Search results for: discrete dynamical system

Authors: Vandana N. Purav

Abstract:

The Dynamical systems concept is a mathematical formalization for any fixed rule that describes the time dependence of a points position in its ambient space. e.g. pendulum of a clock, the number of fish each spring in a lake, the number of rabbits spring in an enclosure, etc. The Dynamical system theory used to describe the complex nature that is dynamical systems with differential equations called continuous dynamical system or dynamical system with difference equations called discrete dynamical system. The concept of dynamical system has its origin in Newtonian mechanics.

Keywords: dynamical systems, Fibonacci numbers, Newtonian mechanics, discrete dynamical system

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Authors: Eugene Stepanov, Arkadi Ponossov

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Ordinary differential equations with switching nonlinearities constitute a very useful tool in many applications. The solutions of such equations can usually be calculated analytically if they cross the discontinuities transversally. Otherwise, one has trajectories that slides along the discontinuity, and the calculations become less straightforward in this case. For instance, one of the problems one faces is non-uniqueness of the sliding modes. In the presentation, it is proposed to apply the theory of hybrid dynamical systems to calculate the solutions that are ‘hidden’ in the discontinuities. Roughly, one equips the underlying switched system with an explicitly designed discrete dynamical system (‘automaton’), which governs the dynamics of the switched system. This construction ‘splits’ the dynamics, which, as it is shown in the presentation, gives uniqueness of the resulting hybrid trajectories and at the same time provides explicit formulae for them. Projecting the hybrid trajectories back onto the original continuous system explains non-uniqueness of its trajectories. The automaton is designed with the help of the attractors of the specially constructed adjoint dynamical system. Several examples are provided in the presentation, which supports the efficiency of the suggested scheme. The method can be of interest in control theory, gene regulatory networks, neural field models and other fields, where switched dynamics is a part of the analysis.

Keywords: hybrid dynamical systems, singular perturbation analysis, sliding modes, switched dynamics

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Authors: Chao-Qing Dai

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Two families of exact travelling solutions for the discrete sine-Gordon equation are constructed based on the variable-coefficient Jacobian elliptic function method and different transformations. When the modulus of Jacobian elliptic function solutions tends to 1, soliton solutions can be obtained. Some soliton solutions degenerate into the known solutions in literatures. Moreover, dynamical properties of exact solutions are investigated. Our analysis and results may have potential values for certain applications in modern nonlinear science and textile engineering.

Keywords: exact solutions, variable-coefficient Jacobian elliptic function method, discrete sine-Gordon equation, dynamical behaviors

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Authors: Lana Horvat Dmitrović

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The main purpose of this paper is to study the box dimension as fractal property of bifurcations of discrete dynamical systems in higher dimensions. The paper contains the fractal analysis of the orbits near the hyperbolic and non-hyperbolic fixed points in discrete dynamical systems. It is already known that in one-dimensional case the orbit near the hyperbolic fixed point has the box dimension equal to zero. On the other hand, the orbit near the non-hyperbolic fixed point has strictly positive box dimension which is connected to the non-degeneracy condition of certain bifurcation. One of the main results in this paper is the generalisation of results about box dimension near the hyperbolic and non-hyperbolic fixed points to higher dimensions. In the process of determining box dimension, the restriction of systems to stable, unstable and center manifolds, Lipschitz property of box dimension and the notion of projective box dimension are used. The analysis of the bifurcations in higher dimensions with one multiplier on the unit circle is done by using the normal forms on one-dimensional center manifolds. This specific change in box dimension of an orbit at the moment of bifurcation has already been explored for some bifurcations in one and two dimensions. It was shown that specific values of box dimension are connected to appropriate bifurcations such as fold, flip, cusp or Neimark-Sacker bifurcation. This paper further explores this connection of box dimension as fractal property to some specific bifurcations in higher dimensions, such as fold-flip and flip-Neimark-Sacker. Furthermore, the application of the results to the unit time map of continuous dynamical system near hyperbolic and non-hyperbolic singularities is presented. In that way, box dimensions which are specific for certain bifurcations of continuous systems can be obtained. The approach to bifurcation analysis by using the box dimension as specific fractal property of orbits can lead to better understanding of bifurcation phenomenon. It could also be useful in detecting the existence or nonexistence of bifurcations of discrete and continuous dynamical systems.

