Browsing by Author "Perera, B. B. U. P."

Now showing 1 - 6 of 6

Dynamical behaviour and stability control of new hyperchaotic finance system
(Faculty of Science, University of Kelaniya Sri Lanka, 2024) Nimansa, K. H. K.; Perera, B. B. U. P.
In economic activities, chaos is undesirable as it can lead to an economic crisis and should be addressed promptly. In chaotic situations, it is desirable to control the chaotic orbits to a stable state, a periodic orbit, or to synchronize the chaotic system. Various control and synchronization methods have been developed and tested to achieve this. The new hyperchaotic finance system proposed by Yu, Cai, and Li in 2012 is a non-linear coupled system of four differential equations. Linear control method, speed feedback method, recursive backstepping control method, backstepping sliding mode control method, and adaptive control strategies have already been applied to synchronize and stabilize new hyperchaotic finance system. However, most of these methods lack one or more desired features of a control method, such as simplicity, feasibility, and efficiency. Therefore, in this work, we investigate the feasibility of two different control methods for addressing the chaos of the new hyperchaotic finance system. Firstly, the dynamical behaviours of the new hyperchaotic finance system were examined using waveforms, phase portraits, equilibrium points, Lyapunov exponents, and bifurcation diagrams. The results indicate that this system exhibits intricate features and evolves into chaotic, hyperchaotic and periodic phenomena based on parameter variations. Secondly, we propose and implement the two control methods for the new hyperchaotic finance system, namely the active control method and the active backstepping technique, which possess the desirable properties of a control method. The validity and effectiveness of the two control methods are illustrated using numerical simulations. The Active Backstepping technique was utilized to stabilize the unstable motion of trajectories globally at the origin. The computer simulation results have been compared with the existing speed feedback control method, showing that the designed controllers regulate the new hyperchaotic finance system to the origin at t = 5s and the speed feedback controller at t = 30s. The comparison shows that the performance of the proposed control method is better than the speed feedback control method for stabilizing of the hyperchaotic finance system as it converges faster. The active and backstepping control techniques have been utilized to synchronize the two new hyperchaotic finance systems. The numerical simulations have proven that over time, the Slave hyperchaotic finance system can replicate the chaotic dynamics of the Master hyperchaotic finance system after activating the designed controllers. Additionally, the Slave system synchronized with the Master System by eliminating the error between the state variables of the two systems. Finally, it was concluded that the designed controllers are suitable for controlling the new hyperchaotic finance system to track any trajectory that is a smooth function of time, as well as for stabilizing at any position effectively.
Numerical solutions of fuzzy differential equations using higher order methods
(Faculty of Science, University of Kelaniya Sri Lanka, 2023) Kumara, H. H. D.; Perera, B. B. U. P.
Fuzzy differential equations (FDEs) are useful for modelling real-world problems in science and engineering and have been studied by many researchers. Fuzzy solutions are required for certain problems due to the inherent uncertainty of the initial conditions. For instance, Fuzzy Predator Prey (FPP) models and fuzzy RLC-circuit models. The predator-prey model and RLC-Circuit model consist of a system of first-order differential equations. When the initial conditions are imprecise, they can be modelled as triangular Fuzzy numbers. In this study, four fuzzy numerical methods: namely, fuzzy Euler method, 2nd and 3rd-order fuzzy Taylor methods, and 4th-order fuzzy Runge-Kutta method were used. Those methods were implemented and validated for the solutions of certain Fuzzy Predator Prey (FPP) models and Fuzzy RLC-circuit models using MATLAB. The absolute and relative errors were compared with the analytical solution when available and among the methods. It was observed that the 3rd-order fuzzy Taylor method gives a better approximation (and minimum error) for the models. In addition, stability analysis, qualitative error analysis, sensitivity analysis of the crisp parameters, and fuzzy initial populations of the above FPP model were studied. From the stability analysis, it was observed that there is a fuzzy stable equilibrium point for the FPP model. According to the sensitivity analysis of the crisp parameter in the FPP example, it can be concluded that when the natural growth rate increases, the prey population increases, and the maximum number of predators also increases in the shorter period, as the death rate per encounter of preys due to predation increases the maximum number of prey and predators decrease. The first peak in predators occurs sooner, and as the natural death rate of predators in the absence of the prey increases, the maximum number of prey increases, and as the death rate goes to zero, the predator population remains constant. When the reproduction rate of predators for each prey captured increases, the prey population decreases, and it does not show a significant variation in the peak of the predator populations. By the sensitivity analysis of the initial population in the FPP model, it was observed that the maximum number of preys is larger than the maximum number of predators and the fuzzy phase plane is closed or open. Also, it can be concluded that as in the unfuzzy version of the algorithms, there’s a trade-off between step size, accuracy, and computational cost. In particular, if the step size is too large, the solution fails to converge.
Off-line signature verification system using artificial neural networks
(Research Symposium on Pure and Applied Sciences, 2018 Faculty of Science, University of Kelaniya, Sri Lanka, 2018) Ambegoda, A. L. T. P.; Perera, B. B. U. P.
