No Arabic abstract
Universal regularities of the Fourier spectrum of signal, generated by complex analytic map at the period-tripling bifurcations accumulation point are considered. The difference between intensities of the subharmonics at the values of frequency corresponding to the neighbor hierarchical levels of the spectrum is characterized by a constant $gamma=21.9$ dB?, which is an analogue of the known value $gamma_F=13.4$ dB, intrinsic to the Feigenbaum critical point. Data of the physical experiment, directed to the observation of the spectrum at period-tripling accumulation point, are represented.
Accumulation point of period-tripling bifurcations for complexified Henon map is found. Universal scaling properties of parameter space and Fourier spectrum intrinsic to this critical point is demonstrated.
Spectral properties and transition to instability in neutral delay differential equations are investigated in the limit of large delay. An approximation of the upper boundary of stability is found and compared to an analytically derived exact stability boundary. The approximate and exact stability borders agree quite well for the large time delay, and the inclusion of a time-delayed velocity feedback improves this agreement for small delays. Theoretical results are complemented by a numerically computed spectrum of the corresponding characteristic equations.
An epidemic model with distributed time delay is derived to describe the dynamics of infectious diseases with varying immunity. It is shown that solutions are always positive, and the model has at most two steady states: disease-free and endemic. It is proved that the disease-free equilibrium is locally and globally asymptotically stable. When an endemic equilibrium exists, it is possible to analytically prove its local and global stability using Lyapunov functionals. Bifurcation analysis is performed using DDE-BIFTOOL and traceDDE to investigate different dynamical regimes in the model using numerical continuation for different values of system parameters and different integral kernels.
The Fourier spectrum at a fractional period is often examined when extracting features from biological sequences and time series. It reflects the inner information structure of the sequences. A fractional period is not uncommon in time series. A typical example is the 3.6 period in protein sequences, which determines the $alpha$-helix secondary structure. Computing the spectrum of a fractional period offers a high-resolution insight into a time series. It has thus become an important approach in genomic analysis. However, computing Fourier spectra of fractional periods by the traditional Fourier transform is computationally expensive. In this paper, we present a novel, fast algorithm for directly computing the fractional period spectrum (FPS) of time series. The algorithm is based on the periodic distribution of signal strength at periodic positions of the time series. We provide theoretical analysis, deduction, and special techniques for reducing the computational costs of the algorithm. The analysis of the computational complexity of the algorithm shows that the algorithm is much faster than traditional Fourier transform. Our algorithm can be applied directly in computing fractional periods in time series from a broad of research fields. The computer programs of the FPS algorithm are available at https://github.com/cyinbox/FPS.
We study the presence in the Lozi map of a type of abrupt order-to-order and order-to-chaos transitions which are mediated by an attractor made of a continuum of neutrally stable limit cycles, all with the same period.