Chaotic Dynamics Research Articles

TESTING OF PSEUDORANDOM SIGNAL GENERATORS BASED ON THE LORENTZ SYSTEM

This work presents results of the testing of modified pseudorandom signal generators. Two generators developed on the basis of the Lorentz system with chaotic dynamics are considered. The first generator is described by the Lorentz system implemented over a Galois field, and the second is described by the Lorentz system in which digital sequence formation occurs by periodic sampling from signal signatures. As an additional confirmation of the relevance of the results obtained, the article valuates a generator developed on the basis of the standard Matlab function for binary random sequence generating. Generator testing was carried out using a test battery developed by the National Institute of Standards and Technology, which includes 15 different tests. The results obtained are compared with the results obtained by testing reference generators. The results of this research can be applied in the development of cryptographic and data transmission systems.

Analysis of New 4D Hyperchaotic Systems MACM and Implementation Using Runge-kutta

In general, hyperchaotic system is defined as a chaotic system with more than one positive Lyapunov exponent, this implies that its chaotic dynamics extend in several different directions simultaneously. The discovery of reliable and effective methods for solving hyperchaotic systems is carried out extensively. The aim of this research is to implement nonlinear hyperchaotic using runge-kutta. The NHS system is built by inspecting, modifying and adding one state to the MACM chaotic system. In this study, we use the same initial conditions x_0=y_0=z_0=w_0=1 and the parameters a = 2,b = 2,c = 0.5, and d = 14.5. At first we experiments modified approaches, namely Classic Runge-Kutta Method 3 Order, Runge-Kutta Method Based On Arithmetic Mean, Metode Runge-Kutta Method Based On Geometric Mean, Modified Runge-Kutta Geometric Mean Based Method-1, RKLCM Approaches Instead of RKGM using MAPLE software as a calculation tool. In this work, we observe that the Runge-Kutta Method Based on Arithmetic Mean has the smallest error value with an average variable x of 0.000606953, variable y of 0.009280181, variable z of 0.000378639 and variable w of 0.60781585. Then also in RKLCM Approaches Instead of RKGM it has the largest error value with an average variable x of 0.483789793, variable y of 0.620803849, variable z of 0.92097139, and variable w of 0.596137549.

On the Effects of Type II Left Censoring in Stable and Chaotic Compartmental Models for Infectious Diseases: Do Small Sample Estimates Survive Censoring?

In this paper, we discuss a selection of tools from dynamical systems and order statistics, which are most often utilized separately, and combine them into an algorithm to estimate the parameters of mathematical models for infectious diseases in the case of small sample sizes and left censoring, which is relevant in the case of rapidly evolving infectious diseases and remote populations. The proposed method relies on the analogy between survival functions and the dynamics of the susceptible compartment in SIR-type models, which are both monotone decreasing in time and are both determined by a dual variable: the hazard function in survival prediction and the number of infected people in SIR-type models. We illustrate the methodology in the case of a continuous model in the presence of noisy measurements with different distributions (Normal, Poisson, Negative Binomial) and in a discrete model, reminiscent of the Ricker map, which admits chaotic dynamics. This estimation procedure shows stable results in experiments based on a popular benchmark dataset for SIR-type models and small samples. This manuscript illustrates how classical theoretical statistical methods and dynamical systems can be merged in interesting ways to study problems ranging from more fundamental small sample situations to more complex infectious disease and survival models, with the potential that this tools can be applied in the presence of a large number of covariates and different types of censored data.

Elastoviscoplasticity intensifies the unstable flows through a micro-contraction geometry

