Time Scaling of Chaotic Systems: Application to Secure Communications

The paper deals with time-scaling transformations of dynamical systems. Such scaling functions operate a change of coordinates on the time axis of the system trajectories preserving its phase portrait. Exploiting this property, a chaos encryption technique to transmit a binary signal through an analog channel is proposed. The scheme is based on a suitable time-scaling function which plays the role of a private key. The encoded transmitted signal is proved to resist known decryption attacks offering a secure and reliable communication.Comment: 15 pages, 7 figure

A Tutorial on Time-Evolving Dynamical Bayesian Inference

In view of the current availability and variety of measured data, there is an increasing demand for powerful signal processing tools that can cope successfully with the associated problems that often arise when data are being analysed. In practice many of the data-generating systems are not only time-variable, but also influenced by neighbouring systems and subject to random fluctuations (noise) from their environments. To encompass problems of this kind, we present a tutorial about the dynamical Bayesian inference of time-evolving coupled systems in the presence of noise. It includes the necessary theoretical description and the algorithms for its implementation. For general programming purposes, a pseudocode description is also given. Examples based on coupled phase and limit-cycle oscillators illustrate the salient features of phase dynamics inference. State domain inference is illustrated with an example of coupled chaotic oscillators. The applicability of the latter example to secure communications based on the modulation of coupling functions is outlined. MatLab codes for implementation of the method, as well as for the explicit examples, accompany the tutorial.Comment: Matlab codes can be found on http://py-biomedical.lancaster.ac.uk

Symmetry in Chaotic Systems and Circuits

Symmetry can play an important role in the field of nonlinear systems and especially in the design of nonlinear circuits that produce chaos. Therefore, this Special Issue, titled “Symmetry in Chaotic Systems and Circuits”, presents the latest scientific advances in nonlinear chaotic systems and circuits that introduce various kinds of symmetries. Applications of chaotic systems and circuits with symmetries, or with a deliberate lack of symmetry, are also presented in this Special Issue. The volume contains 14 published papers from authors around the world. This reflects the high impact of this Special Issue

Electromagnetic Radiation Control for Nonlinear Dynamics of Hopfield Neural Networks

© 2024 Author(s). Published under an exclusive license by AIP Publishing. This is the accepted manuscript version of an article which has been published in final form at https://doi.org/10.1063/5.0194928Electromagnetic radiation (EMR) affects the dynamical behavior of the nervous system, and appropriate EMR helps to study the dynamic mechanism of the nervous system. This paper uses a sophisticated four-dimensional Hopfield neural network (HNN) model augmented with one or more memristors to simulate the effects of EMR. We focus on the chaotic dynamics of HNN under the influence of EMR. Complex dynamical behaviors are found and transient chaotic phenomena have the same initial value sensitivity, showing how transient chaos is affected by EMR. Multiperiodic phenomena induced by quasi-periodic alternations are found in the dual EMR, as well as the suppression properties of the dual EMR for system chaos. This implies that the dynamical behavior of the HNN system can be controlled by varying the amount of EMR or the number of affected neurons in the HNN. Finally, a strong validation of our proposed model is provided by Multisim and FPGA hardware.Peer reviewe

Anti-deterministic behavior of discrete systems that are less predictable than noise

We present a new type of deterministic dynamical behaviour that is less predictable than white noise. We call it anti-deterministic (AD) because time series corresponding to the dynamics of such systems do not generate deterministic lines in Recurrence Plots for small thresholds. We show that although the dynamics is chaotic in the sense of exponential divergence of nearby initial conditions and although some properties of AD data are similar to white noise, the AD dynamics is in fact less predictable than noise and hence is different from pseudo-random number generators.Comment: 6 pages, 5 figures. See http://www.chaosandnoise.or

Computational Intelligence and Complexity Measures for Chaotic Information Processing

