha, Huynh Trung Luong, M. H. Aziz, Tharinan Rattanalert

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Now-a-days, numbers of simulation software are being used all over the world to solve Computational Fluid Dynamics (CFD) related problems. In this present study, a commercial CFD simulation software namely STAR-CCM+ is applied to analyze the airflow characteristics inside a 2.5" hard disk drive. Each step of the software is described adequately to obtain the output and the data are verified with the theories to justify the robustness of the simulation outcome. This study gives an insight about the accuracy level of the CFD simulation software to compute CFD related problems although it largely depends upon the computer speed. Also this study will open avenues for further research.

Keywords: Computational fluid dynamics, Hard disk drive, Meshing, Recirculation filter, and Filter physics parameter.

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Authors: C. E. Madubueze, S. C. Madubueze, S. Ajama

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Cholera is a disease that is predominately common in developing countries due to poor sanitation and overcrowding population. In this paper, a deterministic model for the dynamics of cholera is developed and control measures such as health educational message, therapeutic treatment, and vaccination are incorporated in the model. The effective reproduction number is computed in terms of the model parameters. The existence and stability of the equilibrium states, disease free and endemic equilibrium states are established and showed to be locally and globally asymptotically stable when R0 < 1 and R0 > 1 respectively. The existence of backward bifurcation of the model is investigated. Furthermore, numerical simulation of the model developed is carried out to show the impact of the control measures and the result indicates that combined control measures will help to reduce the spread of cholera in the population.

Keywords: Backward bifurcation, cholera, equilibrium, dynamics, stability.

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Authors: José Carlos Rodríguez, Mario Gómez, Medardo Serna

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This paper analyses managing higher education institutions in emerging economies. The paper investigates the case of postgraduate studies development at public universities. In so doing, it adopts the complex theory approach to evaluate how postgraduate studies have evolved in these countries. The investigation suggests that the postgraduate studies sector at public universities can be seen as a complex adaptive system (CAS). Therefore, the paper adopts system dynamics (SD) methods to develop this analysis. The case of postgraduate studies at Universidad Michoacana de San Nicolás de Hidalgo in Mexico is investigated in this paper.

Keywords: Higher education institutions, complex adaptive systems, system dynamics, Mexico.

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Authors: Quy Dang Nguyen, Reza Nakhaie Jazar

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The paper investigates vibration dynamics of tire-road separation for a quarter-car model; this separation model is developed to be close to the real situation considering the tire is able to separate from the ground plane. A set of piecewise linear mathematical models is developed and matches the in-contact and no-contact states to be considered as mother models for further investigations. The bound dynamics are numerically simulated in the time response and phase portraits. The separation analysis may determine which values of suspension parameters can delay and avoid the no-contact phenomenon, which results in improving ride comfort and eliminating the potentially dangerous oscillation. Finally, model verification is carried out in the MSC-ADAMS environment.

Keywords: Quarter-car vibrations, tire-road separation, separation analysis, separation dynamics, ride comfort, ADAMS validation.

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Authors: Inese Bula

Abstract:

Symbolic dynamics studies dynamical systems on the basis of the symbol sequences obtained for a suitable partition of the state space. This approach exploits the property that system dynamics reduce to a shift operation in symbol space. This shift operator is a chaotic mapping. In this article we show that in the symbol space exist other chaotic mappings.

Keywords: Infinite symbol space, prefix metric, chaotic mapping, generator function, jump mapping.

