Narrow Results

The limit cycle bifurcation of a Z₂ equivariant quintic planar Hamiltonian vector field under Z₂ equivariant quintic perturbation is studied. We prove that the given system can have at least 27 limit cycles. This is an improved lower bound on the possible number of limit cycles that can bifurcate from a quintic planar Hamiltonian system under quintic perturbation.

The limit cycle bifurcations of a Z₂ equivariant planar Hamiltonian vector field of degree 7 under Z₂ equivariant degree 7 perturbation is studied. We prove that the given system can have at least 53 limit cycles. This is an improved lower bound for the weak formulation of Hilbert's 16th problem for degree 7, i.e. on the possible number of limit cycles that can bifurcate from a degree 7 planar Hamiltonian system under degree 7 perturbation.

In this study, we highlight some fundamental issues of knowledge management and cast them in the setting of Granular Computing (GrC). We show how its formal constructs — information granules are instrumental in knowledge representation and specification of its level of abstraction.

In this paper, by using Sengupta and Pal's method of comparison of interval numbers and a new set of arithmetic operations for interval numbers, we propose a theory for the study of arithmetic operations on interval numbers.

In this paper we develop a new graphical representation of fuzzy numbers, which we then employ to propose a geometrical approach to their defuzzification. The calculations involved in the proposed method and the resultant representation use Moore's semiplane for intervals and therefore are far simpler than those involved in other approaches.

We start by representing triangular and trapezoidal fuzzy numbers in Moore's semiplane. Then we extend this work to any fuzzy number. Although this extension has to be undertaken in ${ℝ}^3$, it preserves all the properties we study for trapezoidal and triangular fuzzy numbers in Moore's semiplane.

Traditionally, game theory problems were considered for exact data, and the decisions were based on known payoffs. However, this assumption is rarely true in practice. Uncertainty in measurements and imprecise information must be taken into account. The interval-based approach for handling such uncertainties assumes that one has lower and upper bounds on payoffs. In this paper, interval bimatrix games are studied. Especially, we focus on three kinds of support set invariancy. Support of a mixed strategy consists of that pure strategies having positive probabilities. Given an interval-valued bimatrix game and supports for both players, the question states as follows: Does every bimatrix game instance have an equilibrium with the prescribed support? The other two kinds of invariancies are slight modifications: Has every bimatrix game instance an equilibrium being a subset/superset of the prescribed support? It is computationally difficult to answer these questions: the first case costs solving a large number of linear programs or mixed integer programs. For the remaining two cases a sufficient condition and a necessary condition are proposed, respectively.

For a mobile robot to operate in its environment it is crucial to determine its position with respect to an external reference frame using noisy sensor readings. A scenario in which the robot is moved to another position during its operation without being told, known as the kidnapped robot problem, complicates global localisation. In addition to that, sensor malfunction and external influences of the environment can cause unexpected errors, called outliers, that negatively affect the localisation process. This paper proposes a method based on the fusion of a particle filter with bounded-error localisation, which is able to deal with outliers in the measurement data. The application of our algorithm to solve the kidnapped robot problem using simulated data shows an improvement over conventional probabilistic filtering methods.

This paper deals a bimatrix game with payoffs as closed intervals. Existence of equilibrium point of this game is discussed by using suitable interval quadratic programming problem. Further, a methodology is proposed for finding optimal strategies for each player of the game. The methodology is illustrated by numerical example.

Game theory-based models are widely used to solve multiple competitive problems such as oligopolistic competitions, marketing of new products, promotion of existing products competitions, and election presage. The payoffs of these competitive models have been conventionally considered as deterministic. However, these payoffs have ambiguity due to the uncertainty in the data sets. Interval analysis-based approaches are found to be efficient to tackle such uncertainty in data sets. In these approaches, the payoffs of the game model lie in some closed interval, which are estimated by previous information. The present paper considers a multiple player game model in which payoffs are uncertain and varies in a closed intervals. The necessary and sufficient conditions are explained to discuss the existence of Nash equilibrium point of such game models. Moreover, Nash equilibrium point of the model is obtained by solving a crisp bi-linear optimization problem. The developed methodology is further applied for obtaining the possible optimal strategy to win the parliament election presage problem.