This study focuses on the two-dimensional numerical investigation of complex fluid flows through a micro-contraction geometry in the creeping flow regime, specifically examining elastoviscoplastic (EVP) fluids. These fluids exhibit a combination of viscous, elastic, and plastic behaviors. The governing equations, including mass and momentum, are solved using a finite volume method-based discretization technique. Saramito’s constitutive model is utilized to accurately represent the viscous, elastic, and plastic responses of the EVP fluid. The present results demonstrate significant differences in flow dynamics, such as vortex dynamics and transitions between flow regimes (e.g., steady to unsteady), when compared to simple Newtonian and non-Newtonian viscoelastic (VE) or viscoplastic (VP) fluids. This study reveals that when the yield strain (ϵy) exceeds a critical value, approximately ranging from 0.79 to 0.89, the flow transits from a steady to an unsteady state for the EVP fluids. Importantly, the present study shows that EVP fluids exhibit intensified chaotic flow dynamics and increased instability compared to VE and VP fluids under similar flow conditions. However, the presence of shear-thinning behavior in EVP fluids suppresses this instability. The analysis of local velocity fields and flow deformation in this study highlights the impact on the stretching of fluid microstructure and elastic stresses, which ultimately contribute to the origin of this intensified unstable flow condition for EVP fluids. The finding from this study holds significant potential for enhancing heat or mass transfer rates and mixing efficiency in micro-scale systems, where the prevailing steady and laminar flow conditions often hinder these transport processes.

The impact of radio-chemotherapy on tumour cells interaction with optimal control and sensitivity analysis

Oncologists and applied mathematicians are interested in understanding the dynamics of cancer-immune interactions, mainly due to the unpredictable nature of tumour cell proliferation. In this regard, mathematical modelling offers a promising approach to comprehend this potentially harmful aspect of cancer biology. This paper presents a novel dynamical model that incorporates the interactions between tumour cells, healthy tissue cells, and immune-stimulated cells when subjected to simultaneous chemotherapy and radiotherapy for treatment. We analysed the equilibria and investigated their local stability behaviour. We also study transcritical, saddle–node, and Hopf bifurcations analytically and numerically. We derive the stability and direction conditions for periodic solutions. We identify conditions that lead to chaotic dynamics and rigorously demonstrate the existence of chaos. Furthermore, we formulated an optimal control problem that describes the dynamics of tumour-immune interactions, considering treatments such as radiotherapy and chemotherapy as control parameters. Our goal is to utilize optimal control theory to reduce the cost of radiotherapy and chemotherapy, minimize the harmful effects of medications on the body, and mitigate the burden of cancer cells by maintaining a sufficient population of healthy cells. Cost-effectiveness analysis is employed to identify the most economical strategy for reducing the disease burden. Additionally, we conduct a Latin hypercube sampling-based uncertainty analysis to observe the impact of parameter uncertainties on tumour growth, followed by a sensitivity analysis. Numerical simulations are presented to elucidate how dynamic behaviour of model is influenced by changes in system parameters. The numerical results validate the analytical findings and illustrate that a multi-therapeutic treatment plan can effectively reduce tumour burden within a given time frame of therapeutic intervention.

Stability, bifurcation, and control: Modeling interaction of the predator-prey system with Alles effect

In this study, we investigate the resilience and bifurcation properties of a discrete-time predator-prey model incorporating an Allee effect, the transfer of prey to predator energy, and a necessary threshold for prey consumption prior to predator reproduction. The discrete-time formulation involves iteratively adjusting the predator and prey populations at distinct time steps. The model encompasses equations that depict the fluctuations in population sizes, considering processes like breeding, predation, death, and the influence of the Allee effects. The investigation results indicate that the system's boundary equilibrium goes through a period-doubling bifurcation. Moreover, when the system is in close proximity to the exclusive positive equilibrium influenced by the prey-predator population, it exhibits Neimark-Sacker bifurcations. It's worth noting that our observations suggest that the predator population might approach extinction or achieve a stable equilibrium when chaotic dynamics appear in the prey population. The validity of this phenomenon is confirmed using the Lyapunov exponents calculation, which verifies the range of diverse dynamic behaviors. The outcomes and analysis of this research offer intriguing insights in both mathematical and biological realms. Implementing methods for chaos control could be beneficial in stabilizing or mitigating the occurrence of chaotic dynamics. Additionally, the theoretical results are supported by numerical simulations, phase portraits, bifurcation diagrams, graphs depicting chaos control, and maximum Lyapunov exponent graphs.

Chaotic dynamics in X-ray free-electron lasers with an optical undulator.

