Attractor Research Articles

Construction and analysis of symmetric Sprott B multi-attractors with electric implementation

Abstract In chaos theory, a number of systems and models which apparently contain simple ordinary differential equations (ODEs) turn out to show a dynamic characterized by complicated behaviors and complex trajectories. One of such systems is the Sprott B model. We construct some set of multi-attractors based on the Sprott B model where additional parameters and operators are considered. After summarizing important preliminaries relevant to simple chaotic differential systems, the model is firstly solved analytically and numerically, then graphical simulations are provided. The later show coexistence and evolution of two chaotic attractors in a symmetrical representation. Lastly, similar results and expected outcomes are recovered via an electrical circuit implementation, realized using the Field Programmable Gate Array (FPGA) board, the Digital-to-Analog Converter (DAC) and the Rigol Oscilloscope. They also show progressing sets of coexisting multi-attractors.

Dynamic Analysis and FPGA Implementation of Fractional-Order Hopfield Networks with Memristive Synapse

Memristors have become important components in artificial synapses due to their ability to emulate the information transmission and memory functions of biological synapses. Unlike their biological counterparts, which adjust synaptic weights, memristor-based artificial synapses operate by altering conductance or resistance, making them useful for enhancing the processing capacity and storage capabilities of neural networks. When integrated into systems like Hopfield neural networks, memristors enable the study of complex dynamic behaviors, such as chaos and multistability. Moreover, fractional calculus is significant for their ability to model memory effects, enabling more accurate simulations of complex systems. Fractional-order Hopfield networks, in particular, exhibit chaotic and multistable behaviors not found in integer-order models. By combining memristors with fractional-order Hopfield neural networks, these systems offer the possibility of investigating different dynamic phenomena in artificial neural networks. This study investigates the dynamical behavior of a fractional-order Hopfield neural network (HNN) incorporating a memristor with a piecewise segment function in one of its synapses, highlighting the impact of fractional-order derivatives and memristive synapses on the stability, robustness, and dynamic complexity of the system. Using a network of four neurons as a case study, it is demonstrated that the memristive fractional-order HNN exhibits multistability, coexisting chaotic attractors, and coexisting limit cycles. Through spectral entropy analysis, the regions in the initial condition space that display varying degrees of complexity are mapped, highlighting those areas where the chaotic series approach a pseudo-random sequence of numbers. Finally, the proposed fractional-order memristive HNN is implemented on a Field-Programmable Gate Array (FPGA), demonstrating the feasibility of real-time hardware realization.

Three-Dimensional Space Chaotic Scrolls Generated by Memristive Hopfield Neural Network and Its Application in Robot Path Planning

Abstract In this paper, a novel three-dimensional space-scroll chaotic attractor is constructed by introducing piece-wise linear memristors into a three-neurons-based Hopfield neural network to simulate the electromagnetic influence on neural dynamics. The dynamical behaviors of the novel memristive Hopfield neural network are discussed by means of numerical methods including the Lyapunov exponent spectrum, bifurcation diagrams, phase plots and dynamic maps. Furthermore, we also investigated different neurons subject to the memristive electromagnetic influence, where the two-dimensional grid-scroll chaotic attractors can be generated by one neuron subjects to the electromagnetic influence and three-dimensional space-scroll chaotic attractors can be generated by all of the neurons subject to the memristive electromagnetic influence. Moreover, we present a three-dimensional space-scroll scroll attractor-based chaotic mobile robot system. Superiority of the proposed space-scroll-based chaotic mobile robot system compared to existing methods has been demonstrated through numerical simulations.

