Discrete Map Research Articles

Dynamical analysis of a tabu learning neuron through the discrete implicit mapping method

The complex dynamics of neuron model including bifurcation behaviors and firing patterns, especially unstable firing patterns, furnish a favorable point of view for the research of neurons, which may be of great importance in the prevention of brain-based diseases. To study this issue in depth, the discrete implicit mapping method is employed to evaluate a tabu learning neuron system, which is coupled with electromagnetic excitation and applied driving current in this paper. The bifurcation orbits of periodic motion with rich dynamic phenomena are accurately shown with varying the amplitudes of external input. The evolutions of period-5 to period-20 and period-6 to period-12 can be obtained via period doubling bifurcations and saddle bifurcation. The subcritical pitchfork bifurcation is also exhibited through the complex coexistence behavior of stable and unstable periodic orbits in such a model, which cannot be obtained by the traditional numerical method due to its accumulative errors. The corresponding stabilities and bifurcation of the periodic motions are determined by means of eigenvalues. Therefore, using such a method, the neural spiking events can be developed through phase diagram and time history of membrane potential. Moreover, the tabu learning system is implemented via FPGA (i.e., field programmable gate array), and the unstable and stable phase diagrams are observed completely from oscilloscope, where the experiments results verify the feasibility of semi-analytical solutions. Such investigation will also benefit the development of artificial intelligence and the advances in Brain Medicine.

Offset boosting in a discrete system.

Offset boosting plays an important role in chaos application in electronic engineering. A direct variable substitution typically will destroy the dynamics of a discrete map even though the initial condition is well considered. The internal fundamental reason is that the left-hand side of a discrete system does not have the dimension of variable differentiation (DVD) like the one of a continuous system. When the key property of DVD is completely preserved, the offset boosting based on a parameter or the initial condition can be reasonably achieved like in a differential system. Consequently, by the initial condition-oriented offset boosting, flexible multistability like attractor self-reproducing or attractor doubling can be further realized. A circuit experiment is completed for the verification of reliable offset boosting. The systematic exploration of offset boosting in a map will cast a new light on chaos regulation and attractor transportation in a discrete map. As a simple case, a two-dimensional Hénon map is taken as the example demonstrating the achievement of offset boosting via the parameter or initial condition.

An adaptive energy regulation in a memristive map linearized from a circuit with two memristive channels

Nonlinear circuits can show multistability when a magnetic flux-dependent memristor (MFDM) or a charge-sensitive memristor (CSM) is incorporated into a one branch circuit, which helps estimate magnetic or electric field effects. In this paper, two different kinds of memristors are incorporated into two branch circuits composed of a capacitor and a nonlinear resistor, thus a memristive circuit with double memristive channels is designed. The circuit equations are presented, and the dynamics in this oscillator with two memristive terms are discussed. Then, the memristive oscillator is converted into a memristive map by applying linear transformation on the sampled time series for the memristive oscillator. The Hamilton energy function for the memristive oscillator is obtained by using the Helmholtz theorem, and it can be mapped from the field energy of the memristive circuit. An energy function for the dual memristive map is suggested by imposing suitable weights on the discrete energy function. The dynamical behaviors of the new memristive map are investigated, and an adaptive law is proposed to regulate the firing mode in the memristive map. This work will provide a theoretical basis and experimental guidance for oscillator-to-map transformation and discrete map energy calculation.

Assessing sigmoidal function on memristive maps

Memristors offer a crucial element for constructing discrete maps that have garnered significant attention in complex dynamics and various potential applications. In this study, we have integrated memristive and sigmoidal function to propose innovative mapping techniques. Our research confirms that the amalgamation of memristor and sigmoidal functions represents a promising approach for creating both 2D and 3D maps. Particularly noteworthy are the chaotic maps featuring multiple sigmoidal functions and multiple memristors, as highlighted in our findings. Specifically focusing on the novel STMM1 map, we delve into its dynamics and assess its feasibility. Intriguingly, the introduction of sigmoidal functions leads to alterations in the quantity of fixed points and the symmetry of the map.