Keywords: bifurcation, box dimension, invariant manifold, orbit near fixed point

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Authors: Eugene Stepanov, Arcady Ponosov

Abstract:

To model gene regulatory networks one uses ordinary differential equations with switching nonlinearities, where the initial value problem is known to be well-posed if the trajectories cross the discontinuities transversally. Otherwise, the initial value problem is usually ill-posed, which lead to theoretical and numerical complications. In the presentation, it is proposed to apply the theory of hybrid dynamical systems, rather than switched ones, to regularize the problem. 'Hybridization' of the switched system means that one attaches a dynamic discrete component ('automaton'), which follows the trajectories of the original system and governs its dynamics at the points of ill-posedness of the initial value problem making it well-posed. The construction of the automaton is based on the classification of the attractors of the specially designed adjoint dynamical system. Several examples are provided in the presentation, which support the suggested analysis. The method can also be of interest in other applied fields, where differential equations contain switchings, e.g. in neural field models.

Keywords: hybrid dynamical systems, ill-posed problems, singular perturbation analysis, switching nonlinearities

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Authors: Jixiang Zhang

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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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Authors: Jixiang Zhang, Yanhua Wang

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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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Authors: Taha Benarbia, Kay W. Axhausen, Anugrah Ilahi

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Electric car sharing system as ecologic transportation increasing in the world. The complexity of managing electric car sharing systems, especially one-way trips and battery autonomy have direct influence to on supply and demand of system. One must be able to precisely model the demand and supply of these systems to better operate electric car sharing and estimate its effect on mobility management and the accessibility that it provides in urban areas. In this context, our work focus to develop performances optimization model of the system based on discrete event simulation and stochastic Petri net. The objective is to search optimal decisions and management parameters of the system in order to fulfil at best demand while minimizing undesirable situations. In this paper, we present new model of electric cars sharing with relocation based on monitoring system. The proposed approach also help to precise the influence of battery charging level on the behaviour of system as important decision parameter of this complex and dynamical system.

Keywords: electric car-sharing systems, smart mobility, Petri nets modelling, discrete event simulation

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Authors: Javad Khazaei, Rick 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: T. K. Dutta, K. K. Das, N. Dutta

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The salient feature of this paper is primarily concerned with Ricker’s population model: f(x)=x e^(r(1-x/k)), where r is the control parameter and k is the carrying capacity, and some fruitful results are obtained with the following objectives: 1) Determination of bifurcation values leading to a chaotic region, 2) Development of Statistical Methods and Analysis required for the measure of Fractal dimensions, 3) Calculation of various fractal dimensions. These results also help that the invariant probability distribution on the attractor, when it exists, provides detailed information about the long-term behavior of a dynamical system. At the end, some open problems are posed for further research.

Keywords: Feigenbaum universality, chaos, Lyapunov exponent, fractal dimensions

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Authors: Jorge Gonzalez Camus, Valentin Keyantuo, Mahamadi Warma

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In this work, we obtain representation results for solutions of a time-fractional differential equation involving the discrete fractional Laplace operator in terms of generalized Wright functions. Such equations arise in the modeling of many physical systems, for example, chain processes in chemistry and radioactivity. The focus is on the linear problem of the simplified Moore - Gibson - Thompson equation, where the discrete fractional Laplacian and the Caputo fractional derivate of order on (0,2] are involved. As a particular case, we obtain the explicit solution for the discrete heat equation and discrete wave equation. Furthermore, we show the explicit solution for the equation involving the perturbed Laplacian by the identity operator. The main tool for obtaining the explicit solution are the Laplace and discrete Fourier transforms, and Stirling's formula. The methodology mainly is to apply both transforms in the equation, to find the inverse of each transform, and to prove that this solution is well defined, using Stirling´s formula.

Keywords: discrete fractional Laplacian, explicit representation of solutions, fractional heat and wave equations, fundamental

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Authors: Tomas Menard

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The synthesis of output-feedback control law has been investigated by many researchers since the last century. While many results exist for the case of Linear Time Invariant systems whose measurements are continuously available, nowadays, control laws are usually implemented on micro-controller, then the measurements are discrete-time by nature. This fact has to be taken into account explicitly in order to obtain a satisfactory behavior of the closed-loop system. One considers here a general class of systems corresponding to an observability normal form and which is subject to uncertainties in the dynamics and sampling of the output. Indeed, in practice, the modeling of the system is never perfect, this results in unknown uncertainties in the dynamics of the model. We propose here an output feedback algorithm which is based on a linear state feedback and a continuous-discrete time observer. The main feature of the proposed control law is that only discrete-time measurements of the output are needed. Furthermore, it is formally proven that the state of the closed loop system exponentially converges toward the origin despite the unknown uncertainties. Finally, the performances of this control scheme are illustrated with simulations.