Controllable multi-scroll chaotic attractors with multiple wings in Chua’s system

Controllable multi-scroll chaotic attractors with multiple wings in Chua’s system

Universal dynamical function behind all genetic codes: [formula omitted]-adic attractor dynamical model

The genetic code is a map which gives the correspondence between codons in DNA and amino acids. In the attractor dynamical model (ADM), genetic codes can be described as the sets of the cyclic attractors of discrete dynamical systems - the iterations of functions acting in the ring of 2-adic integers Z2. This ring arises from representation of nucleotides by binary vectors and hence codons by triples of binary vectors. We construct a Universal Function B such that the dynamical functions for all known genetic codes can be obtained from B by simple transformations on the set of codon cycles - the “Addition” and “Division” operations. ADM can be employed for study of phylogenetic dynamics of genetic codes. One can speculate that the “common ancestor genetic code” was caused by B. We remark that this function has 24 cyclic attractors which distribution coincides with the distribution for the hypothetical pre-LUCA code. This coupling of the Universal Function with the pre-LUCA code assigns the genetic codes evolution perspective to ADM. All genetic codes are generated from B through the special chains of the “Addition” and “Division” operations. The challenging problem is to assign the biological meaning to these mathematical operations.

Host-Parasitoid Systems are Vulnerable to Extinction via P-Tipping: Forest Tent Caterpillar as an Example.

Continuous-time predator-prey models admit limit cycle solutions that are vulnerable to the phenomenon of phase-sensitive tipping (P-tipping): The predator-prey system can tip to extinction following a rapid change in a key model parameter, even if the limit cycle remains a stable attractor. In this paper, we investigate the existence of P-tipping in an analogous discrete-time system: a host-parasitoid system, using the economically damaging forest tent caterpillar as our motivating example. We take the intrinsic growth rate of the consumer as our key parameter, allowing it to vary with environmental conditions in ways consistent with the predictions of global warming. We find that the discrete-time system does admit P-tipping, and that the discrete-time P-tipping phenomenon shares characteristics with the continuous-time one: Both require an Allee effect on the resource population, occur in small subsets of the phase plane, and exhibit stochastic resonance as a function of the autocorrelation in the environmental variability. In contrast, the discrete-time P-tipping phenomenon occurs when the environmental conditions switch from low to high productivity, can occur even if the magnitude of the switch is relatively small, and can occur from multiple disjoint regions in the phase plane.

Multi-wing chaotic system based on meminductor and its application in image encryption

Meminductor is a novel type of nonlinear device following the memristor, characterized by its memory properties. Currently, research on meminductors is still in its infancy, with their physical devices yet to be formally realized. Therefore, conducting fundamental research on their nonlinear circuit properties and applications is of great significance. In this paper, a new multi-wing chaotic system is proposed based on the mathematical model of a magnetically controlled meminductor. By varying the values of its parameters, the system can generate two-wing, three-wing, and four-wing chaotic attractors. Various analytical methods are employed to study the dynamical behaviours of the proposed chaotic system. The results demonstrate that the system is highly sensitive to its initial conditions and control parameters, which makes it suitable for image encryption. Based on the new system, we propose a new algorithm for image encryption that combines the newly established four-dimensional multi-wing chaotic system with bit plane decomposition technique, firstly, the high four-bit planes containing 94% image information are disordered by S-type permutation, then the disordered bit planes perform operation of XOR with the random matrix generated by chaotic sequences, and finally, the encrypted image is obtained by merging the bit planes.

Chaos-driven detection of methylene blue in wastewater using fractional calculus and laser systems

Abstract Methylene blue (MB) concentrations in residual water were detected using fractional calculus, the Rössler chaotic attractor and laser systems. A Nd:YVO4 nanosecond pulsed laser at 532 nm, with pulse energies ranging from 2 µJ to 7 µJ, was applied to irradiate different water samples containing MB concentrations from 20 µl to 100 µl. Fractional calculus was employed with the purpose of modeling the temperature distribution in the samples, with the Caputo fractional derivative describing photothermal effects induced by laser irradiation. Different MB concentrations were detected by using the Rössler chaotic attractor, it monitored variation on concentrations, associating attractor shapes with MB concentrations. Lower concentrations showed a weaker attractor response, whereas higher concentrations manifest stronger attractor shapes in magnitude. Raman spectroscopy confirmed the detection of MB in residual water from the Requena dam, located in Tepeji del Río de Ocampo, Hidalgo, Mexico. The application of fractional calculus improved the prediction of heat distribution in the samples, by incorporating numerical simulation. The results suggest that this approach is suitable for real-time monitoring, as it associates MB concentrations with distinct chaotic attractor shapes. This technique shows promise for the detection of other contaminants as well. Future research should focus on refining this method and expanding its application to develop innovative monitoring solutions.

