Strange Attractors Research Articles

A Review of Mathematical and Computational Methods in Cancer Dynamics.

Cancers are complex adaptive diseases regulated by the nonlinear feedback systems between genetic instabilities, environmental signals, cellular protein flows, and gene regulatory networks. Understanding the cybernetics of cancer requires the integration of information dynamics across multidimensional spatiotemporal scales, including genetic, transcriptional, metabolic, proteomic, epigenetic, and multi-cellular networks. However, the time-series analysis of these complex networks remains vastly absent in cancer research. With longitudinal screening and time-series analysis of cellular dynamics, universally observed causal patterns pertaining to dynamical systems, may self-organize in the signaling or gene expression state-space of cancer triggering processes. A class of these patterns, strange attractors, may be mathematical biomarkers of cancer progression. The emergence of intracellular chaos and chaotic cell population dynamics remains a new paradigm in systems medicine. As such, chaotic and complex dynamics are discussed as mathematical hallmarks of cancer cell fate dynamics herein. Given the assumption that time-resolved single-cell datasets are made available, a survey of interdisciplinary tools and algorithms from complexity theory, are hereby reviewed to investigate critical phenomena and chaotic dynamics in cancer ecosystems. To conclude, the perspective cultivates an intuition for computational systems oncology in terms of nonlinear dynamics, information theory, inverse problems, and complexity. We highlight the limitations we see in the area of statistical machine learning but the opportunity at combining it with the symbolic computational power offered by the mathematical tools explored.

Non-Linear Models in Economic and Social Research

The article shows differences of presented models from models used in exact science. It analyzes the reasons why for mathematic models of economic and social systems we cannot reach the quantitative correlation of modeling results with indicators of real processes. The article discusses the qualitative identity of non-linear effects in mathematic models and real processes in natural scientific, economic and social systems. Irremovable uncertainty of economic and social system behavior and connected with this uncertainty chaotic component were shown as an integral feature of complicated system evolution. For simple natural scientific systems chaotic behavior is only one of possible forms of the dynamic process in line with determined behavior. For complicated systems a chaotic component is dominating. The author explains the impossibility and inefficiency of detailing and formalizing of economic and social system functioning above a certain level. The impact of initial data quality on results of modeling systems with different level of complexity was shown. The author proposes to use the concept of the strange attractor for substantiating the depth of planning and reporting in economic and social systems. Issues of self-organization and depth of forecasting for complicated system are studied. This information is considered to be a tool for managing economic and social systems. Promising variants of developing mathematic modeling in economic and social systems are analyzed.

Strange nonchaotic dynamics in a discrete FitzHugh-Nagumo neuron model with sigmoidal recovery variable.

We report the appearance of strange nonchaotic attractors in a discrete FitzHugh-Nagumo neuron model with discontinuous resetting. The well-known strange nonchaotic attractors appear in quasiperiodically forced continuous-time dynamical systems as well as in a discrete map with a small intensity of noise. Interestingly, we show that a discrete FitzHugh-Nagumo neuron model with a sigmoidal recovery variable and discontinuous resetting generates strange nonchaotic attractors without external force. These strange nonchaotic attractors occur as intermittency behavior (locally unstable behavior in laminar flow) in the periodic dynamics. We use various characterization techniques to validate the existence of strange nonchaotic attractors in the considered system.

Phantoms and strange attractors in cosmology

We study a cosmological model of gravity coupled to three, self-interacting scalar fields, one of them with negative kinetic term.The theory has cosmological solutions described by three-dimensional quadratic autonomous equations.Remarkably, the dynamical system has strange attractors, which are in fact very similar to the classic Lorenz attractor. The associated chaotic cosmologies exhibit highly fluctuating periods of contraction and expansion, alternating with long, steady periods in a de Sitter-like phase.