Keywords: dynamical systems, output feedback control law, sampling, uncertain systems

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Authors: Mohsin Raza, Arne Bilberg, Thomas Ditlev Brunø, Ann-Louise Andersen, Filip SKärin

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Shifting from a dedicated or flexible manufacturing system to a reconfigurable manufacturing system (RMS) requires a significant amount of time, money, and effort. Therefore, it is vital to verify beforehand that the potential reconfigurable solution will be able to achieve the organizational objectives. Discrete event simulation offers the opportunity of assessing several reconfigurable alternatives against the set objectives. This study signifies the importance of using discrete-event simulation as a tool to verify several reconfiguration options. Two different industrial cases have been presented in the study to elaborate on the role of discrete event simulation in the implementation methodology of RMSs. The study concluded that discrete event simulation is one of the important tools to consider in the RMS implementation methodology.

Keywords: reconfigurable manufacturing system, discrete event simulation, Tecnomatix plant simulation, RMS

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Authors: Tomas Menard

Abstract:

The design of control-law for engineering systems has been investigated for many decades. While many results are concerned with continuous systems with continuous output, nowadays, many controlled systems have to transmit their output measurements through network, hence making it discrete-time. But it is well known that the sampling of a system whose control-law is based on the continuous output may render the system unstable, especially when this sampling period is long compared to the system dynamics. The control design then has to be adapted in order to cope with this issue. In this paper, we consider systems which can be modeled as double integrator with uncertainty on the input since many mechanical systems can be put under such form. We present a control scheme based on an observer using only discrete time measurement and which provides continuous time estimation of the state, combined with a continuous control law, which stabilized a system with second-order dynamics even in the presence of uncertainty. It is further shown that arbitrarily long sampling periods can be dealt with properly setting the control scheme parameters.

Keywords: dynamical system, control law design, sampled output, observer design

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Authors: Mohammad Reza Rahimi Khoygani, Reza Ghasemi

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In this paper, designing direct adaptive neural controller is applied for a class of a nonlinear pendulum dynamic system. The radial basis function (RBF) is used for the Neural network (NN). The adaptive neural controller is robust in presence of external and internal uncertainties. Both the effectiveness of the controller and robustness against disturbances are the merits of this paper. The promising performance of the proposed controllers investigates in simulation results.

Keywords: adaptive control, pendulum dynamical system, nonlinear control, adaptive neural controller, nonlinear dynamical, neural network, RBF, driven pendulum, position control

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Authors: Farhad Asadi, Mohammad Javad Mollakazemi

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In this paper, Bayesian online inference in models of data series are constructed by change-points algorithm, which separated the observed time series into independent series and study the change and variation of the regime of the data with related statistical characteristics. variation of statistical characteristics of time series data often represent separated phenomena in the some dynamical system, like a change in state of brain dynamical reflected in EEG signal data measurement or a change in important regime of data in many dynamical system. In this paper, prediction algorithm for studying change point location in some time series data is simulated. It is verified that pattern of proposed distribution of data has important factor on simpler and smother fluctuation of hazard rate parameter and also for better identification of change point locations. Finally, the conditions of how the time series distribution effect on factors in this approach are explained and validated with different time series databases for some dynamical system.

Keywords: time series, fluctuation in statistical characteristics, optimal learning, change-point algorithm

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Authors: Joon-Hoon Park

Abstract:

In this paper the rationalized Haar transform is applied for distributed parameter system identification and estimation. A distributed parameter system is a dynamical and mathematical model described by a partial differential equation. And system identification concerns the problem of determining mathematical models from observed data. The Haar function has some disadvantages of calculation because it contains irrational numbers, for these reasons the rationalized Haar function that has only rational numbers. The algorithm adopted in this paper is based on the transform and operational matrix of the rationalized Haar function. This approach provides more convenient and efficient computational results.

Keywords: distributed parameter system, rationalized Haar transform, operational matrix, system identification

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Authors: Muhammad Fiaz

Abstract:

The article in hand is the study of complex features like Zero Hopf Bifurcation, Chaos and Synchronization of integer and fractional order version of a new 3D finance system. Trusted tools of averaging theory and active control method are utilized for investigation of Zero Hopf bifurcation and synchronization for both versions respectively. Inventiveness of the paper is to find the answer of a question that is it possible to find a chaotic system which can be synchronized with any other of the same dimension? Based on different examples we categorically develop a theory that if a couple of master and slave chaotic dynamical system is synchronized by selecting a suitable gain matrix with special conditions then the master system is synchronized with any chaotic dynamical system of the same dimension. With the help of this study we developed generalized theorems for synchronization of n-dimension dynamical systems for integer as well as fractional versions. it proposed that this investigation will contribute a lot to control dynamical systems and only a suitable gain matrix with special conditions is enough to synchronize the system under consideration with any other chaotic system of the same dimension. Chaotic properties of fractional version of the new finance system are also analyzed at fractional order q=0.87. Simulations results, where required, also provided for authenticity of analytical study.