Regulation and analysis of multi-scroll attractors in a class of memristive Hopfield neural networks with high Lyapunov exponents

Multi-scroll chaotic systems have gained significant attention in recent years. However, there is little research on the chaotic systems of memristive neural networks with multiple complex structural attractors. A class of multi-scroll chaotic systems are designed in this paper, which are based on Hopfield neural network by using memristors as synapses. Implementing synaptic control with memristors at various coupling positions allows for the creation of multiple chaotic attractors with distinct topological structures. The proposed systems exhibit special and irregular shaped attractors, and can generate a fixed number of scrolls. Through analysis we find that the systems have high Lyapunov exponents, indicating strong sensitivity and randomness. Meanwhile, the systems exhibit extremely rich dynamic behaviors, such as symmetric coexistence phenomena. Especially, we observe signal amplitude control and offset boosting phenomena in the systems. In addition, simulation circuits are designed based on such chaotic systems to verify the physical feasibility of the systems.

Dynamic Investigations of Shared Bicycle Operators’ Competition Based on Profit Maximization

With the rise of the sharing economy, shared bicycles have become an important component of urban transportation. This paper explores the nonlinear dual oligopoly system for the Cournot model in the bike-sharing market; both operators have maximized profits as their competitive goals. The analysis of pivotal factors influencing passenger preferences, including pricing discounts and comfort levels, is meticulously depicted by a bifurcation diagram. A new chaotic attractor—the shared bicycle attractor—is discovered. The research results indicate that larger discounts and adjustment speeds can cause the system to be in a chaotic state, which is not conducive to the long-term development of operators, although discounts can indeed attract more passengers to a certain extent. On the other hand, the increase in the marginal cost of comfort loss can also make it difficult for enterprises to operate, which requires continuous technological innovation to improve the comfort of cycling.

Halvorsen chaotic system based microwave absorber modelling for fighter jet stealth technologies

This study focuses on the development and detailed analysis of a broadband microwave absorber utilizing Halvorsen chaotic dynamics, that focuses at enhancing stealth capabilities for fighter jets. The investigation begins by exploring the mathematical formulation of the Halvorsen chaotic system and conducting a parametric sweep of its control parameters to generate distinct two-dimensional and three-dimensional chaotic attractor plots. These plots are then post-processed using Julia set theory to develop intricate fractal patterns, which serve as the foundation for the absorber design. Image processing techniques, including filtering and thresholding, are employed to refine the patterns by removing artifacts and noise, ensuring they are suitable for practical implementation. The refined fractal patterns are then imported into a computational electromagnetic simulation environment where they are patterned onto a 0.035 mm thick copper sheet. Moreover, the Magtrex 555 substrate with a thickness of 1.52 mm, is selected for its high permittivity and low-loss characteristics. A comprehensive series of parametric studies are conducted to evaluate the influence of various design parameters such as side length, unit cell geometry, and substrate thickness on the absorber’s electromagnetic performance. Important parameters include the effects of chaotic control parameter optimization and the electromagnetic boundary conditions applied during the simulations. Extensive simulations are performed across the 2–20 GHz frequency range to evaluate absorption efficiency, focusing on key metrics like absorptivity, surface current distribution, and electric field distribution. The final design achieves over 90 % absorption efficiency within the target frequency band when the chaotic control parameters are optimized. Finally, comparative analysis using different commercially available substrates, including FR-4 and Rogers RO3003, reveals that Magtrex 555 offers superior absorption performance. The study concludes with a detailed presentation of the final absorber design, alongside an indepth discussion of the frequency range, parametric variations, and the impact of chaotic system dynamics on the absorption properties. This research provides crucial insights into the design and optimization of chaotic-system-based microwave absorbers, that advances the development of stealth technology in military applications.

RBFNN-PSO Intelligent Synchronisation Method for Sprott B Chaotic Systems with External Noise and Its Application in an Image Encryption System.

