Modeling of Chaotic Processes by Means of Antisymmetric Neural ODEs

The main goal of this work is to construct an algorithm for modeling chaotic processes using special neural ODEs with antisymmetric matrices (antisymmetric neural ODEs) and power activation functions (PAFs). The central part of this algorithm is to design a neural ODEs architecture that would guarantee the generation of a stable limit cycle for a known time series. Then, one neuron is added to each equation of the created system until the approximating properties of this system satisfy the well-known Kolmogorov theorem on the approximation of a continuous function of many variables. In addition, as a result of such an addition of neurons, the cascade of bifurcations that allows generating a chaotic attractor from stable limit cycles is launched. We also consider the possibility of generating a homoclinic orbit whose bifurcations lead to the appearance of a chaotic attractor of another type. In conclusion, the conditions under which the found attractor adequately simulates the chaotic process are discussed. Examples are given.

A New Approach to Optimize the Relative Clearance for Cylindrical Joints Manufactured by FDM 3D Printing Using a Hybrid Genetic Algorithm Artificial Neural Network and Rational Function

Nowadays, FDM technology permits obtaining functional prototypes or even end parts. The process parameters, such as layer thickness, building orientation, fill density, type of support, etc., have great influence on the quality, functionality and behavior of the obtained parts during their lifetime. In this paper, we present a study concerning the possibilities of obtaining certain values for clearance in revolute joints of non-assembly mechanisms manufactured by FDM 3D Printing. To ensure the functioning of the assembly, one must know the relationship between the imposed and measured clearances by taking into account the significant input data. One way is to use the automat learning method with an artificial neuronal network (ANN). The data necessary for the training, testing, and validation of ANN were experimentally obtained, using a complete L 27 Taguchi experimental plan. A total of 27 samples were printed with different values of the following parameters: the infill density, the imposed clearance between the shaft and the hole, and the layer thickness. ANN architecture corresponds to the Hecht–Kolmogorov theorem. Genetic algorithms (GA) were used for the optimization of the output. The Neural Network Toolbox from MATLAB was used for training the network and a hybrid tool genetic algorithm artificial neural network (GA-ANN) was used to minimize the value of the absolute relative clearance (arc). The minimum value of the absolute relative clearance established by GA-ANN was 0.0385788. This value was validated experimentally, with a relative difference of 4%. We also introduced a rational function to approximate the correlation between the input and output parameters. This function fulfills some frontier conditions resulted from practice. In addition, the function may be used to establish the designed clearance in order to obtain an imposed one.