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Authors: Yao Jie, Yeo Khoon Seng

Abstract:

In recent decades, flapping wing aerodynamics has attracted great interest. Understanding the physics of biological flyers such as birds and insects can help improve the performance of micro air vehicles. The present research focuses on the aerodynamics of insect-like flapping wing flight with the approach of numerical computation. Insect model of hawkmoth is adopted in the numerical study with rigid wing assumption currently. The numerical model integrates the computational fluid dynamics of the flow and active control of wing kinematics to achieve stable flight. The computation grid is a hybrid consisting of background Cartesian nodes and clouds of mesh-free grids around immersed boundaries. The generalized finite difference method is used in conjunction with single value decomposition (SVD-GFD) in computational fluid dynamics solver to study the dynamics of a free hovering hummingbird hawkmoth. The longitudinal dynamics of the hovering flight is governed by three control parameters, i.e., wing plane angle, mean positional angle and wing beating frequency. In present work, a PID controller works out the appropriate control parameters with the insect motion as input. The controller is adjusted to acquire desired maneuvering of the insect flight. The numerical scheme in present study is proven to be accurate and stable to simulate the flight of the hummingbird hawkmoth, which has relatively high Reynolds number. The PID controller is responsive to provide feedback to the wing kinematics during the hovering flight. The simulated hovering flight agrees well with the real insect flight. The present numerical study offers a promising route to investigate the free flight aerodynamics of insects, which could overcome some of the limitations of experiments.

Keywords: Aerodynamics, flight control, computational fluid dynamics, flapping-wing flight.

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Authors: Theodore N.S. Karfakis, Anna Andrade

Abstract:

The aim of this study was to examine the dynamics of functional composition of a non flooded Amazonian forest in response to drought stress in terms of diameter growth, recruitment and mortality. The survey was carried out in the continuous forest of the Biological dynamics of forest fragments project 90 km outside the city of Manaus, state of Amazonas Brazil. All stems >10 cm dbh where identified to species level and monitored in 18 one hectare permanent sample plots from 1981 to 2004.For statistical analysis all species where aggregated in three ecological guilds. Two distinct drought events occurred in 1983 and 1997. Results showed that more early successional species performed better than later successional ones. Response was significant for both events but for the 1997 event this was more pronounced possibly because of the fact that the event was in the middle of the dry rather than the wet period as was the 1983 one.

Keywords: Brazil, functional composition, drought, Amazonian non flooded forest.

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Authors: Valeri A. Makarov, Nazareth P. Castellanos

Abstract:

Statistical analysis of electrophysiological recordings obtained under, e.g. tactile, stimulation frequently suggests participation in the network dynamics of experimentally unobserved “hidden" neurons. Such interneurons making synapses to experimentally recorded neurons may strongly alter their dynamical responses to the stimuli. We propose a mathematical method that formalizes this possibility and provides an algorithm for inferring on the presence and dynamics of hidden neurons based on fitting of the experimental data to spike trains generated by the network model. The model makes use of Integrate and Fire neurons “chemically" coupled through exponentially decaying synaptic currents. We test the method on simulated data and also provide an example of its application to the experimental recording from the Dorsal Column Nuclei neurons of the rat under tactile stimulation of a hind limb.

Keywords: Integrate and fire neuron, neural network models, spike trains.

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Authors: Teewin Plangsrinont, Wasawat Nakkiew

Abstract:

In this study, Computational Fluid Dynamics (CFD) was utilized to simulate and predict the path of water from water soot blower through an ambient flow field in 300-megawatt tangentially burned pulverized coal boiler that utilizes a water soot blower as a cleaning device. To predict the position of the impact of water on the opposite side of the water soot blower under identical conditions, the nozzle size and water flow rate were fixed in this investigation. The simulation findings demonstrated a high degree of accuracy in predicting the direction of water flow to the boiler's water wall tube, which was validated by comparison to experimental data. Results show maximum deviation value of the water jet trajectory is 10.2%.

Keywords: Computational fluid dynamics, tangentially fired boiler, thermal power plant, water soot blower.