Strange Nonchaotic Attractors in Memristor-Based Shimizu–Morioka Oscillator

The existence of strange nonchaotic attractors (SNAs) is examined using a memristor-based Shimizu–Morioka oscillator. In particular, we analyze the birth of SNAs by varying one of the frequency of external quasi-periodic forces. The occurrence and its mechanism are further analyzed through rational approximation theory. Following that, we also perform different nonlinear dynamical characterizations, including Poincaré surface sections, singular continuous spectrum, separation of nearby trajectories, 0–1 test, as well as finite-time Lyapunov exponents for more precise validation. Finally, the experimental evidence for the occurrence of SNA is inspected through a real-time analog circuit in the laboratory.

A New 6D Chaotic Generator: Computer Modelling and Circuit Design

The objective of this study aims at using the Matlab-Simulink environment and the LabVIEW software environment to build computer models of a six-dimensional (6D) chaotic dynamic system. For the fixed system’s parameters, the spectrum of Lyapunov exponents and the Kaplan-York dimension are calculated. The presence of two positive Lyapunov exponents demonstrates the hyperchaotic behavior of the system. The fractional Kaplan-York dimension indicates the fractal structure of strange attractors. An active control method is extended to achieve global chaotic synchronization of two identical novel 6D chaotic systems with unknown system parameters. Based on the results obtained in Matlab-Simulink and LabVIEW models, a chaotic signal generator for the 6D chaotic system is implemented in the MultiSim environment. The experimental results show that the chaotic behavior simulation in the MultiSim environment is similar to those in the Matlab-Simulink and LabVIEW models. The simulation results demonstrate that the Pecora-Carroll method is a simple way of chaotic masking and signal decoding.

A New Robust Method to Investigate Dynamic Instability of FTV for the Double Tripod Industrial Driveshafts in the Principal Parametric Resonance Region

The present work aims to design a robust method to detect and certify the deterministic chaos or ergodic process for the forced torsional vibrations (FTV) of a double tripod industrial driveshaft (DTID) in transition through the principal parametric resonance region (PPRR) which is considered by the researchers in the field as one of the most important resonance regions for the systems having parametric excitations. The DTID’s model for FTV considers the following effects: nonuniformities of inertial characteristics of the DTID’s elements, the harmonic torque excitation induced by the asynchronous electrical motor used for a heavy-duty grain mill, and the harmonic reaction torque generated by different granulation of the substance needed to be milled. Based on these aspects, a model of the FTV for the DTID was designed which was a modified, physically consistent model already used by the authors to investigate the FTV of automotive driveshafts (homokinetic transmission). For the DTID elements, the dynamic instability for nonstationary FTV in the PPRR using time–history analysis (THA) was analyzed—THA represents the phase portraits. Time–history analysis is a detection method for possible chaotic dynamic behavior for the nonstationary FTV (NFTV) in transition through PPRR. If this dynamic behavior was seen, a new robust method LEA–PM was created to certify and confirm the deterministic chaos for the NFTV of DTID. The new method, LEA–PM, is composed of the Lyapunov exponent’s approach (LEA) coupled with the Poincaré Map (PM) applied to the global system of differential equations that describe the FTV of DTID in the PPRR. This new robust method, which embeds LEA and PM, LEA–PM, establishes if the mechanical system has a deterministic chaotic dynamic behavior (strange attractor) or an ergodic dynamic process in this resonant region. LEA represents a new method that includes not only the maximal Lyapunov exponent method (MLEM) but also new mathematical criteria that is “the sum of all Lyapunov exponents has to be negative” which, coupled with MLEM, indicates the presence of deterministic chaos (strange attractors). THA–LEA–PM had been used for the NFTV of DTID computing the phase portraits, the Lyapunov exponents, and representing the Poincaré Maps of the NFTV for the DTID’s elements in transition through PPRR, founding deterministic chaos or ergodic dynamic behavior. Based on the obtained results, numerical simulations revealed the pitting manifestations of the DTID’s elements, typical for the geared systems transmission, mentioned recently in experimental data research for the homokinetic transmissions. Using the new robust method, THA–LEA–PM (time–history analysis coupled with LEA–PM) can be used in future research for chaotic dynamic analysis of DTID’s NFTV transition through superharmonic resonances, subharmonic resonances, combination resonances, and internal resonances. Time–history analysis as a detection method for chaos and LEA–PM as a certifying method for deterministic chaos can be integrated as a design tool for DTID’s FTV control of the homokinetic transmission.