In this work, the chaotic motions of relativistic electrons in X-ray free-electron lasers are investigated using an optical undulator in the presence of a magnetized ion-channel background. To miniaturize X-ray light sources, the optical undulator is a promising concept. The optical undulator provides higher optical gain than conventional magnetostatic undulators due to its micrometer wavelength. In addition, it reduces the required electron beam energy from several GeV to the multi-MeV range to produce X-ray pulses. The interaction of an optical undulator with an intense relativistic electron beam is a highly non-linear phenomenon that can lead to chaotic dynamics. At synchrotron radiation sources, the possibility of chaos control for X-ray FELs can be critical for certain classes of experimental studies. The equations of motion for a relativistic electron propagating through the optical undulator in the presence of a magnetized ion-channel can be derived from the Hamiltonian of the interaction region. Simulation results revealed that the intensity of the perturbation route from orderly behavior to chaos depends on the beam density, axial magnetic field strength, ion-channel density parameter, and pump laser undulator. Specific values of parameters were obtained for the transition from regular to chaotic paths. Bifurcation diagrams of the system were plotted to demonstrate the origin of chaos at a critical point, and Poincaré maps were created to distinguish between chaotic and orderly motions of electrons. The proposed new scheme can help to improve X-ray FELs, which have potential usages in basic sciences, medicine, and industry.

A nonautonomous model for the effect of rarity value on the dynamics of a predator–prey system with variable harvesting

The human inclination to value rarity contributes to the exploitation of many wildlife species by driving up their market value when their numbers are scarce. This research investigates a model to explore how the value attached to rarity impacts the harvesting of predator species and subsequently influences the dynamics of predator–prey interactions in an ecological system. To broaden the model's scope, it's extended to its nonautonomous form by allowing the harvesting rate of predator species and the degree of neutrality in consumers to respond for the rarity of predator species to change over time. Both the autonomous and nonautonomous systems are explored through analytical and numerical investigation. Numerical findings reveal that as the prey population's growth rate gradually increases, the unforced system shifts from stable boundary equilibrium (harvesting absent) to stable interior equilibrium, then to oscillatory interior equilibrium, followed by period doubling oscillations leading to chaotic behavior. The autonomous system exhibits limit cycle oscillations around the boundary equilibrium when predator species are minimally harvested. Moderate harvesting stabilizes the system, while excessive harvesting induces instability. However, the chaotic oscillations can be eliminated through period‐halving route upon decreasing the consumers' response to predator rarity. Furthermore, the seasonally forced model demonstrates periodic solutions, higher order periodic solutions, and chaotic dynamics contingent upon different settings of prey growth rate, the strength of seasonality in predator harvesting, and the consumers' response to predator rarity.

Predicting nonsmooth chaotic dynamics by reservoir computing.

Reservoir computing (RC) has been widely applied to predict the chaotic dynamics in many systems. Yet much broader areas related to nonsmooth dynamics have seldom been touched by the RC community which have great theoretical and practical importance. The generalization of RC to this kind of system is reported in this paper. The numerical work shows that the conventional RC with a hyperbolic tangent activation function is not able to predict the dynamics of nonsmooth systems very well, especially when reconstructing attractors (long-term prediction). A nonsmooth activation function with a piecewise nature is proposed. A kind of physics-informed RC scheme is established based on this activation function. The feasibility of this scheme has been proven by its successful application to the predictions of the short- and long-term (reconstructing chaotic attractor) dynamics of four nonsmooth systems with different complexity, including the tent map, piecewise linear map with a gap, both noninvertible and discontinuous compound circle maps, and Lozi map. The results show that RC with the new activation function is efficient and easy to run. It can make perfectly both short- and long-term predictions. The precision of reconstructing attractors depends on their complexity. This work reveals that, to make efficient predictions, the activation function of an RC approach should match the smooth or nonsmooth nature of the dynamical systems.

Chaotic Dynamics in a Class of Delay Controlled Partial Difference Equations

Chaotic Dynamics in a Class of Delay Controlled Partial Difference Equations

Time-delay signature suppression in delayed-feedback semiconductor lasers as a paradigm for feedback control in complex physiological networks.