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Authors: Y. V. Kim

Abstract:

This article presents an approach with regards to the Functional Testing of Space System (SS) that could be a space vehicle (spacecraft-S/C) and/or its equipment and components – S/C subsystems. This test should finalize the Space Qualification Tests (SQT) campaign. It could be considered as a generic test and used for a wide class of SS that, from the point of view of System Dynamics and Control Theory, may be described by the ordinary differential equations. The suggested methodology is based on using semi-natural experiment laboratory stand that does not require complicated, precise and expensive technological control-verification equipment. However, it allows for testing totally assembled system during Assembling, Integration and Testing (AIT) activities at the final phase of SQT, involving system hardware (HW) and software (SW). The test physically activates system input (sensors) and output (actuators) and requires recording their outputs in real time. The data are then inserted in a laboratory computer, where it is post-experiment processed by the MATLAB/Simulink Identification Toolbox. It allows for estimating the system dynamics in the form of estimation of its differential equation coefficients through the verification experimental test and comparing them with expected mathematical model, prematurely verified by mathematical simulation during the design process. Mathematical simulation results presented in the article show that this approach could be applicable and helpful in SQT practice. Further semi-natural experiments should specify detail requirements for the test laboratory equipment and test-procedures.

Keywords: system dynamics, space system ground tests, space qualification, system dynamics identification, satellite attitude control, assembling integration and testing

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Authors: A. K. Kamath, N. M. Singh, R. Pasumarthy

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This paper investigates the control of a bouncing ball using Model Predictive Control. Bouncing ball is a benchmark problem for various rhythmic tasks such as juggling, walking, hopping and running. Humans develop intentions which may be perceived as our reference trajectory and tries to track it. The human brain optimizes the control effort needed to track its reference; this forms the central theme for control of bouncing ball in our investigations.

Keywords: Bouncing Ball, impact dynamics, intermittent control, model predictive control.

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Authors: Belkacem Meziane

Abstract:

The original 3D Lorenz-Haken equations -which describe laser dynamics- are converted into 2-second-order differential equations out of which the so far missing mathematics is extracted. Leaning on high-order trigonometry, important outcomes are pulled out: A fundamental result attributes chaos to forbidden periodic solutions, inside some precisely delimited region of the control parameter space that governs self-pulsing.

Keywords: chaos, Lorenz-Haken equations, laser dynamics, nonlinearities

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Authors: Vigen Arakelian

Abstract:

This paper deals with the study of devices for displacement of the mould base of blow-molding machines. The displacement of the mould in the studied case is carried out by a linear actuator, which ensures the descent of the mould base and by extension springs, which return the letter in the initial position. The aim of this paper is to study the inverse dynamics of the device for displacement of the mould base of blow-molding machines and to determine its optimum parameters for higher rate of production. In the other words, it is necessary to solve the inverse dynamic problem to find the equation of motion linking applied forces with displacements. This makes it possible to determine the stiffness coefficient of the spring to turn the mold base back to the initial position for a given time. The obtained results are illustrated by a numerical example. It is shown that applying a spring with stiffness returns the mould base of the blow molding machine into the initial position in 0.1 sec.

Keywords: Design, blow-molding machines, dynamics.

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Authors: Wenhua Hou, Xing Liu, Deqiang Chen

Abstract:

The ubiquitous payment problems within construction industry of China are notoriously hard to be resolved, thus lead to a series of impacts to the industry chain. Among of them, the most direct result is affecting the normal operation of contractors negatively. A wealth of research has already discussed reasons of the payment problems by introducing a number of possible improvement strategies. But the causalities of these problems are still far from harsh reality. In this paper, the authors propose a model for cash flow system of construction projects by introducing System Dynamics techniques to explore causal facets of the payment problem. The effects of payment arrears on both cash flow and profitability of project are simulated into four scenarios by using data from real projects. Simulating results show visible clues to help contractors quantitatively determining the consequences for the construction project that arise from payment delay.

Keywords: payment problems, cash flow, profitability, system dynamics.

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Authors: A. K. Abdel-Fattah, A. B. El-Tawil, N. A. Harraz

Abstract:

This paper focuses on the operational and strategic planning decisions related to the quayside of container terminals. We introduce an integrated operational research (OR) and system dynamics (SD) approach to solve the Berth Allocation Problem (BAP) and the Quay Crane Assignment Problem (QCAP). A BAP-QCAP optimization modeling approach which considers practical aspects not studied before in the integration of BAP and QCAP is discussed. A conceptual SD model is developed to determine the long-term effect of optimization on the system behavior factors like resource utilization, attractiveness to port, number of incoming vessels to port and port profits. The framework can be used for improving the operational efficiency of container terminals and providing a strategic view after applying optimization.