This dissertation investigates the application of computational intelligence methods in the analysis of nonlinear chaotic systems in the framework of many known and newly designed complex systems. Parallel comparisons are made between these methods. This provides insight into the difficult challenges facing nonlinear systems characterization and aids in developing a generalized algorithm in computing algorithmic complexity measures, Lyapunov exponents, information dimension and topological entropy. These metrics are implemented to characterize the dynamic patterns of discrete and continuous systems. These metrics make it possible to distinguish order from disorder in these systems. Steps required for computing Lyapunov exponents with a reorthonormalization method and a group theory approach are formalized. Procedures for implementing computational algorithms are designed and numerical results for each system are presented. The advance-time sampling technique is designed to overcome the scarcity of phase space samples and the buffer overflow problem in algorithmic complexity measure estimation in slow dynamics feedback-controlled systems. It is proved analytically and tested numerically that for a quasiperiodic system like a Fibonacci map, complexity grows logarithmically with the evolutionary length of the data block. It is concluded that a normalized algorithmic complexity measure can be used as a system classifier. This quantity turns out to be one for random sequences and a non-zero value less than one for chaotic sequences. For periodic and quasi-periodic responses, as data strings grow their normalized complexity approaches zero, while a faster deceasing rate is observed for periodic responses. Algorithmic complexity analysis is performed on a class of certain rate convolutional encoders. The degree of diffusion in random-like patterns is measured. Simulation evidence indicates that algorithmic complexity associated with a particular class of 1/n-rate code increases with the increase of the encoder constraint length. This occurs in parallel with the increase of error correcting capacity of the decoder. Comparing groups of rate-1/n convolutional encoders, it is observed that as the encoder rate decreases from 1/2 to 1/7, the encoded data sequence manifests smaller algorithmic complexity with a larger free distance value

On the Application of PSpice for Localised Cloud Security

The work reported in this thesis commenced with a review of methods for creating random binary sequences for encoding data locally by the client before storing in the Cloud. The first method reviewed investigated evolutionary computing software which generated noise-producing functions from natural noise, a highly-speculative novel idea since noise is stochastic. Nevertheless, a function was created which generated noise to seed chaos oscillators which produced random binary sequences and this research led to a circuit-based one-time pad key chaos encoder for encrypting data. Circuit-based delay chaos oscillators, initialised with sampled electronic noise, were simulated in a linear circuit simulator called PSpice. Many simulation problems were encountered because of the nonlinear nature of chaos but were solved by creating new simulation parts, tools and simulation paradigms. Simulation data from a range of chaos sources was exported and analysed using Lyapunov analysis and identified two sources which produced one-time pad sequences with maximum entropy. This led to an encoding system which generated unlimited, infinitely-long period, unique random one-time pad encryption keys for plaintext data length matching. The keys were studied for maximum entropy and passed a suite of stringent internationally-accepted statistical tests for randomness. A prototype containing two delay chaos sources initialised by electronic noise was produced on a double-sided printed circuit board and produced more than 200 Mbits of OTPs. According to Vladimir Kotelnikov in 1941 and Claude Shannon in 1945, one-time pad sequences are theoretically-perfect and unbreakable, provided specific rules are adhered to. Two other techniques for generating random binary sequences were researched; a new circuit element, memristance was incorporated in a Chua chaos oscillator, and a fractional-order Lorenz chaos system with order less than three. Quantum computing will present many problems to cryptographic system security when existing systems are upgraded in the near future. The only existing encoding system that will resist cryptanalysis by this system is the unconditionally-secure one-time pad encryption

Practical Chaos: Using Dynamical Systems to Encrypt Audio and Visual Data

Although dynamical systems have a multitude of classical uses in physics and applied mathematics, new research in theoretical computer science shows that dynamical systems can also be used as a highly secure method of encrypting data. Properties of Lorenz and similar systems of equations yield chaotic outputs that are good at masking the underlying data both physically and mathematically. This paper aims to show how Lorenz systems may be used to encrypt text and image data, as well as provide a framework for how physical mechanisms may be built using these properties to transmit encrypted wave signals