Physiological networks, as observed in the human organism, involve multi-component systems with feedback loops that contribute to self-regulation. Physiological phenomena accompanied by time-delay effects may lead to oscillatory and even chaotic dynamics in their behaviors. Analogous dynamics are found in semiconductor lasers subjected to delayed optical feedback, where the dynamics typically include a time-delay signature. In many applications of semiconductor lasers, the suppression of the time-delay signature is essential, and hence several approaches have been adopted for that purpose. In this paper, experimental results are presented wherein photonic filters utilized in order to suppress time-delay signatures in semiconductor lasers subjected to delayed optical feedback effects. Two types of semiconductor lasers are used: discrete-mode semiconductor lasers and vertical-cavity surface-emitting lasers (VCSELs). It is shown that with the use of photonic filters, a complete suppression of the time-delay signature may be affected in discrete-mode semiconductor lasers, but a remnant of the signature persists in VCSELs. These results contribute to the broader understanding of time-delay effects in complex systems. The exploration of photonic filters as a means to suppress time-delay signatures opens avenues for potential applications in diverse fields, extending the interdisciplinary nature of this study.

Preliminary evidence for chaotic signatures in host-microbe interactions.

Host-microbe interactions constitute dynamical systems that can be represented by mathematical formulations that determine their dynamic nature and are categorized as deterministic, stochastic, or chaotic. Knowing the type of dynamical interaction is essential for understanding the system under study. Very little experimental work has been done to determine the dynamical characteristics of host-microbe interactions, and its study poses significant challenges. The most straightforward experimental outcome involves an observation of time to death upon infection. However, in measuring this outcome, the internal parameters and the dynamics of each particular host-microbe interaction in a population of interactions are hidden from the experimentalist. To investigate whether a time-to-death (time-to-event) data set provides adequate information for searching for chaotic signatures, we first determined our ability to detect chaos in simulated data sets of time-to-event measurements and successfully distinguished the time-to-event distribution of a chaotic process from a comparable stochastic one. To do so, we introduced an inversion measure to test for a chaotic signature in time-to-event distributions. Next, we searched for chaos in the time-to-death of Caenorhabditis elegans and Drosophila melanogaster infected with Pseudomonas aeruginosa or Pseudomonas entomophila, respectively. We found suggestions of chaotic signatures in both systems but caution that our results are preliminary and highlight the need for more fine-grained and larger data sets in determining dynamical characteristics. If validated, chaos in host-microbe interactions would have important implications for the occurrence and outcome of infectious diseases, the reproducibility of experiments in the field of microbial pathogenesis, and the prediction of microbial threats.IMPORTANCEIs microbial pathogenesis a predictable scientific field? At a time when we are dealing with coronavirus disease 2019, there is intense interest in knowing about the epidemic potential of other microbial threats and new emerging infectious diseases. To know whether microbial pathogenesis will ever be a predictable scientific field requires knowing whether a host-microbe interaction follows deterministic, stochastic, or chaotic dynamics. If randomness and chaos are absent from virulence, there is hope for prediction in the future regarding the outcome of microbe-host interactions. Chaotic systems are inherently unpredictable, although it is possible to generate short-term probabilistic models, as is done in applications of stochastic processes and machine learning to weather forecasting. Information on the dynamics of a system is also essential for understanding the reproducibility of experiments, a topic of great concern in the biological sciences. Our study finds preliminary evidence for chaotic dynamics in infectious diseases.

Joint Analysis and Morphological Characterization of HFSWR Echo Properties during Severe Typhoon Muifa

Investigating the dynamic evolution process of the ocean and ionosphere in sudden sea conditions poses a challenging problem. To address this objective, this study utilizes actual data from high-frequency surface wave radar (HFSWR) to analyze, validate, summarize, and characterize the echo properties of the ocean and ionosphere during the severe Typhoon Muifa. By employing the short-time Fourier transform (STFT) method, the HFSWR ocean and ionosphere echoes stimulated by typhoon-induced gravity waves are observed, and the joint gravity wave features of the ocean and ionosphere echoes at different time scales are extracted. Additionally, the phase-space reconstruction method is employed to characterize the dynamical evolution of the joint gravity wave features in higher-dimensional space. Furthermore, the chaotic dynamics behavior of the joint gravity wave features is analyzed using the largest Lyapunov exponents. By combining the gravity wave features with chaotic dynamics, this study introduces a method to characterize the joint gravity wave features. The extraction of joint gravity wave features in HFSWR echoes stimulated by typhoons, along with the construction of a chaotic characterization scheme for the gravity wave features, provides an innovative approach and a solid technical foundation for studying the ocean and ionosphere using HFSWR under sudden sea conditions.