Keywords: complex analysis, chaos, generalized synchronization, control dynamics, fractional order analysis

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Authors: Alica Miller

Abstract:

A semiflow is a triple consisting of a Hausdorff topological space $X$, a commutative topological monoid $T$ and a continuous monoid action of $T$ on $X$. The acting monoid $T$ is usually either the discrete monoid $\N_0$ of nonnegative integers (in which case the semiflow can be defined as a pair $(X,f)$ consisting of a phase space $X$ and a continuous function $f:X\to X$), or the monoid $\R_+$ of nonnegative real numbers (the so-called one-parameter monoid). However, it turns out that there are real-life situations where it is useful to consider the acting monoids that are a combination of discrete and continuous monoids. That, for example, happens, when we are observing certain dynamical system at discrete moments, but after some time realize that it would be beneficial to continue our observations in real time. The acting monoid in that case would be $T=\{0, t_0, 2t_0, \dots, (n-1)t_0\} \cup [nt_0,\infty)$ with the operation and topology induced from real numbers. This partly explains the motivation for the level of generality which is pursued in our research. We introduce the PSP monoids, which include all but ``pathological'' monoids, and most of our statements hold for them. The topic of our presentation are some recent results about chaos-related properties in semiflows, indecomposability and sensitivity of semiflows in the described general context.

Keywords: chaos, indecomposability, PSP monoids, semiflow, sensitivity

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Authors: Graziano Chesi

Abstract:

The topological entropy plays a key role in linear dynamical systems, allowing one to establish the existence of stabilizing feedback controllers for linear systems in the presence of communications constraints. This paper addresses the determination of a robust value of the topological entropy in nonlinear dynamical systems, specifically the largest value of the topological entropy over all linearized models in a region of interest of the state space. It is shown that a sufficient condition for establishing upper bounds of the sought robust value of the topological entropy can be given in terms of a semidefinite program (SDP), which belongs to the class of convex optimization problems.

Keywords: non-linear system, communication constraint, topological entropy

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Authors: Nidal F. Shilbayeh, Belal AbuHaija, Zainab N. Al-Qudsy

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In this paper, a hybrid blind digital watermarking system using Discrete Wavelet Transform (DWT) and Contourlet Transform (CT) has been implemented and tested. The implemented combined digital watermarking system has been tested against five common types of image attacks. The performance evaluation shows improved results in terms of imperceptibility, robustness, and high tolerance against these attacks; accordingly, the system is very effective and applicable.

Keywords: discrete wavelet transform (DWT), contourlet transform (CT), digital image watermarking, copyright protection, geometric attack

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Authors: Ibrahim Alsendid, Rob Sturman, Benjamin Sharp

Abstract:

In this paper, we consider a family of 2-dimensional nonlinear area-preserving transformations on the torus. A single parameter η varies between 0 and 1, taking the transformation from a hyperbolic toral automorphism to the “Cerbelli-Giona” map, a system known to exhibit multifractal properties. Here we study the multifractal properties of the family of maps. We apply a box-counting method by defining a grid of boxes Bi(δ), where i is the index and δ is the size of the boxes, to quantify the distribution of stable and unstable manifolds of the map. When the parameter is in the range 0.51< η <0.58 and 0.68< η <1 the map is ergodic; i.e., the unstable and stable manifolds eventually cover the whole torus, although not in a uniform distribution. For accurate numerical results, we require correspondingly accurate construction of the stable and unstable manifolds. Here we use the piecewise linearity of the map to achieve this, by computing the endpoints of line segments that define the global stable and unstable manifolds. This allows the generalized fractal dimension Dq, and spectrum of dimensions f(α), to be computed with accuracy. Finally, the intersection of the unstable and stable manifold of the map will be investigated and compared with the distribution of periodic points of the system.

Keywords: Discrete-time dynamical systems, Fractal geometry, Multifractal behaviour of the Perturbed map, Multifractal of Dynamical systems

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Authors: Iftikhar Ahmad, A. Benallegue, A. El Hadri

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In this paper, we have proposed an observer for the reconstruction and a control law for the rejection application of unknown bounded external disturbance in a dynamical system. The strategy of both the observer and the controller is designed like a second order sliding mode with a proportional-integral (PI) term. Lyapunov theory is used to prove the exponential convergence and stability. Simulations results are given to show the performance of this method.