Air traffic inefficiencies and predictability evaluation using route mapping—the Tokyo International Airport case

Air traffic inefficiencies lead to excess fuel burn, emissions and air traffic controller (ATCo) workload. Various stakeholders have developed metrics to assess the operation performance. Most metrics compare the actual trajectories to some benchmark ones to calculate excess time or distance. This research is inspired by cellular automata (CA) and develops a combined time-distance lateral inefficiency and predictability metric using discrete space and time mapping on published flight routes. The analysis is focused on Tokyo International Airport, but uses only track data and published routes, which makes it easily applicable to any other hub airport worldwide. The mapping and velocity analyses are used to investigate when and where ATCos are most likely to intervene to provide save separation. A metric which can be adjusted to evaluate both traffic flow predictability and efficiency is proposed. This metric can be applied to better understand current traffic and enable future improvements towards seamless air traffic flow management.

An efficient image encryption algorithm using a discrete memory-based logistic map with deep neural network

In the last few years, multimedia technology has made tremendous strides. These days, the Web is frequently used to transfer multimedia content, including audio, video, and photos. However, the Internet is a very vulnerable medium with many security holes. To ensure that multimedia content carried across unprotected channels, like the Internet, is secure and private, several encryption techniques have been proposed. New encryption strategies must be developed because multimedia data streams cannot be encrypted using traditional methods. Therefore, the main goal of the recommended system is to present an analytical research approach for introducing a sophisticated framework wherein the suggested encryption technologies' efficacy is increased through the use of deep neural networks (DNNs). The robustness of the DNN principle is coupled with a discrete memristor-based logistic chaotic map notion for enhanced security performance. In this paper, three distinct encryption algorithms—Arnie cat with an artificial neural network (ANN), Henon map with an ANN, and logistic map with a DNN—are compared for security and performance with the suggested algorithm. Correlation coefficients, information entropy, number of pixels changing rate (NPCR), encryption quality, and encryption duration are the cryptographic analysis parameters examined here. The results show that the recommended implementation enhances security performance without degrading image quality. The proposed algorithm achieves 35.9% of UACI, 99.95% of NPCR, and 7.997231 of entropy.

Fractional-order quantum game chaotic map and its synchronization with application

In this paper, according to the knowledge of discrete fractional calculus and quantum game chaotic map, we first get the fractional quantum game chaotic map, which describes the dynamics of a quantum duopoly game. Then, the phase space, maximum Lyapunov exponent, and bifurcation diagram of the proposed map are obtained for the purpose of observing dynamic behaviors compared with the original quantum game chaotic map. After that, the synchronous controller is designed and the chaos synchronization of the fractional quantum game chaotic map is observed. Finally, we develop the fractional quantum game chaotic map into the chaotic sequence generator and image encryption scheme.

Stability analysis of a discrete chaotic map in superior orbit

Stability analysis of a discrete chaotic map in superior orbit

Linear coupling of transverse betatron oscillations: Dynamic stability and invariants of motion

Based on the technique of the discrete one-turn transfer maps, the problem of linear coupling between horizontal and vertical betatron oscillations in an accelerator has been treated exactly and entirely in explicit form. The stability region in the fractional part of the horizontal and the vertical betatron tune space as a function of the linear coupling strength has been obtained, and the increment/decrement of the horizontal and the vertical betatron oscillations in the case of the linear sum resonance has been shown to be approximately equal to the half of the coupling strength. The normal form parameterization of the one-turn linear map with horizontal-to-vertical coupling has been developed in detail in the spirit of Edwards and Teng formalism. The motion in the normal mode in the new normal form coordinates is decoupled by implying that two independent Courant-Snyder invariants exist, which have been found explicitly. Published by the American Physical Society 2024

Recurrent and chaotic outbreaks in SIR model

Abstract We examine several extensions to the basic Susceptible-Infected-Recovered model, which are able to induce recurrent outbreaks (the basic Susceptible-Infected-Recovered model by itself does not exhibit recurrent outbreaks). We first analyse how slow seasonal variations can destabilise the endemic equilibrium, leading to recurrent outbreaks. In the limit of slow immunity loss, we derive asymptotic thresholds that characterise this transition. In the outbreak regime, we use asymptotic matching to obtain a two-dimensional discrete map which describes outbreak times and strength. We then analyse the resulting map using linear stability and numerics. As the frequency of forcing is increased, the map exhibits a period-doubling route to chaos which alternates with periodic outbreaks of increasing frequency. Other extensions that can lead to recurrent outbreaks include the addition of noise, state-dependent variation and fine-graining of model classes.