TOWARDS UNDERSTANDING CRITICAL TRANSITIONS IN WARFARE

In recent decades, the systems approach has become the dominant school of thought to the theoretical understanding of warfare. Within systems theory, the insights of complex (adaptive) systems theory have proved particularly important, enabling us to deepen our understanding of warfare by drawing on knowledge that has been applied in many other disciplines. This paper seeks to improve the understanding of warfare and upgrade it with the concepts of attractor and critical transition. The main idea is that a degradation of a crucial subsystem leads to a phase transition (a significant decrease of combat power) and thus to a shift from one to another strange attractor, i.e., to an alternative relatively stable state. The theoretical framework has been supported by a case study of the first phase of the Russian invasion of Ukraine in 2022.

Nonlinear Dynamics Analysis of a Multistage Planetary Gear Transmission System

A nonlinear dynamic model for a multistage planetary gear transmission system, which consists of two-stage planetary gear plus one-stage parallel shaft gear, is proposed. The time-varying meshing stiffness, comprehensive meshing errors and backlash between gear pairs are taken into account in the model, and the connections between the gear stages are characterized by coupling stiffness. The dimensionless vibration differential equations of the system are derived and solved numerically. By means of global bifurcation diagram, largest Lyapunov exponent (LLE), phase diagram and Poincaré map, the stability of the system is studied with the bifurcation parameters variation including excitation frequency and comprehensive meshing errors. The results demonstrate that the system presents strange attractors with rich forms under different parameter combinations. With the increase of the excitation frequency, the meshing state of the system changes, showing a complex motion and indicating the sensitivity of the system to external excitation. Under the variation of the bifurcation parameter of comprehensive meshing error, the complex dynamic behavior of the system is observed, it is found that the increase of comprehensive meshing error has a negative impact on the stability of the system.

Chaotic and Hyperchaotic Dynamics of a Clapp Oscillator

This paper describes recent findings achieved during a numerical investigation of the circuit known as the Clapp oscillator. By considering the generalized bipolar transistor as an active element and after applying the search-for-chaos optimization approach, parameter regions that lead to either chaotic or hyperchaotic dynamics were discovered. For starters, the two-port that represents the transistor was firstly assumed to have a polynomial-forward trans-conductance; then the shape of trans-conductance changes into the piecewise-linear characteristics. Both cases cause vector field symmetry and allow the coexistence of several different attractors. Chaotic and hyperchaotic behavior were deeply analyzed by using standard numerical tools such as Lyapunov exponents, basins of attraction, bifurcation diagrams, and solution sensitivity. The structural stability of strange attractors observed numerically was finally proved via a real practical experiment: a flow-equivalent chaotic oscillator was constructed as the lumped electronic circuit, and desired attractors were captured and provided as oscilloscope screenshots.

Persistent strange attractors in 3D polymatrix replicators

We introduce a one-parameter family of polymatrix replicators defined in a three-dimensional cube and study its bifurcations. For a given interval of parameters, each element of the family can be C2-approximated by a vector field whose flow exhibits suspended horseshoes and persistent strange attractors. The proof relies on the numerically observed Shilnikov homoclinic cycle to the interior equilibrium. We also describe the phenomenological steps responsible for the transition from regular to chaotic dynamics in the family (route to chaos).

Dynamic investigations in a Stackelberg model with differentiated products and bounded rationality

This work focuses on a Stackelberg duopoly game characterized by firms with differentiated products and bounded rationality. Both firms adopt myopic mechanism to update their production decisions in each discrete period. For the purpose of stressing hierarchy between both firms, the leader firm is supposed to estimate marginal profit by only thinking over his own interfering factors. Since global stability analysis of Nash equilibrium point becomes desirable in complex behaviors research, we cover it as well besides local stability. Numerical simulations are carried out to verify some essential appearances of previous theoretical analyses. Such aspects include maximum Lyapunov exponent, strange attractor, basin of attraction and sensitivity on initial conditions. In addition, we calculate fractal dimension of the non-integral dimension shapes in chaotic state. As to the chaos emerged under certain conditions, effective control method has been utilized to force them back towards stability.

