The paper is dedicated to the numerical and experimental study of nonlinear oscillations exhibited by the Vilnius chaotic generator. The need for practical applicability of the mentioned generator in chaotic communication systems defined the necessity to study the operation of the circuit in a wide parameter range. The bifurcation maps and corresponding one-parameter diagrams, obtained and analysed utilising the Method of Complete Bifurcation Groups, reveal the complex nonlinear dynamics and regions of robust chaotic oscillations. Laboratory experiments allow the verification of the numerical results and identification of the most practically relevant phenomena.

A novel kind of power spectrum is constructed, the inter-spike spectrum, which transforms any signal into its spike-frequency domain. This method clearly shows the apparent cycles in the data and overcomes the problems for spike-train-like signals when using the obvious idea of Fourier-transforming it. We invent this instructive approach with the idea of transforming the τ-recurrence rate of a recurrence plot (RP), which often has a spiky appearance. The τ-recurrence rate is the density of recurrence points along diagonals of the RP, which are parallel to the main diagonal with a distance of τ. In this context the inter-spike spectrum can be interpreted as a nonlinear power spectrum of a potentially high dimensional system which constitutes the RP. The proposed measure is robust to noise and is able to detect and analyze bifurcations.

The behaviour and bifurcations of solutions to three-dimensional (three-phase) quadratic polynomial dynamical systems (DSs) are considered. The integrability in elementary functions is proved for a class of autonomous polynomial DSs. The occurrence of bifurcations of the type twisted fold is discovered on the basis and within the frames of the elements of the developed DS qualitative theory. The discriminant criterion applied originally to two-phase quadratic polynomial DSs is extended to three-phase DSs investigated in terms of their coefficient matrices. Specific classes of D- and S-vectors are introduced and complete description of the symmetry relations inherent to the DS coefficient matrices is performed using the discriminant criterion.

Fundus image registration plays a crucial role in the clinical evaluation of ocular diseases, such as diabetic retinopathy and macular degeneration, necessitating meticulous monitoring. The alignment of multiple fundus images enables the longitudinal analysis of patient progression, widening the visual scope, or augmenting resolution for detailed examinations. Currently, prevalent methodologies rely on feature-based approaches for fundus registration. However, certain methods exhibit high feature point density, posing challenges in matching due to point similarity. This study introduces a novel fundus image registration technique integrating U-Net for the extraction of feature points employing FIVES for its training and evaluation, a novel and large dataset for blood vessels segmentation, prioritizing point distribution over abundance. Subsequently, the method employs medial axis transform and pattern detection to obtain feature points characterized by the Fast Retina Keypoint (FREAK) descriptor, facilitating matching for transformation matrix computation. Assessment of the vessel segmentation achieves 0.7559 for Intersection Over Union (IoU), while evaluation on the Fundus Image Registration Dataset (FIRE) demonstrates the method’s comparative performance against existing methods, yielding a registration error of 0.596 for area under the curve, refining similar earlier methods and suggesting promising performance comparable to prior methodologies.

As paradigm of complex behavior, Self-organized Criticality (SOC) reflects the ability of nonequilibrium dynamical systems to self-adjust into metastable states that are scale independent. The goal of this report is to tentatively show that the hierarchy of Standard Model masses and mixing angles follows from the universal scaling attributes of SOC.

The Standard Model (SM) fails to account for either the triplication of fermion families or chiral symmetry breaking in the electroweak sector. Here we show that both phenomena arise from the approach to chaos of quantum theory near the Fermi scale.

This work is a top-level summary of several contributions published in the last three decades. It makes the case that complex dynamics of nonlinear systems lies at the heart of foundational physics.