Keywords: non-linear systems, sliding mode observer, disturbance rejection, nonlinear control

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Authors: Sergey Kuznetsov, Yuri Urman

Abstract:

We present an approach to investigate non-linear dynamical effects occurring in the noncontact superconducting and diamagnetic suspensions, when levitated body has finite size. This approach is based on the calculation of interaction energy between spherical finite size superconducting or diamagnetic body with external magnetic field. Effects of small deviations from spherical shape may be also taken into account by introducing small corrections to the energy. This model allows investigating dynamical effects important for practical applications, such as nonlinear resonances, change of vibration plane, coupling of rotational and translational motions etc. We also show how the geometry of suspension affects various dynamical effects and how an inverse problem may be formulated to enforce or diminish various dynamical effects.

Keywords: levitation, non-linear dynamics, superconducting, diamagnetic stability

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Authors: Mohammad Javad Mollakazemi, Farhad Asadi, Aref Ghafouri

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In this paper, we considered and applied parametric modeling for some experimental data of dynamical system. In this study, we investigated the different distribution of output measurement from some dynamical systems. Also, with variance processing in experimental data we obtained the region of nonlinearity in experimental data and then identification of output section is applied in different situation and data distribution. Finally, the effect of the spanning the measurement such as variance to identification and limitation of this approach is explained.

Keywords: Gaussian process, nonlinearity distribution, particle filter, system identification

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Authors: R. Ghasemi, M. R. Rahimi Khoygani

Abstract:

This paper proposes the designing direct adaptive neural controller to apply for a class of a nonlinear pendulum dynamic system. The radial basis function (RBF) neural adaptive controller is robust in presence of external and internal uncertainties. Both the effectiveness of the controller and robustness against disturbances are importance of this paper. The simulation results show the promising performance of the proposed controller.

Keywords: adaptive neural controller, nonlinear dynamical, neural network, RBF, driven pendulum, position control

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Authors: Weihua Ruan, Kuan-Chou Chen

Abstract:

This paper is concerned with a system of Hamilton-Jacobi-Bellman equations coupled with an autonomous dynamical system. The mathematical system arises in the differential game formulation of political economy models as an infinite-horizon continuous-time differential game with discounted instantaneous payoff rates and continuously and discretely varying state variables. The existence of a weak solution of the PDE system is proven and a computational scheme of approximate solution is developed for a class of such systems. A model of democratization is mathematically analyzed as an illustration of application.

Keywords: Hamilton-Jacobi-Bellman equations, infinite-horizon differential games, continuous and discrete state variables, political-economy models

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Authors: Amin Behradfar

Abstract:

‎Tourism has a direct impact on the national revenue for all touristic countries. It creates work opportunities‎, ‎industries‎, ‎and several investments to serve and raise nations performance and cultures. ‎This paper is devoted to analyze dynamical behaviour of a four-dimensional non-linear tourism-based social-ecological system by using the codimension two bifurcation theory‎. ‎In fact we investigate the cusp bifurcation of that‎. ‎Implications of our mathematical results to the tourism‎ ‎industry are discussed‎. Moreover, profitability‎, ‎compatibility and sustainability of the tourism system are shown by the aid of cusp bifurcation and numerical techniques‎.

Keywords: tourism-based social-ecological dynamical systems, cusp bifurcation, center manifold theory, profitability, ‎compatibility, sustainability

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Authors: Kazuyoshi Mori

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In this paper, we consider the parametrization of the discrete-time systems without the unit-delay element within the framework of the factorization approach. In the parametrization, we investigate the number of required parameters. We consider single-input single-output systems in this paper. By the investigation, we find, on the discrete-time systems without the unit-delay element, three cases that are (1) there exist plants which require only one parameter and (2) two parameters, and (3) the number of parameters is at most three.

Keywords: factorization approach, discrete-time system, parameterization of stabilizing controllers, system without unit-delay

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Authors: Elimhan N. Mahmudov

Abstract:

The paper concerns the necessary and sufficient conditions of optimality for Cauchy problem of fourth order discrete (PD) and discrete-approximate (PDA) inclusions. The main problem is formulation of the fourth order adjoint discrete and discrete-approximate inclusions and transversality conditions, which are peculiar to problems including fourth order derivatives and approximate derivatives. Thus the necessary and sufficient conditions of optimality are obtained incorporating the Euler-Lagrange and Hamiltonian forms of inclusions. Derivation of optimality conditions are based on the apparatus of locally adjoint mapping (LAM). Moreover in the application of these results we consider the fourth order linear discrete and discrete-approximate inclusions.

Keywords: difference, optimization, fourth, approximation, transversality

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