In the past two decades, research in the field of chaotic synchronization has attracted extensive attention from scholars, and at the same time, more synchronization methods, such as chaotic master-slave synchronization, projection synchronization, sliding film synchronization, fractional-order synchronization and so on, have been proposed and applied to chaotic secure communication. In this paper, based on radial basis function neural network theory and the particle swarm optimisation algorithm, the RBFNN-PSO synchronisation method is proposed for the Sprott B chaotic system with external noise. The RBFNN controller is constructed, and its parameters are used as the particle swarm particle optimisation parameters, and the optimal values of the controller parameters are obtained by the PSO training method, which overcomes the influence of external noise and achieves the synchronisation of the master-slave system. Then, it is shown by numerical simulation and analysis that the scheme has a good performance against external noise. Because the Sprott B system has multiple chaotic attractors with richer dynamics, the synchronization system based on Sprott B chaos is applied to the image encryption system. In particular, the Zigzag disambiguation method for top corner rotation and RGB channel selection is proposed, and the master-slave chaotic system synchronisation sequences are diffused to the disambiguated data streams, respectively. Therefore, the encryption and decryption of image transmission are implemented and the numerical simulation results are given, the random distribution characteristics of encrypted images are analysed using histogram and Shannon entropy methods, and the final results achieve the expected results.

Analytical and dynamical analysis of nonlinear Riemann wave equation in plasma systems

The Riemann wave equation presents appealing nonlinear equations applicable in sea-water and tsunami wave propagation, ion and magneto-sound waves in plasmas, electromagnetic waves in transmission lines, and homogeneous stationary media. This study focuses on deriving soliton solutions in optics and exploring their physical properties. A wave transformation is used to convert a partial differential equation into an ordinary differential equation, from which soliton solutions are obtained using the generalized Riccati equation mapping approach. The solutions encompass various types of solitons, including bright, dark, periodic, and kink solitons. A comparison of solutions from this analytical method enhances the understanding of the nonlinear model’s behavior, with implications in plasma physics, fluid dynamics, optics, and communication technology. Additionally, 2D and 3D graphs illustrate the physical phenomena of the solutions using appropriate constant parameters. The qualitative analysis of the undisturbed planar system involves examining phase portraits in bifurcation theory, followed by introducing an outward force to induce disruption and reveal chaotic phenomena. Chaotic trajectories in the perturbed system are detected through various plots, including 3D, 2D, power spectrum, and chaotic attractor, alongside Lyapunov exponents. Stability analysis under different initial conditions is conducted, and sensitivity assessments are performed using the Runge–Kutta method. The findings are innovative and have not been previously explored for this system, highlighting the reliability, simplicity, and effectiveness of these techniques in analyzing nonlinear models in mathematical physics and engineering.

Comparative Electroencephalographic Profile of a New Anesthetic and Anticonvulsant That Is Selective for the GABAAR Slow Receptor Subtype

BACKGROUND: Anesthetics like propofol increase electroencephalography (EEG) power in delta frequencies (0.1–4 Hz), with a decrease of power in bandwidths >30 Hz. Propofol is nonselective for gamma amino butyric acid type A receptor subtypes (GABAAR) as it enhances all 3 GABAAR subtypes (slow, fast, and tonic). Our newly developed anesthetic class selectively targets GABAAR-slow synapses to depress brain responsiveness. We hypothesized that a selective GABAAR-slow agonist, KSEB 01-S2, would produce a different EEG signature compared to the broad-spectrum GABAAR agonist (propofol), and tested this using rat EEG recordings. METHODS: Male rats were studied after Institutional Animal Care and Use Committees (IACUC) approval from the US Army Medical Research Institute of Chemical Defense and the University of Michigan. Rats were anesthetized using isoflurane (3%–5% induction, 1%–3% maintenance) with oxygen at 0.5 to 1.0 L/min. Stainless steel screws were placed in the skull and used to record subcranial cortical EEG signals. After recovery, either propofol or KSEB 01-S2 was administered and effects on EEG signals were analyzed. RESULTS: As previously reported, propofol produced increased power in delta frequencies (0.1–4 Hz) compared to predrug recordings and produced a decrease in EEG power >30 Hz but no significant changes were seen within ±20 seconds of losing the righting reflex. By contrast, KSEB 01-S2 produced a significant increase in theta frequency percent power (median 14.7%, 16.2/13.8, 75/25 confidence interval; to 34.7%, 35/31.8; P < .015) and a significant decrease in low gamma frequency percent power (16.9%, 18.6/15.8; to 5.45%, 5.5/5.39; P < .015) for all rats at ± 20 seconds of loss of consciousness (LOC). Both anesthetics produced a flattening of chaotic attractor plots from nonlinear dynamic analyses, like that produced by volatile and dissociative anesthetics at LOC. CONCLUSIONS: KSEB 01-S2 produced a markedly different EEG pattern, with a selective increase observed in the theta frequency range. KSEB 01-S2 also differs markedly in its activity at the GABAAR-slow receptor subtype, suggesting a possible mechanistic link between receptor subtype specificity and EEG frequency band signatures. Increased theta together with depressed gamma frequencies is interesting because GABAAR slow synapses have previously been suggested to underlie theta frequency oscillations, while fast synapses control gamma activity. These reciprocal effects support a previous model for theta and nested gamma oscillations based on inhibitory connections between GABAAR fast and slow interneurons. Although each anesthetic produced a unique EEG response, propofol and KSEB 01-S2 both increased slow wave activity and flattened chaotic attractor plots at the point of LOC.