Uncertainty plays a fundamental role in structural engineering since it may affect both external excitations and structural parameters. In this study, the analysis of linear structures with slight variations of the structural parameters subjected to stochastic excitation is addressed. It is realistically assumed that sufficient data are available to model the external excitation as a Gaussian random process, while only fragmentary or incomplete information about the structural parameters are known. Under this assumption, a nonprobabilistic approach is pursued and the fluctuating properties are modeled as uncertain-but-bounded parameters via interval analysis. A method for evaluating the lower and upper bounds of the second-order statistics of the response is presented. The proposed procedure basically consists in combining random vibration theory with first-order interval Taylor series expansion of the mean-value and covariance vectors of the response. After some algebra, the sets of first-order ordinary differential equations ruling the nominal and first-order sensitivity vectors of response statistics are derived. Once such equations are solved, the bounds of the mean-value and covariance vectors of the response can be evaluated by handy formulas.

To validate the procedure, numerical results concerning two different structures with uncertain-but-bounded stiffness properties under seismic excitation are presented.

Uncertainties in structural parameters and measurements can be accounted for by incorporating interval analysis into the updating scheme of finite element models using a response-surface function. To facilitate the interval arithmetic operation, two different strategies are proposed in this paper to transform the response-surface function into a corresponding interval response-surface function. These strategies minimize the inherent interval overestimation that can arise from the variable dependency of the surrogate model. In the first strategy, the natural extension and centered-form extension methods are used to mitigate the interval overestimation of the surrogate model, which may or may not contain interaction terms. In the second strategy, the natural extensión method is also adopted to realize the interval transformation of the surrogate model containing interaction terms but an affine arithmetic is further introduced to minimize the interval overestimation. To demonstrate the efficacy of the proposed method, model parameters are determined from an instrumented model of a cable-stayed bridge tested on a shaking table. Results show that the proposed updating method is feasible and effective for applications to finite element models of complex bridge structures.

A hybrid method for plotting 2-dimensional curves, defined implicitly by equations of the form f(x,y) = 0 is presented. The method is extremely robust and reliable and consists of Space Covering techniques, Continuation principles and Interval analysis (i.e. SCCI). The space covering, based on iterated subdivision, guarantees that no curve branches or isolated curve parts or even points are lost (which can happen if grid methods are used). The continuation method is initiated in a subarea as soon as it is proven that the subarea contains only one smooth curve. Such a subarea does not need to be subdivided further so that the computation is accelerated as far as possible with respect to the subdivision process. The novelty of the SCCI-hybrid method is the intense use of the implicit function theorem for controlling the steps of the method. Although the implicit function theorem has a rather local nature, it is empowered with global properties by evaluating it in an interval environment. This means that the theorem can provide global information about the curve in a subarea such as existence, non-existence, uniqueness of the curve or even the presence of singular points. The information gained allows the above-mentioned control of the subarea and the decision of its further processing, i.e. deleting it, subdividing it, switching to the continuation method or preparing the plotting of the curve in this subarea. The curves can be processed mathematically in such a manner, that the derivation of the plotted curve from the exact curve is as small as desired (modulo the screen resolution).

In this paper, the interval analysis method is introduced to calculate the bounds of the structural displacement responses with small uncertain levels' parameters. This method is based on the first-order Taylor expansion and finite element method. The uncertain parameters are treated as the intervals, not necessary to know their probabilistic distributions. Through dividing the intervals of the uncertain parameters into several subintervals and applying the interval analysis to each subinterval combination, a subinterval analysis method is then suggested to deal with the structures with large uncertain levels' parameters. However, the second-order truncation error of the Taylor expansion and the linear approximation of the second derivatives with respect to the uncertain parameters, two error estimation methods are given to calculate the maximum errors of the interval analysis and subinterval analysis methods, respectively. A plane truss structure is investigated to demonstrate the efficiency of the presented method.

In this paper, an inverse method that combines the interval analysis with regularization is presented to stably identify the bounds of dynamic load acting on the uncertain structures. The uncertain parameters of the structure are treated as intervals and hence only their bounds are needed. Using the first-order Taylor expansion, the identified load can be approximated as a linear function of the uncertain parameters. In this function, it is assumed that the load at the midpoint of the uncertain parameters can be expressed as a series of impulse kernels. The finite element method (FEM) is used to obtain the response function of the impulse kernel and the response to the midpoint load is expressed in a form of convolution. In order to deal with the ill-posedness arising from the deconvolution, two regularization methods are adopted to provide the numerically efficient and stable solution of the desired unknown midpoint load. Then, a sensitivity analysis is suggested to calculate the first derivative of the identified load with respect to each uncertain parameter. Applying the interval extension in interval mathematics, the lower and upper bounds of identified load caused by the uncertainty can be finally determined. Numerical simulation demonstrates that the present method is effective and robust to stably determine the range of the load on the uncertain structures from the noisy measured response in time domain.