Parametrized family of pseudo-arc attractors: Physical measures and prime end rotations.

The main goal of this paper is to study topological and measure‐theoretic properties of an intriguing family of strange planar attractors. Building toward these results, we first show that any generic Lebesgue measure‐preserving map f generates the pseudo‐arc as inverse limit with f as a single bonding map. These maps can be realized as attractors of disc homeomorphisms in such a way that the attractors vary continuously (in Hausdorff distance on the disc) with the change of bonding map as a parameter. Furthermore, for generic Lebesgue measure‐preserving maps f the background Oxtoby–Ulam measures induced by Lebesgue measure for f on the interval are physical on the disc and in addition there is a dense set of maps f defining a unique physical measure. Moreover, the family of physical measures on the attractors varies continuously in the weak* topology; that is, the parametrized family is statistically stable. We also find an arc in the generic Lebesgue measure‐preserving set of maps and construct a family of disk homeomorphisms parametrized by this arc which induces a continuously varying family of pseudo‐arc attractors with prime ends rotation numbers varying continuously in [0,1/2]. It follows that there are uncountably many dynamically non‐equivalent embeddings of the pseudo‐arc in this family of attractors.

Application of The Classical Soliton Theory to Study the Dynamics in PassiveDispersed and Active-Relaxation Media During the Formation of a Twist in A Toroidal Plasma

In this paper we consider the possibility of using analog models to study the behavior of the evolution and dynamics of toroidal vortex flows. The current model based on coupled van der Pol generators is used as the main one. In this robot, based on nonlinear dynamics display Bernoulli presented the evolution of the Torus. The use of analog models makes it possible to trace the dynamics of vortex flows. In General, the presented results allow us to confidently assert that both in the experiment and in the theoretical consideration we are dealing with the same object - a strange attractor of the Smale – Williams type in a non-Autonomous oscillatory system. The available data suggest that it is an attractor of hyperbolic type, although, strictly speaking, this statement needs mathematical proof. The appearance of an example of a physical system with a hyperbolic chaotic attractor is of fundamental importance for the further development of nonlinear dynamics and its applications. This is, in a sense, a "breakthrough into the hyperbolic realm." Based on the inherent property of coarseness of hyperbolic attractors, we can build other examples of systems with hyperbolic chaos. The presence of such physical systems opens up opportunities for the application of a deeply studied branch of mathematics – hyperbolic theory, and also translates into practice the problem of comparative study of hyperbolic and non-hyperbolic chaos in theory and experiment. This system can be used to study toroidal vortex processes.

Langmuir Probe Perturbations during In Situ Monitoring of Pulsed Laser Deposition Plasmas.

The recent advancements in pulsed laser deposition (PLD) control via plasma diagnostics techniques have been positive and raised questions on the limitation of some techniques, such as the Langmuir probe (LP). The particularities of laser-produced plasma can lead to incorrect interpretation of collected electrical signal. In this paper, we explored the limitations of LP as a technique for in situ PLD control by performing investigations on several metallic plasmas, expanding in various Ar atmosphere conditions. Sub-microsecond modulation was seen in the reconstructed IV characteristics attributed to non-equilibrium dynamics of the ejected charges. A perturbative regime was recorded for Ar pressures higher than 2 Pa, where ionic bursts were observed in the electron saturation region. This perturbation was identified as a plasma fireball. A non-linear multifractal model was developed here to explore these new regimes of the LP. The strange attractors characterizing each fireball were reconstructed, and their evolution with the Ar pressure is discussed. Both short- and long-time non-linear behavior were correlated via probe bias, and the pressure effect on the strange attractor’s defining the fireball-like behavior was investigated. A good correlation was noticed between the simulated data and experimental findings.

