Logistic map with a delayed feedback: Stability of a discrete time-delay control of chaos

External low energy electromagnetic fields affect heart dynamics: surrogate for system synchronization, chaos control and cancer patient's health

All cells in the human body, including cancer cells, possess specific electrical properties crucial for their functions. These properties are notably different between normal and cancerous cells. Cancer cells are characterized by autonomous oscillations and damped electromagnetic field (EMF) activation. Cancer reduces physiological variability, implying a systemic disconnection that desynchronizes bodily systems and their inherent random processes. The dynamics of heart rate, in this context, could reflect global physiological network instability in the sense of entrainment. Using a medical device that employs an active closed-loop system, such as administering specifically modulated EMF frequencies at targeted intervals and at low energies, we can evaluate the periodic oscillations of the heart. This procedure serves as a closed-loop control mechanism leading to a temporary alteration in plasma membrane ionic flow and the heart's periodic oscillation dynamics. The understanding of this phenomenon is supported by computer simulations of a mathematical model, which are validated by experimental data. Heart dynamics can be quantified using difference logistic equations, and it correlates with improved overall survival rates in cancer patients.

A quantitative model of relation between respiratory-related blood pressure fluctuations and the respiratory sinus arrhythmia

In order to propose an interpretation of recent experimental findings concerning short-term variability of arterial blood pressure (ABP), heart rate variability (HRV), and their dependence on body posture, we develop a qualitative dynamical model of the short-term cardiovascular variability at respiratory frequency (HF). It shows the respiratory-related blood pressure fluctuations in relation to the respiratory sinus arrhythmia (RSA). Results of the model-based analysis show that the observed phenomena may be interpreted as buffering of the respiratory-related ABP fluctuations by heart rate (HR) fluctuations, i.e., the respiratory sinus arrhythmia. A paradoxical enhancement (PE) of the fluctuations of the ABP in supine position, that was found in experiment, is explained on the ground of the model, as an ineffectiveness of control caused by the prolonged phase shift between the the peak of modulation of the pulmonary flow and the onset of stimulation of the heart. Such phasic changes were indeed observed in some other experimental conditions. Up to now, no other theoretical or physiological explanation of the PE effect exists, whereas further experiments were not performed due to technical problems. Better understanding of the short-term dynamics of blood pressure may improve medical diagnosis in cardiology and diseases which alter the functional state of the autonomous nervous system.

Graphical Abstract

A Novel Delay Linear Coupling Logistics Map Model for Color Image Encryption

With the popularity of the Internet, the transmission of images has become more frequent. It is of great significance to study efficient and secure image encryption algorithms. Based on traditional Logistic maps and consideration of delay, we propose a new one-dimensional (1D) delay and linearly coupled Logistic chaotic map (DLCL) in this paper. Time delay is a common phenomenon in various complex systems in nature, and it will greatly change the dynamic characteristics of the system. The map is analyzed in terms of trajectory, Lyapunov exponent (LE) and Permutation entropy (PE). The results show that this map has wide chaotic range, better ergodicity and larger maximum LE in comparison with some existing chaotic maps. A new method of color image encryption is put forward based on DLCL. In proposed encryption algorithm, after various analysis, it has good encryption performance, and the key used for scrambling is related to the original image. It is illustrated by simulation results that the ciphered images have good pseudo randomness through our method. The proposed encryption algorithm has large key space and can effectively resist differential attack and chosen plaintext attack.

Controlling intermediate dynamics in a family of quadratic maps

The intermediate dynamics of composed one-dimensional maps is used to multiply attractors in phase space and create multiple independent bifurcation diagrams which can split apart. Results are shown for the composition of k-paradigmatic quadratic maps with distinct values of parameters generating k-independent bifurcation diagrams with corresponding k orbital points. For specific conditions, the basic mechanism for creating the shifted diagrams is the prohibition of period doubling bifurcations transformed in saddle-node bifurcations.

Exact period-four solutions of a family of n-dimensional quadratic maps via harmonic balance and Gröbner bases

Analytical solutions of the period-four orbits exhibited by a classical family of n-dimensional quadratic maps are presented. Exact expressions are obtained by applying harmonic balance and Gröbner bases to a single-input single-output representation of the system. A detailed study of a generalized scalar quadratic map and a well-known delayed logistic model is included for illustration. In the former example, conditions for the existence of bistability phenomenon are also introduced.

Impact of heterogeneous delays on cluster synchronization

We investigate cluster synchronization in coupled map networks in the presence of heterogeneous delays. We find that while the parity of heterogeneous delays plays a crucial role in determining the mechanism of cluster formation, the cluster synchronizability of the network gets affected by the amount of heterogeneity. In addition, heterogeneity in delays induces a rich cluster pattern as compared to homogeneous delays. The complete bipartite network stands as an extreme example of this richness, where robust ideal driven clusters observed for the undelayed and homogeneously delayed cases dismantle, yielding versatile cluster patterns as heterogeneity in the delay is introduced. We provide arguments behind this behavior using a Lyapunov function analysis. Furthermore, the interplay between the number of connections in the network and the amount of heterogeneity plays an important role in deciding the cluster formation.

Effect of delayed feedback on the dynamics of a scalar map via a frequency-domain approach

The effect of delayed feedback on the dynamics of a scalar map is studied by using a frequency-domain approach. Explicit conditions for the occurrence of period-doubling and Neimark-Sacker bifurcations in the controlled map are found analytically. The appearance of a 1:2 resonance for certain values of the delay is also formalized, revealing that this phenomenon is independent of the system parameters. A detailed study of the well-known logistic map under delayed feedback is included for illustration.

Description of stochastic and chaotic series using visibility graphs

Nonlinear time series analysis is an active field of research that studies the structure of complex signals in order to derive information of the process that generated those series, for understanding, modeling and forecasting purposes. In the last years, some methods mapping time series to network representations have been proposed. The purpose is to investigate on the properties of the series through graph theoretical tools recently developed in the core of the celebrated complex network theory. Among some other methods, the so-called visibility algorithm has received much attention, since it has been shown that series correlations are captured by the algorithm and translated in the associated graph, opening the possibility of building fruitful connections between time series analysis, nonlinear dynamics, and graph theory. Here we use the horizontal visibility algorithm to characterize and distinguish between correlated stochastic, uncorrelated and chaotic processes. We show that in every case the series maps into a graph with exponential degree distribution P(k)∼exp(-λk), where the value of λ characterizes the specific process. The frontier between chaotic and correlated stochastic processes, λ=ln(3/2) , can be calculated exactly, and some other analytical developments confirm the results provided by extensive numerical simulations and (short) experimental time series.

Characteristics of a delayed system with time-dependent delay time

The characteristics of a time-delayed system with time-dependent delay time is investigated. We demonstrate that the nonlinearity characteristics of the time-delayed system are significantly changed depending on the properties of time-dependent delay time and especially that the reconstructed phase trajectory of the system is not collapsed into simple manifold, differently from the delayed system with fixed delay time. We discuss the possibility of a phase space reconstruction and its applications.