Mathematics
http://nur.nu.edu.kz:80/handle/123456789/511
2021-05-03T11:58:16ZSynchronization of Coupled Nonlinear Oscillators with Applications to Photonic Arrays
http://nur.nu.edu.kz:80/handle/123456789/4200
Synchronization of Coupled Nonlinear Oscillators with Applications to Photonic Arrays
Zharas, Banu
Bountis, Anastasios
In recent years, the study of synchronization of coupled oscillators have been the subject of intense research interest, leading to many new and unexpected phenomena. Our research is first focused on the analysis of a network of coupled nonlinear oscillators exhibiting the breakdown of synchronization into fascinating “chimera states” exhibiting the coexistence of synchronized and unsynchronized parts. We then apply these ideas to laser arrays of photonic “oscillators”, which have numerous applications in optical communications, sensing and imaging. First of all, we demonstrate the occurrence of synchronization and chimera states in a simpler problem, consisting of a ring of coupled 4D simplified Lorenz systems, in which each oscillator is described by a Li-Sprott oscillator [1]. An interesting feature of each oscillator is the coexistence of a limit cycle and two symmetric strange attractors for some specific range of parameters, which influences the global synchronization dynamics and leads to the formation of chimera states. Inspired
by this model, we study some fascinating oscillatory phenomena of coupled photonic oscillators consisting of dimers of semiconductor lasers, each of which is capable of performing limit cycle oscillations. Coupling in an appropriate way a large number of dimers in long arrays we find that they can exhibit combinations of oscillatory patterns involving long amplitude oscillations (LAO) and also localized oscillations of very small amplitude close to the fixed points (LOCFP). As preliminary results of this investigation, we show the coexistence of LOA and LOCFP patterns reminiscent of “chimera–like” states and LOCFP “breather– like” phenomena. Both of these behaviors are shown to be spatially robust, when we calculate the Discrete Laplacian of their amplitudes for long times.
Master of Science Thesis in Applied mathematics
Department of Mathematics, School of Science and Technology Nazarbayev University
2019
2019-05-01T00:00:00ZThe Dynamics of Hamiltonian Lattices With Application to Hollomon Oscillators
http://nur.nu.edu.kz:80/handle/123456789/4199
The Dynamics of Hamiltonian Lattices With Application to Hollomon Oscillators
Zholmaganbetova, Aigerim
Bountis, Anastasios
Many problems in theoretical physics are expressed in the form of Hamiltonian systems. Among these the first to be extensively studied were low-dimensional, possessing as few as two (or three) degrees of freedom. In the last decades, however, great attention has been devoted to Hamiltonian systems of high dimensionality. The most famous among them are the ones that deal with the dynamics and statistics of a large number N of mass particles connected with nearest neighbor interactions. At low energies E,
these typically execute quasiperiodic motions near some fundamental stable periodic orbits (SPOs) which
represent nonlinear continuations of the N normal mode solutions of the corresponding linear system.
However, as the energy is increased, these solutions destabilize causing the motion in their vicinity to drift into chaotic domains, thus giving rise to important questions concerning the systems behavior in the thermodynamic limit, where E and N diverge with E=N = constant. One of the open problems in Hamiltonian dynamics, therefore, examines the relation between local (linear) stability properties of simple periodic solutions of Hamiltonian systems, and the more “global” dynamics. In this thesis, after reviewing the main results on these topics for the case of N-particle Fermi-Pasta-Ulam Hamiltonians, I proceed to apply the corresponding methods to a lattice of Hollomon oscillators, which are of interest to applications in problems of nonlinear elasticity.
Master of Science Thesis in Applied Mathematics
Department of Mathematics, School of Science and Technology Nazarbayev University
Astana 010000, Kazakhstan
2019-05-29T00:00:00ZConvergence Rate of Fourier Neural Networks
http://nur.nu.edu.kz:80/handle/123456789/4198
Convergence Rate of Fourier Neural Networks
Zhumekenov, Abylay
Assylbekov, Zhenisbek
The paper investigates a convergence rate for 2-layer feedforward Fourier Neural Network
(FNN). Such networks are motivated by the approximation properties of wellknown
Fourier series. Several implementations of FNNs were proposed since 1990’s:
by Gallant and White; A. Silvescu; Tan, Zuo and Cai; Liu. The main focus of this
research is Silvescu’s FNN, because such activation function does not fit into the category
of networks, where the linearly transformed input is exposed to activation. The
latter ones were extensively described by Hornik in 1989. In regard to non-trivial
Silvescu’s FNN, its convergence rate is proven to be of order 𝑂(1/𝑛). The paper
continues investigating classes of functions approximated by Silvescu FNN, which
appeared to be from Schwartz space and space of positive definite functions.
2019-04-26T00:00:00ZExplorations on chaotic behaviors of Recurrent Neural Networks
http://nur.nu.edu.kz:80/handle/123456789/4197
Explorations on chaotic behaviors of Recurrent Neural Networks
Myrzakhmetov, Bagdat
Assylbekov, Zhenisbek; Takhanov, Rustem
In this thesis work we analyzed the dynamics of the Recurrent Neural Network architectures.
We explored the chaotic nature of state-of-the-art Recurrent Neural Networks:
Vanilla Recurrent Network, Recurrent Highway Networks and Structurally
Constrained Recurrent Network. Our experiments showed that they exhibit chaotic
behavior in the absence of input data. We also proposed a way of removing chaos
chaos from Recurrent Neural Networks. Our findings show that initialization of the
weight matrices during the training plays an important role, as initialization with
the matrices whose norm is smaller than one will lead to the non-chaotic behavior
of the Recurrent Neural Networks. The advantage of the non-chaotic cells is stable
dynamics. At the end, we tested our chaos-free version of the Recurrent Highway
Networks (RHN) in a real-world application.
In a sequence-to-sequence modeling experiments, particularly in the language
modeling task, chaos-free version of RHN perform on par with the original version by
using the same hyperparameters.
Submitted to the Department of Mathematics on Apr 29, 2019, in partial fulfillment of the
requirements for the degree of Master of Science in Applied Mathematics
2019-04-29T00:00:00ZNumerical computations of complexification of Legendrian knots
http://nur.nu.edu.kz:80/handle/123456789/4196
Numerical computations of complexification of Legendrian knots
Yerzhigit, Bauyrzhan
Lawrence, Mark
With the recent interest in knots, it is interesting to study their complexification. We
have chosen to study Legendrian knots as they have the property that we can reconstruct
the original knot from its projection. This property is especially useful in the case of the
complexification of a knot as in this case the diagram of the projection of the knot is no
longer real. In this paper we show a way to compute complex rational functions that have a Legendrian knot as an image under unit circle.
2019-04-29T00:00:00ZMultigrid method for Mild-Slope equation in Coastal Wave Modelling
http://nur.nu.edu.kz:80/handle/123456789/4195
Multigrid method for Mild-Slope equation in Coastal Wave Modelling
Tabarek, Rysbergen
Erlangga, Yogi
In this thesis we propose and study an efficient iterative multigrid method for the time independent modified mild slope equation with and without energy dissipation term.
The algorithm relies on a multigrid method preconditioned with shifted-Laplacian
preconditioner and solved by Bi-CGSTAB algorithm. Multigrid analysis results are
shown by numerical experiments. Numerical experiments are conducted in depth
sloped elliptic shoal introduced by Berkhoff et. al
Submitted to the Department of Mathematics on Apr 19, 2019, in partial fulfillment of the
requirements for the degree of Master of Applied Mathematics
2019-04-19T00:00:00Z