Interesting and complex behaviour of Duffing equations within the frame of Caputo fractional operator

The coupled system exemplifying the damped and driven oscillators (namely, Duffing equations) is examined with a familiar and robust numerical method. In the framework, we hired a reliable and most cited Caputo fractional operator to capture essential and stimulating behaviours of the hired physical model. The existence of the solution for the considered model is presented, and we captured the nature of the strange attractor for the Duffing equations with a period of the driving force. The effect of chaotic nature associated parameter is illustrated. The essence of generalizing the classical models is presented with plots, and associated consequences are demonstrated for the substantial time period.

Strange attractors in a dynamical system inspired by a seasonally forced SIR model

We analyze a multiparameter periodically-forced dynamical system inspired in the SIR endemic model. We show that the condition on the basic reproduction numberR0<1 is not sufficient to guarantee the elimination of Infectious individuals due to a backward bifurcation. Using the theory of rank-one attractors, for an open subset in the space of parameters where R0<1, the flow exhibits persistent strange attractors. These sets are not confined to a tubular neighborhood in the phase space, shadow the ghost of a two-dimensional invariant torus and are numerically observable. Although numerical experiments have already suggested that periodically-forced biological models may exhibit observable chaos, a rigorous proof was not given before. Our results agree well with the empirical belief that intense seasonality induces chaos.This work provides a preliminary investigation of the interplay between seasonality, deterministic dynamics and the prevalence of strange attractors in a nonlinear forced system inspired by biology.

New fractional modelling and control analysis of the circumscribed self-excited spherical strange attractor

The purpose of this study is to present and examine a novel non-integer model of the circumscribed self-excited spherical strange attractor, which has not been worked yet. To design the fractional-order model, we use the Caputo-Fabrizio derivative. In order to ensure the existence of the solution Picard-Lindel and fixed-point theories are provided. Moreover, the stability of the considered fractional-order model is shown using the Picard iteration and fixed point theory approach. To get the approximate solutions of the proposed fractional model an efficient numerical scheme called the fractional Euler method(FEM) is used. To see the performance of the used method, the behavior of the numerical solutions of the model is examined under various initial conditions(ICs) and fractional orders. Considerable chaotic behaviors of the solutions are obtained which prove the accuracy and reliability of FEM.

Modeling and Dynamic Analysis for a Motion of Mounted-Based Axisymmetric Rigid Body under Self-Excited Vibrations in an Attractive Newtonian Field

In this article, the mathematical modeling and dynamic analysis for a motion of a rigid body mounted on a vibrating base affected by a strongly nonlinear damping and an attractive Newtonian field are investigated. The governing equations are derived, and specific conditions for asymptotic stability of the motion are stated. Results of analytical analysis are used to identify some distinct types of motion, including the cases that are apparently regular or chaotic. A reduction technique via averaging is used to obtain the amplitude equation and the response diagrams to identify regions where the jump phenomenon may occur. Moreover, a simple analytical relationship between the parameters of the system, describing whether the jump phenomenon will be possible, is obtained. Homoclinic bifurcation diagrams are used to argue apparently that the chaotic behavior is slowly approaching a strange attractor at specific values of system parameters. Results of numerical solutions using the fourth-order Runge-Kutta method are closely coincided with the analytical ones to verify all types of motion.

TUNING OF TWO-DEGREE-OF-FREEDOM IMC BASED ON A STRANGE NONCHAOTIC OPTIMIZATION APPROACH FOR LARGE TIME-DELAY PROCESSES

The internal model control (IMC) has a good control effect for large time-delay systems, but the parameter tuning of the controller is a challenge for researchers. As a typical fractal (strange) attractor, a strange nonchaotic attractor has the dynamic characteristics of randomness and ergodicity. Based on these characteristics, an optimization method is proposed to tune the parameters of an internal model controller for a large time-delay process, which is named a strange nonchaotic optimization algorithm (SNOA). SNOA has the advantages of easy implementation, short execution time and having robust mechanisms to avoid local optimization. It can find the optimal parameters of the internal model controller in large time-delay systems. The results show that SNOA can effectively improve the dynamic stability of the large time-delay systems. It has also proved to be superior to chaotic optimization (CO) algorithm and particle swarm optimization (PSO) algorithm.