A discrete Chialvo–Rulkov neuron network coupled with a novel memristor model: Design, Dynamical analysis, DSP implementation and its application

In this paper, a novel discrete flux-controlled memristor model combining square and cosine function is proposed and its properties are analyzed. Then, the discrete memristor is used to couple discrete Chialvo and Rulkov neuron forming a simple neuron network. The memristive coupled bi-neuron(C-M-R) map is constructed to simulate the connection of two different neurons by a synaptic in biological organizations. The equilibrium of this discrete map is studied. Dynamical behaviors of the C-M-R map are studied through a variety of methods with different parameters. The analysis methods adopted include phase diagram, iterative diagram, bifurcation diagram, Lyapunov Exponent spectrum(LEs), and Spectral Entropy(SE) complexity. Multiple types of hyperchaotic, chaotic, periodic attractors are observed in the C-M-R map. Various hidden firing modes and coexisting firing modes also exist in the map. Furthermore, state transition and attractor coexistence concerning different types of hyperchaos are found. The DSP hardware implementation is accomplished, proving the physical feasibility of the C-M-R map. Finally, the map is used in a simple image encryption scheme. This research imitates the principle of biological neuron coupling and can be applied to secrecy and encryption systems.

Chaotic discrete map of pulse oscillator dynamics with threshold nonlinear rate coding

Chaotic discrete map of pulse oscillator dynamics with threshold nonlinear rate coding

Mesoscopic damage study of red sandstone considering initial defects under drying-wetting cycles

Abstract In order to study the mechanical properties and microscopic damage of red sandstone by dry and wet cycling, this paper establishes a 3-dimensional uniaxial compression model of red sandstone by using PFC3D software. On this basis, microscopic cracks were introduced to analyze the influence of different cracks on the mesoscopic cracking of the model. The results show that the error between the simulation results and the test results is less than 10% for the model established by PFC3D software. Based on the discrete element model, displacement and velocity maps were obtained, and the values are larger at the contact with the load and smaller at the distance from the load. With the increase of the initial crack angle, the peak stress gradually increases, and with the decrease of the crack size, the peak stress gradually decreases, and the contact force also decreases.

Color image encryption based on discrete memristor logistic map and DNA encoding

This paper proposes a color image encryption scheme with new features based on two-dimensional discrete memristor logistic map (2D-MLM), Deoxyribonucleic acid (DNA) encoding and multi-wing hyperchaotic systems. Firstly, the discrete memristor map and four-wing hyperchaotic system are used to generate various pseudo-random sequences. Secondly, the original image is processed in blocks, and DNA encoding and operation are performed on the sub-block image using multiple pseudo-random sequences. Finally, combining the pseudo-random sequence to diffuse the sub-block image and DNA decoding, we can get the encrypted image. The scheme generates a one-time key for each image, and uses multiple different sub-keys to increase the key space. The use of diverse pseudo-random sequences for image scrambling effectively avoids the security risks associated with using a single chaotic sequence. The key space of this algorithm reaches 2425, the information entropy of the ciphertext image is 7.9993, the number of pixels change rate (NPCR) and the uniformed average change intensity (UACI) reach 99.6095 % and 33.5089 %, and the inter-pixel correlation is close to zero. The simulation results prove that the proposed algorithm has good security against many potential attacks.

Dynamical analysis and implementation of novel discrete memristive chaotic maps with hidden attractors

Dynamical analysis and implementation of novel discrete memristive chaotic maps with hidden attractors

From retraction maps to symplectic-momentum numerical integrators

Retraction maps have been generalized to discretization maps in (Barbero Linan and Martín de Diego, 2022). Discretization maps are used to systematically derive numerical integrators that preserve the symplectic structure, as well as the discrete momemtum map under the assumption of symmetric preservation for the discretization map. The procedure described here gives a geometrical construction that can be easily adapted to discretize dynamics on more general structures and open the door to reduction processes.