Evidence theory has a strong capacity to deal with epistemic uncertainty, in view of the overestimation in interval analysis, the responses of structural-acoustic problem with epistemic uncertainty could be untreated. In this paper, a numerical method is proposed for structural-acoustic system response analysis under epistemic uncertainties based on evidence theory. To improve the calculation accuracy and reduce the computational cost, the interval analysis technique and radial point interpolation method are adopted to obtain the approximate frequency response characteristics for each focal element, and the corresponding formulations of structural-acoustic system for interval response analysis are deduced. Numerical examples are introduced to illustrate the efficiency of the proposed method.

For vehicle-bridge system, structural uncertainties, especially the interval variables with correlation, have a great influence on dynamic response. Therefore, this paper proposes an effective uncertainty analysis method for vehicle-bridge system based on multidimensional parallelepiped (MP) model, which can reasonably deal with the correlation of interval variables. First, the vehicle-bridge system is simplified as a four degrees-of-freedom mass-spring vehicle model running on a simply supported beam. MP model is adopted to describe the uncertainties of all the interval variables. Second, via affine coordinate system transform, the interval variables with correlation are transformed as the independent variables, which is very convenient for uncertainty analysis. Finally, the uncertain dynamic response is approximated through the first-order Taylor interval expansion, and the upper and lower bounds can be calculated using the dynamic response at midpoints and the partial difference multiplied by interval radius. Because the correlation is sufficiently considered, the uncertainty analysis results on vehicle–bridge interaction system will be much more accurate than the traditional interval analysis method (IAM). Numerical example demonstrates the correctness and effectiveness of the proposed method.

Uncertainties in parameters can affect racing car performance. In this study, a nonlinear interval suspension damping optimization method is proposed to improve the road holding of a racing car. To evaluate the dynamic responses of racing cars under a random road input and a bump input with interval uncertain parameters, a quarter car model with a two-stage asymmetric damper is established. Then, a quadratic approximation model with second derivative terms is developed by second-order Taylor series expansion and dimension reduction to calculate the nonlinear dynamic response of the vehicle. Interval analysis of the objective function and constraints is carried out using interval arithmetic to eliminate nesting optimization and make the optimization efficient. The results show that the proposed optimization method can improve road holding performance, effectively suppress the fluctuation range of the road holding performance evaluation index, and ensure the robustness of the design scheme.

Due to the uncertainties of material, geometrical and load parameters, dynamic responses of actual engineering structures are uncertain. The paper investigates dynamic responses of uncertain frameworks by combining the interval method with the traveling wave method. The uncertainties of material, physical dimension and loads are firstly characterized by interval parameters. The waveguide and transmission equations of uncertain framework structures with interval parameters are then proposed based on the traveling wave method. The uncertain junction scattering equation is extracted from the force equilibrium conditions and displacement compatibility conditions and the interval form of dynamic responses are developed using interval arithmetic rules. Finally, the numerical examples including a single beam and a planar frame structure have been presented to verify the feasibility and validity of the proposed method.

One of the fundamental tasks of robotics is to solve the localization problem, in which a robot must determine its true pose without any knowledge on its initial location. In underwater environments, this is specially hard due to sensors restrictions. For instance, many times, the localization process must rely on information from acoustic sensors, such as transponders. We propose a method to deal with this scenario, that consists in a hybridization of probabilistic and interval approaches, aiming to overcome the weaknesses found in each approach and improve the precision of results. In this paper, we use the set inversion via interval analysis (SIVIA) technique to reduce the region of uncertainty about robot localization, and a particle filter to refine the estimates. With the information provided by SIVIA, the distribution of particles can be concentrated in regions of higher interest. We compare this approach with a previous hybrid approach using contractors instead of SIVIA. Experiments with simulated data show that our hybrid method using SIVIA provides more accurate results than the method using contractors.