Keywords: Operational research, system dynamics, container terminal, quayside operational problems, strategic planning decisions.

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Authors: Myunggon Yoon, Jung-Ho Moon

Abstract:

In this paper, we present a transfer function representation of a general one-dimensional combustor. The input of the transfer function is a heat rate perturbation of a burner and the output is a flow velocity perturbation at the burner. This paper considers a general combustor model composed of multiple cans with different cross sectional areas, along with a non-zero flow rate.

Keywords: Thermoacoustics, dynamics, combustor, transfer function.

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Authors: M. Kaggwa

Abstract:

Technology or lack of it will play an important role in Africa-s effort to achieve inclusive development. Although a key determinant of competitiveness, new technology can exacerbate exclusion of the majority from the mainstream economic activities. To minimise potential technology exclusion while leveraging its critical role in African-s development, requires insight into technology diffusion process. Using system dynamics approach, a technology diffusion model is presented. The frequency of interaction of people exposed to and those not exposed to technology, and the technology adoption rate - the fraction of people who embrace new technologies once they are exposed, are identified as the broad factors critical to technology diffusion to wider society enabling more people to be part of the economic growth process. Based on simulation results, it is recommends that these two broad factors should form part of national policy aimed at achieving inclusive and sustainable development in Africa.

Keywords: Inclusive Development, System Dynamics, Technology, Technology diffusion.

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Authors: R. Pilafkan, M. Kaffash Irzarahimi, S. F. Asbaghian Namin

Abstract:

The biaxial buckling behavior of single-layered graphene sheets (SLGSs) is studied in the present work. To consider the size-effects in the analysis, Eringen’s nonlocal elasticity equations are incorporated into classical plate theory (CLPT). A Generalized Differential Quadrature Method (GDQM) approach is utilized and numerical solutions for the critical buckling loads are obtained. Then, molecular dynamics (MD) simulations are performed for a series of zigzag SLGSs with different side-lengths and with various boundary conditions, the results of which are matched with those obtained by the nonlocal plate model to numerical the appropriate values of nonlocal parameter relevant to each type of boundary conditions.

Keywords: Biaxial buckling, single-layered graphene sheets, nonlocal elasticity, molecular dynamics simulation, classical plate theory.

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Authors: D.Toghraie, A.R.Azimian

Abstract:

Molecular dynamics simulation of annular flow boiling in a nanochannel with 70000 particles is numerically investigated. In this research, an annular flow model is developed to predict the superheated flow boiling heat transfer characteristics in a nanochannel. To characterize the forced annular boiling flow in a nanochannel, an external driving force F ext ranging from 1to12PN (PN= Pico Newton) is applied along the flow direction to inlet fluid particles during the simulation. Based on an annular flow model analysis, it is found that saturation condition and superheat degree have great influences on the liquid-vapor interface. Also, the results show that due to the relatively strong influence of surface tension in small channel, the interface between the liquid film and vapor core is fairly smooth, and the mean velocity along the stream-wise direction does not change anymore.

Keywords: Lennard-Jones Potential, Molecular DynamicsSimulation, Periodic Boundary Conditions (PBC), Non-EquilibriumMolecular Dynamics (NEMD), Annular Flow Boiling

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Authors: M. F. Haroun, T. A. Gulliver

Abstract:

In this paper, an encryption algorithm is proposed for real-time image encryption. The scheme employs a dual chaotic generator based on a three dimensional (3D) discrete Lorenz attractor. Encryption is achieved using non-autonomous modulation where the data is injected into the dynamics of the master chaotic generator. The second generator is used to permute the dynamics of the master generator using the same approach. Since the data stream can be regarded as a random source, the resulting permutations of the generator dynamics greatly increase the security of the transmitted signal. In addition, a technique is proposed to mitigate the error propagation due to the finite precision arithmetic of digital hardware. In particular, truncation and rounding errors are eliminated by employing an integer representation of the data which can easily be implemented. The simple hardware architecture of the algorithm makes it suitable for secure real-time applications.