A state-dependent impulsive system with ratio-dependent action threshold for investigating SIR model

<abstract><p>In general, there is an imperative to amalgamate timely interventions and comprehensive measures for the efficacious control of infectious diseases. The deployment of such measures is intricately tied to the system's state and its transmission rate, presenting formidable challenges for stability and bifurcation analyses. In our pursuit of devising qualitative techniques for infectious disease analysis, we introduced a model that incorporates state-dependent transmission interventions. Through the introduction of state-dependent control, characterized by a non-linear action threshold contingent upon the combination of susceptible population density and its rate of change, we employ analytical methods to scrutinize various facets of the model. This encompasses addressing the existence, stability, and bifurcation phenomena concerning disease-free periodic solutions (DFPS). The analysis of the established Poincaré map leads us to the conclusion that DFPS indeed exists and maintains stability under specific conditions. Significantly, we have formulated a distinctive single-parameter family of discrete mappings, leveraging the bifurcation theorems of discrete maps to dissect the transcritical bifurcations around DFPS with respect to parameters such as $ ET $ and $ \eta_{1} $. Under particular conditions, these phenomena may give rise to effects like backward bifurcation and bistability. Through the analytical methodologies developed in this study, our objective is to unveil a more comprehensive understanding of infectious disease models and their potential relevance across diverse domains.</p></abstract>

DETERMINATION OF THE AVAILABLE PART OF THE EARTH SURFACE TAKING INTO ACCOUNT THE LANDSCAPE AND THE WIND

Modeling the movement of a body along a ballistic trajectory and determining the area of the Earth's surface that a given body can reach is an important task for researchers in various fields. Additional influencing factors – wind and air resistance – complicate calculations. The models are even more difficult to simulate a heterogeneous landscape where there are mountains and depressions and for which adjacent surface sectors have significant elevation differences. In view of this, the development of modeling methods in the specified field, which will be both fast and effective, remains relevant. The purpose of the study is to describe the structure of the data and the working principles of the model for determining the reachable area of the Earth's surface for a body thrown at an angle to the horizon. The methodological basis of the research is the computer simulation method and general scientific methods. An approach to modeling the landscape area reachable from a point by ballistic trajectories is proposed. The basis of the approach is to specify the surface in the form of a matrix, the elements of which are objects with fields of height values. A generator of discrete maps for further processing has been created. The method of calculating body launch angles optimal for reaching target points is determined. A mechanism for determining unreachable areas covered by others and a mechanism for taking into account the effect of wind on the result, which do not require long calculations, are proposed. The matrix task of the surface with height differences allows us to talk about the prospects of research - the definition of dangerous areas for the needs of various spheres of activity.

Multi-layer cross-country maps as a data source for solving the problem of laying a route

Representation of information about the surface of movement in the form of discrete maps of patency is used in problems related to the management of autonomous vehicles. The technology was developed for use on rovers. The areas of autonomous transport, transport and technological machines operating in an artificial and natural environment, mobile robots seem to be promising. The principle of building maps of patency and methods for choosing the trajectory of movement have a great potential for expanding the volume of information entered into the cells of the map, variability of the approach to the criteria for optimality of the trajectory, etc. The basics of the technology for preparing information about the area, the principles of using this information in solving the transport problem are presented, simple examples are given.

Nonlinear Rotations on a 3-Dimensional Lattice

The study of resonances of Hamiltonian systems with a divided phase space is a well-established area of research which attracted the interests of many researchers over the years. We study such resonances over a discrete space focusing on the rotational motion of some orbits in the 3-Dimensional lattice, ?3. In this study, we construct the discrete standard map from 2-Dimensional to 3-Dimensional lattices. Then, we identify and categorize any transformation that could give periodicity for the 3-Dimensional lattice on a specific map. The study aims to determine the points that give the periodicity and at the same time investigate the behavior of the points for the nonlinear stable orbits over a discrete space. For some choice of parameters ???? ?, our findings showed that the orbit of the 3-Dimensional map is periodic depending on the initial conditions. Some arbitrary initial conditions may be periodic in a 3-Dimensional lattice.Keywords: Hamiltonian systems, nonlinear rotations, space discretization, arithmetic dynamics