Symbolic dynamics approach to find periodic windows: The case study of the Rössler system

Modification of a parameter of a chaotic system may lead to the emergence of a periodic attractor. Under certain assumptions periodic windows (regions in the parameter space in which a periodic attractor exists) densely fill a chaotic region. Usually it is very difficult to prove this property. In this work, we propose a systematic procedure to locate and prove the existence of periodic windows. The method combines the symbolic dynamics based approach to find unstable periodic orbits (UPOs), the continuation method to locate periodic windows (PWs), and interval arithmetic tools to prove their existence. The proposed method is applied to the Rössler system. The existence of several thousands of PWs close to the classical parameter values is proved and periodic attractors very close in the parameter space to the classical Rössler attractor are found. Estimates of measures of sets of parameters for which a periodic attractor exists are calculated.

A modified neural network method for computing the Lyapunov exponent spectrum in the nonlinear analysis of dynamical systems

This study presents a modified neural network method for computing the Lyapunov exponent spectrum in non-linear dynamical systems. Its mathematical description is an introduction. The proposed modified neural network method allows the addition of bias and constant neurons to the neural network topology and the use of different numbers of activation functions to suit different cases. Various algorithms for computing Lyapunov exponents, such as the Benettin method, the Wolf method, the Rosenstein method, the Kantz method, the Synchronisation method, the Sano-Sawada algorithm and the proposed modification of the neural network method, are used for classical problems in nonlinear dynamics. These problems include the generalised Hénon map, the chaotic attractor of the Baier-Klein map, and the vibrations of mechanical systems such as the flexible Bernoulli-Euler beam and the flexible functionally graded porous closed cylindrical shell under alternating load. The comparative analyses presented in this study are aimed at validating the accuracy and effectiveness of the methods, and at identifying the most relevant approaches for different types of systems and classes of problems. The proposed method is demonstrated to be superior to existing methods based on time-series evaluation in terms of sample size and accuracy. Furthermore, it does not require the initial system equations.