Keywords: Chaotic systems, image encryption, 3D Lorenz attractor, non-autonomous modulation, FPGA.

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Authors: Mohammad Akbarpour, Reza Nasiri Mahalati, Caro Lucas

Abstract:

The existence of many biological systems, especially human societies, is based on cooperative behavior [1, 2]. If natural selection favors selfish individuals, then what mechanism is at work that we see so many cooperative behaviors? One answer is the effect of network structure. On a graph, cooperators can evolve by forming network bunches [2, 3, 4]. In a research, Ohtsuki et al used the idea of iterated prisoners- dilemma on a graph to model an evolutionary game. They showed that the average number of neighbors plays an important role in determining whether cooperation is the ESS of the system or not [3]. In this paper, we are going to study the dynamics of evolution of cooperation in a social network. We show that during evolution, the ratio of cooperators among individuals with fewer neighbors to cooperators among other individuals is greater than unity. The extent to which the fitness function depends on the payoff of the game determines this ratio.

Keywords: Evolution of cooperation, Iterated prisoner's dilemma, Model dynamics, Social network structure, Intensity of selection.

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Authors: G. Trigui, B. Laroche, M. Miquel, B. Dubreucq, A. Trubuil

Abstract:

The subcellular organelles called oil bodies (OBs) are lipid-filled quasi-spherical droplets produced from the endoplasmic reticulum (ER) and then released into the cytoplasm during seed development. It is believed that an OB grows by coalescence with other OBs and that its stability depends on the composition of oleosins, major proteins inserted in the hemi membrane that covers OBs. In this study, we measured the OB-volume distribution from different genotypes of A. thaliana after 7, 8, 9, 10 and 11 days of seed development. In order to test the hypothesis of OBs dynamics, we developed a simple mathematical model using non-linear differential equations inspired from the theory of coagulation. The model describes the evolution of OB-volume distribution during the first steps of seed development by taking into consideration the production of OBs, the increase of triacylglycerol volume to be stored, and the growth by coalescence of OBs. Fitted parameters values show an increase in the OB production and coalescence rates in A. thaliana oleosin mutants compared to wild type.

Keywords: Biogenesis, coalescence, oil body, oleosin, population dynamics.

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Authors: Marco Kennemann, Steffen C. Eickemeyer, Peter Nyhuis

Abstract:

An increasingly dynamic and complex environment poses huge challenges to production enterprises, especially with regards to logistics. The Logistic Operating Curve Theory, developed at the Institute of Production Systems and Logistics (IFA) of the Leibniz University of Hanover, is a recognized approach to describing logistic interactions, nevertheless, it reaches its limits when it comes to the dynamic aspects. In order to facilitate a timely and optimal Logistic Positioning a method is developed for quickly and reliably identifying dynamic processing states.

Keywords: Dynamics, Logistic Operating Curves, Production Logistics, Production Planning and Control

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Authors: Jixiang Zhang, Guocheng Wang

Abstract:

A dynamic of Bertrand duopoly game is analyzed, where players use different production methods and choose their prices with bounded rationality. The equilibriums of the corresponding discrete dynamical systems are investigated. The stability conditions of Nash equilibrium under a local adjustment process are studied. The stability conditions of Nash equilibrium under a local adjustment process are studied. The stability of Nash equilibrium, as some parameters of the model are varied, gives rise to complex dynamics such as cycles of higher order and chaos. On this basis, we discover that an increase of adjustment speed of bounded rational player can make Bertrand market sink into the chaotic state. Finally, the complex dynamics, bifurcations and chaos are displayed by numerical simulation.

Keywords: Bertrand duopoly model, Discrete dynamical system, Heterogeneous expectations, Nash equilibrium.

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