Handwritten signature recognition method is the most popular recognition method of a personal identity. But it is easy to misuse that the signature forgery has become a great threat to the accuracy of the documentary. In this paper, we present an off-line signature recognition method using an Artificial Neural Network (ANN) created using Matlab (Matrix Laboratory). A signature dataset consisting of 248 signatures (both genuine and forged) of three different owners, used to train the network. First, the signatures were preprocessed enabling extracting their features. Then some geometrical features were extracted from each signature to feed as the inputs to the neural network. Each image was converted to a binary image and after identifying the geometric center, the image was divided into four segments. Again identifying the geometric center of each segment, each segment was divided again into four segments. Geometrical features from each segment were extracted and used as inputs to feed the network. A single neural network was created to execute both authentication and verification steps of the signature recognition. The network topology is optimized to the given dataset. Supplying corresponding target values, the network was trained. The Mean Squared Error (MSE) function was used to determine the performance of the network. Changing the parameters, the network was trained until it gets a favorable output. It showed a favorable performance value of 0.164 and an accuracy greater than 72% for all three subsets: training, validation and test sets in the training dataset. Then the network was tested on an untrained dataset of the same owners. For this untrained dataset, a favorable result was gained with an accuracy of 0.6129 and performance value of 0.3226. The lower the performance value, the better the network. It is assumed that high variability, too simplicity, and consistency of the data affected the results of the network. It is proposed to consider a larger dataset and improve the algorithm to be more sensitive to the above mentioned misleading factors. Although the performance is greater for the testing dataset than for the training dataset, it is concluded that the created network can be enhanced and developed to be applied in practical situations.
Optimal choice of the shape parameter for the radial basis functions method in one-dimensional parabolic inverse problems
(Faculty of Science, University of Kelaniya Sri Lanka, 2024) Wasana, W. B. M. S.; Perera, B. B. U. P.
Inverse problems have many important applications in science, engineering, medicine, and various other fields. Numerous analytical and numerical methods exist for solving these problems. In the present work, we obtain the numerical solution of a one-dimensional inverse parabolic equation with energy overspecification at a point using the Radial Basis Functions (RBF) method. The RBF method is a meshless approach based on the collocation method, which does not require evaluating any integrals. The collocation matrix is severely ill-conditioned, and the performance and accuracy of the method highly depend on the choice of the radial basis function and the shape parameter. We examine the performance and effectiveness of three different RBFs: Gaussian, multiquadric, and inverse multiquadric, with regard to the solutions of the inverse problem. To validate our approach, we compare our results with known analytical solutions for two test problems. Furthermore, we calculate the optimal shape parameter for the three radial basis functions for the one-dimensional parabolic inverse problem with energy overspecification. It was concluded that, in general, the multiquadric RBF (MQRBF) and inverse multiquadric RBF (IMQRBF) are better options than the Gaussian RBF (GRBF) in terms of error in both the inverse and direct problems. We observed that the error in the approximation of the source control function decreases linearly with time and increases with the step size, a behavior consistent across all three RBFs considered in the study. From the analysis carried out in search of the optimum shape parameter, we found that the optimal values of the shape parameters for the GRBF, MQRBF, and IMQRBF are 1, 2, and 2, respectively, considering both error norms. Furthermore, we have observed that IMQRBF is better suited for the current inverse problem due to the smaller condition number of the collocation matrix.
Predator-prey model with fuzzy initial conditions
(Research Symposium on Pure and Applied Sciences, 2018 Faculty of Science, University of Kelaniya, Sri Lanka, 2018) Ravindran, U. M.; Perera, B. B. U. P.
Predator-prey model, an initial value problem which is found in real life describes the relationship between predators and preys in an ecosystem, consisting of two nonlinear, autonomous differential equations. As it is an initial value problem, the initial values - the number of initial populations, directly affect the accuracy of the results. Initial population size depends on many external variables such as environmental conditions and biological factors. Hence taking fixed values for initial conditions at an uncertainty environment leads inaccurate results. Therefore, initial conditions given in fuzzy intervals give solutions more accurately. Fuzziness in the initial conditions is taken as triangular fuzzy numbers. The predator-prey equations are converted into fuzzy differential equations by denoting membership functions and solved numerically using fuzzy Euler method and then using 2nd order fuzzy Taylor method. The solutions converge to crisp solutions. The sensitivity analysis of the parameters of the fuzzy Predator-Prey equations and the fuzzy initial populations are carried out. In a future work, fuzzy Taylor method can be extended to include higher orders.
Solutions of singular Sturm-Liouville problems using exponential integrators
(Faculty of Science, University of Kelaniya Sri Lanka, 2023) Jayasinghe, M. G. P. A.; Perera, B. B. U. P.
Sturm-Liouville problems play a major role in the fields of applied mathematics, physics, and engineering. Therefore, the solutions of SLPs are particularly influential in these study areas. In the mid-1830s, the French mathematician Charles-François Sturm and Joseph Liouville worked independently on the problem of heat conduction through a metal bar, developing techniques for solving a large class of partial differential equations (PDEs) in the process and the study of this Sturm-Liouville problem originated. In combination with Magnus expansion, the Lie group method is used to develop an algorithm to solve the Sturm-Liouville Problem with an arbitrary set of boundary conditions. This study concerns singular Sturm-Liouville problems and their solutions. We use Magnus expansion method, which is an exponential integrator that use the exponential function of the Jacobian for the numerical integration of the problem. The coefficients of the Magnus expansion method of 6th -order were derived using two-point Gaussian Quadrature rule and Lagrange interpolation method. To validate and assess the accuracy and the performance of the method, the results were compared with the analytical solution and the 4th -order method for a particular singular SLP. It was observed that the absolute and relative error values given by the 6th -order method are almost comparable with those of the 4th -order with slight variations which can be attributed to the rounding off errors. As expected, the complexity of the 6th -order method is slightly high as the floating point (FLOPS) count shows. In conclusion, the accuracy of the 4th and 6th order methods are the same although the complexity of 6th -order method is slightly higher. In future work, it is expected to improve the 6th -order method by using the properties of Lie brackets and using different quadrature rule for the integrals.