ON THE CHAOTIC NATURE OF A CAPUTO FRACTIONAL MATHEMATICAL MODEL OF CANCER AND ITS CROSSOVER BEHAVIORS

In this paper, a three-dimensional mathematical model of cancer, incorporating tumor cells, host normal cells, and effector immune cells, is examined. The chaotic nature of this cancer model is explored within the framework of Caputo fractional calculus. The analysis employs local stability assessment using the Matignon criteria, Lyapunov exponents (LEs) for various fractional orders of derivatives, bifurcation plots reflecting changes in the fractional derivative order, simulation outcomes of a novel chaotic attractor, Kaplan–Yorke dimension, and sensitivity to initial conditions. The results demonstrate that the Caputo fractional cancer model exhibits chaotic behavior for selected parameters. Moreover, the fractional derivative orders significantly influence the extent of chaotic behavior. Specifically, as the fractional derivative order increases from zero to one, the intensity of chaos diminishes, evidenced by a decrease in both the LE and the Kaplan–Yorke dimension. A piecewise modeling approach is applied to the cancer model, incorporating deterministic, stochastic, and power-law behaviors. Three distinct scenarios are developed within this framework, revealing several novel crossover behaviors through simulation. Notably, it is observed that when cancer dynamics initially follow a power-law behavior and subsequently transition into a stochastic process, the disease dynamics can stabilize, suggesting a positive outlook for disease control. Conversely, if the cancer dynamics begin with a deterministic process and then shift to a stochastic process, the system may enter a chaotic state, complicating disease management efforts.

Structural Sensitivity of Chaotic Dynamics in Hastings–Powell’s Model

The classical Hastings–Powell model is well known to exhibit chaotic dynamics in a three-species food chain. Chaotic dynamics appear through period-doubling bifurcation of stable coexistence limit cycle around an unstable coexisting equilibrium point. A specific choice of parameter value leads to a situation, where the chaotic attractor disappears through a collision with an unstable limit cycle originated due to subcritical Hopf-bifurcation around the coexistence equilibrium. As a consequence, the top predator goes to extinction. The main objective of this work is to explore the structural sensitivity of this phenomenon by replacing the Holling-type II functional responses with Ivlev functional responses. In this work, we have shown the existence of two Hopf-bifurcation thresholds and numerically verified the existence of an unstable limit cycle. The model with Ivlev functional responses does not indicate any possibility of extinction of the top predator due to any collision of chaotic attractor with the unstable limit cycle for the chosen range of parameter values. Moreover, the model with Ivlev functional responses depicts an interesting scenario of bistable oscillatory coexistence.

Entropic Regression Dynamic Mode Decomposition (ERDMD) Discovers Informative Sparse and Nonuniformly Time Delayed Models

In this work, we present a method that determines optimal multistep Dynamic Mode Decomposition (DMD) models via Entropic Regression (ER), which is a nonlinear information flow detection algorithm. Motivated by the Higher-Order DMD (HODMD) method of [Le Clainche & Vega, 2017], and the ER technique for network detection and model construction found in [Sun et al., 2015; AlMomani et al., 2020], we develop a method that we call ERDMD, which produces high fidelity time-delay DMD models that allow for nonuniformity in the delays. This optimal choice of delays is discovered by maximizing informativity as measured through ER. These models are shown to be highly efficient and robust. We test our method over several data sets generated by chaotic attractors and show that we are able to build excellent reconstructions using relatively minimal models. We likewise are able to better identify multiscale features via our models which enhances the utility of DMD.

Mobile Robot Motion Behavior Based on Hopfield Neural Network Dynamics Under Electromagnetic Radiation

The nonlinear characteristic of activation function is a critical aspect that affects the performance of artificial neural networks (ANNs). Despite its importance, the impact of neuronal nonlinearity variations on the dynamics of ANNs has not been thoroughly studied in previous research. This paper aims to fill this gap by exploring the influence of nonlinearity changes in activation function due to memristive electromagnetic induction on the dynamical characteristics of ANNs. We first propose a memristor model with sinusoidal conductance function. Then, the effects on the biological neuron under the electromagnetic induction are simulated by replacing the gradient of hyperbolic tangent activation function with the conductance of the proposed memristor. Furthermore, a Hopfield neural network (HNN) is constructed by taking the nonlinearity changes of activation function into consideration and its dynamical behaviors are analyzed through Lyapunov exponent spectra, bifurcation diagram, biparameter-based dynamic maps and phase portraits. The numerical simulation results show complex dynamical behaviors such as single-scroll chaotic spiral attractor, double-scroll chaotic attractors and initial-value-dependent offset boosted attractors, demonstrating that nonlinearity changes of activation function induced by the electromagnetic induction have a significant impact on the dynamics of HNN. To further verify our findings, we implement the proposed HNN in hardware and find that the results are in line with our numerical simulations. Finally, we apply the chaotic HNN in the mobile robot path planning task, which achieves higher coverage rate than the random number-based random walk method.