Lytic and temperate phage naturally coexist in a dynamic population model.

When phage infect their bacterial hosts, they may either lyse the cell and generate a burst of new phage, or lysogenize the bacterium, incorporating the phage genome into it. Phage lysis/lysogeny strategies are assumed to be highly optimized, with the optimal tradeoff depending on environmental conditions. However, in nature, phage of radically different lysis/lysogeny strategies coexist in the same environment, preying on the same bacteria. How can phage preying on the same bacteria coexist if one is more optimal than the other? Here, we address this conundrum within a modeling framework, simulating the population dynamics of communities of phage and their lysogens. We find that coexistence between phage of different lysis/lysogeny strategies is a natural outcome of chaotic population dynamics that arise within sufficiently diverse communities, which ensure no phage is able to absolutely dominate its competitors. Our results further suggest a bet-hedging mechanism at the level of the phage pan-genome, wherein obligate lytic (virulent) strains typically outcompete temperate strains, but also more readily fluctuate to extinction within a local community.

Investigation of the hyperchaos and control in the fractional order financial system with profit margin

This research study proposes a novel hyperchaotic finance system with profit margin and then utilizes the Adomian Decomposition Method (ADM) to tackle the solution of the associated Caputo-derivative based fractional-order hyperchaotic finance system with profit margin. Numerical simulations and analyses, such as the evaluation of Lyapunov exponents (LE), the creation of bifurcation diagrams (BD), the complexity analysis (CA), and the 0-1 test, are used to get a full picture of the system. The results of our study demonstrate the occurrence of periodic, chaotic, and hyperchaotic dynamics in the system. Furthermore, the overall criteria for the existence and the uniqueness of the exact solutions for Caputo fractional-order models are presented. In addition, we present a control methodology for the fractional-order hyperchaotic financial system utilizing the Laplace transform and linear feedback control. Significantly, our research showcases a noteworthy correlation between the analytical results and numerical simulations, emphasizing the soundness and effectiveness of our methodology.

Real-time neural identification using a recurrent wavelet first-order neural network of a chaotic system implemented in an FPAA

Recurrent neural networks use sequential or time series data. These algorithms can be used in temporal or ordinal tasks, such as speech recognition, image captioning, and forecasting, among many others. This study introduces a novel approach to the real-time identification of chaotic trajectories using a Recurrent Wavelet First-Order Artificial Neural Network (RWFONN) in conjunction with a Field-Programmable Analog Array (FPAA). The FPAA physically implements the chaotic dynamics of a third-order Jerky equation system with reconfigurable analog electronic components through its voltage output. The trajectory acquisition of the voltage signals is carried out by a dSPACE 1104 board connected to MATLAB & Simulink and Control Desk, which serves as real-time identification data for the RWFONN. The first-order artificial network implements a Morlet wavelet activation function for the online training. It has the particularity to work without the need for multiple epoch training, validation, or dividing the data into batches. Instead, it validates the process by the filtered and mean square errors, affirming the RWFONN’s precision in identifying and reproducing the chaotic behavior of the FPAA-modeled system. This breakthrough in real-time identification holds promise for applications demanding dynamic and unpredictable behaviors, such as random number generators, neural controllers, and cryptographic systems.

Infinite-memory classical wave-particle entities, attractor-driven active particles, and the diffusionless Lorenz equations.

A classical wave-particle entity (WPE) can materialize as a millimeter-sized droplet walking horizontally on the free surface of a vertically vibrating liquid bath. This WPE comprises a particle (droplet) that shapes its environment by locally exciting decaying standing waves, which, in turn, guides the particle motion. At high amplitude of bath vibrations, the particle-generated waves decay very slowly in time and the particle motion is influenced by the history of waves along its trajectory. In this high-memory regime, WPEs exhibit hydrodynamic quantum analogs where quantum-like statistics arise from underlying chaotic dynamics. Exploration of WPE dynamics in the very high-memory regime requires solving an integrodifferential equation of motion. By using an idealized one-dimensional WPE model where the particle generates sinusoidal waves, we show that in the limit of infinite memory, the system dynamics reduce to a 3D nonlinear system of ordinary differential equations (ODEs) known as the diffusionless Lorenz equations (DLEs). We use our algebraically simple ODE system to explore in detail, theoretically and numerically, the rich set of periodic and chaotic dynamical behaviors exhibited by the WPE in the parameter space. Specifically, we link the geometry and dynamics in the phase-space of the DLE system to the dynamical and statistical features of WPE motion, paving a way to understand hydrodynamic quantum analogs using phase-space attractors. Our system also provides an alternate interpretation of an attractor-driven particle, i.e., an active particle driven by internal state-space variables of the DLE system. Hence, our results might also provide new insights into modeling active particle locomotion.

Nonlinear characteristics of variations of Earth’s rotation rate

To study the nonlinear characteristics of changes in the Earth's rotation rate, a comprehensive analysis of the nonlinear characteristics of the length of day (ΔLOD) observations reflecting changes in the Earth’s rotation rate is conducted from multiple perspectives, including periodicity, chaos, and fractal, by using the complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN), recursive quantitative analysis (RAQ), and Grassberger-Procaccia (GP) algorithms. The long-term high-accuracy ΔLOD observations from January 1, 1962 to December 31, 2023, were comprehensively and reliably analyzed and then used as dataset published by the International Earth Rotation and Reference Systems Service, IERS) 14C04 series. The present workfocuses on comparing and analyzing whether there are any significant differences in the ΔLOD characteristics before and after deducting the periodic or chaotic components of ΔLOD time series. The main conclusions obtained are as follows. 1) The ΔLOD time series consists of the well-known trend components, many periodic components, and chaotic components, and therefore can be characterized by obvious multi timescales, chaotic dynamics, and fractal structure. The characteristics were not considered in previous research. 2) The period of the ΔLOD time series after deducting the chaotic components is exactly the same as the period of the original ΔLOD time series, implying that the chaotic components have no effects on reconstruction nor analysis of the periodic components. 3) There is no significant difference in chaotic characteristics between the original ΔLOD time series and its time series after deducting trend and periodic components, but the complexity of the fractal structure of the former is relatively stronger. Not only can this work provide a valuable reference for studying the mechanism of changes in the Earth’s rotation rate, but also model such rotation changes and then predict the chances on different timescales.

Regular and chaotic dynamics in a 2D discontinuous financial market model with heterogeneous traders

We develop a financial market model where three types of traders operate simultaneously: fundamentalists and chartists of two types, namely, trend followers and contrarians. The dynamics of this model is described by a two-dimensional discontinuous map defined by two linear functions, where one acts in the partition between two (parallel) discontinuity lines and the other one acts outside this partition. Our analysis shows that despite the linearity of the map components, its dynamics can be quite complex, with various, possibly coexisting attracting cycles and chaotic attractors. As a first step towards the understanding how the overall bifurcation structure observed in the parameter space of the map is organized, we obtain analytically the boundaries of periodicity regions related to the simplest attracting period-n cycles, n≥3, with one point in the middle partition and n−1 points outside it. These boundaries can be related to border-collision bifurcations (when a point of the cycle collides with a discontinuity line) as well as to degenerate bifurcations (associated with eigenvalues on the unit circle). We show also some elements of period-adding and period-incrementing bifurcation structures for which the cycles mentioned above are basic. From an economic point of view, our study confirms that a fairly simple financial market model with heterogeneous agents is able to produce complicated boom-bust dynamics typical of real financial markets.

A novel delayed exponent coupled chaotic map with countering dynamical degradation

<abstract> <p>While chaotic systems have found extensive applications across diverse scientific domains due to their inherent advantages, they often degrade into cyclic patterns when simulated on hardware with limited computational precision. This results in a pronounced decline in properties related to chaotic dynamics. To address this issue, we introduce the delayed exponent coupled chaotic map (DECCM). This model is designed to enhance the chaotic dynamics of the original map, especially at lower computational precisions. Additionally, DECCM can transform any proficient 1-dimensional seed map into an <italic>N</italic>-dimensional chaotic map. Extensive simulation and performance tests attest to the robust chaotic characteristics of our approach. Furthermore, DECCM holds distinct advantages over premier algorithms, particularly in period analysis experiments. We also introduce various seed maps into DECCM to present 2D and 3D examples, ensuring their generalization through relevant performance evaluations.</p> </abstract>