Dynamical Chaos Research Articles

Mixed dynamics: elements of theory and examples

The main goal of the paper is to present recent results obtained in the mathematical theory of dynamical chaos and related to the discovery of its new, third, form, the so-called mixed dynamics. This type of chaos is very different from its two classical forms, conservative and dissipative chaos, and its main difference is that attractors and repellers can intersect without coinciding. The main results of the paper are related to construction of theoretical schemes aimed to mathematical justification of this phenomenon using the most general methods of topological dynamics. The paper also provides a number of examples of systems from applications in which mixed dynamics is observed. It is shown that such dynamics can be of different types, from close to conservative to strongly dissipative, and also that it can arise as a result of various bifurcation mechanisms.

Embryonic Development in Light of Controlled Chaos Dynamics and Quantum Electrodynami

Embryonic Development in Light of Controlled Chaos Dynamics and Quantum Electrodynami

Modeling and Chaos Dynamics Analysis of Doubly-Fed Induction Generator Based on Incommensurate Fractional-Order

Modeling and Chaos Dynamics Analysis of Doubly-Fed Induction Generator Based on Incommensurate Fractional-Order

RESEARCH ON CHAOTIC CHARACTERISTICS AND SHORT-TERM PREDICTION OF EN-ROUTE TRAFFIC FLOW USING ADS-B DATA

The short-term traffic flow prediction can help to reduce flight delays and optimize resource allocation. Using chaos dynamics theory to analyze the chaotic characteristics of en-route traffic flow is the basis of short-term en-route traffic flow prediction and ensuring the orderly and smooth state of the en-route. This paper takes the time series of en-route traffic flow extracted from Automatic-Dependent Surveillance Broadcast (ADS-B) measured data as the research object, uses the improved C–C method to reconstruct the phase space, and uses the improved small data volume method to calculate the Lyapunov index to identify the chaos phenomenon of en-route traffic flow. In order to avoid the interference of chaos phenomenon on traffic prediction, the Wavelet Neural Network (WNN) model is established to predict the traffic flow at en-route points. The experimental shows that when the number of iterations is 10,000, the average accuracy of WNN prediction is 0.87173, and the average running time is 6.9335334[Formula: see text]s. According to the experimental results, it can be seen that the smaller number of iterations has more advantages in running time, which greatly reduces the overall running time. At the same time, it indicates that appropriately increasing or reducing the number of iterations in this experiment has little effect on the results.

Instability in optical injection locking semiconductors lasers using multiparametric bifurcation analysis.

In this paper, we investigate bifurcations of equilibria and transients by using modified rate equations of semiconductor lasers (SCLs) subjected to optical injection. An analytical study is performed to demonstrate some two-parameter bifurcations, inter alia, Bogdanov-Takens and Gavrilov-Guckenheimer bifurcations. A detailed numerical study based on the multiparametric bifurcation method and using 3D-plots and projections reveal a rich locking dynamics of SCLs. In this way, a so-called zero frequency detuning well is highlighted in the vicinity of a Hopf bifurcation confining minimal states of the larger Lyapunov exponent in injection locking curves. Three-parameter bifurcation curves mainly underscore cusp bifurcation and resizing of its multi-equilibrium region by the specific control parameter defined in this model. The bursting phenomenon observed in the transient regime is discussed by using various numerical approaches wherefrom another quantifying method tapping into two-parameter bifurcation analysis is proposed. Thereafter, metastable chaos dynamics supported by spiraling relaxation oscillations is also investigated as well as planar saddle-node bifurcations with three homoclinic orbits for high positive and negative detunings. At last, zero α-factor effects contribute to drastically shrink the unlocking region of SCLs, twofold increase in Hopf bifurcation along with evidencing of complex chaotic sine-shaped and folded torus-shaped attractors.

Spin Chaos of Exciton Polaritons in a Magnetic Field

The spin properties of exciton polaritons in a micropillar cavity placed in a static magnetic field and excited by a resonant light wave are studied theoretically. Owing to the Zeeman effect, a nonlinear polariton system has two branches of optical response that are characterized by opposite circular polarizations. An indirect mechanism of polarization reversal is predicted, according to which the current state of the system undergoes a transition to dynamical chaos, and then the alternative spin state is established spontaneously. Such spin switches, mediated by a chaotic stage, proceed in both directions near the same critical excitation amplitude, so that the sign of the circular polarization of the cavity radiation is directly determined by the intensity of the optical pump.

The emergence of chaos in productivity distribution dynamics

The distribution of productivity levels, and its evolution over time, is a research topic of utmost importance in empirical and theoretical economics. On the theory side, simple analytical models, involving intertemporal optimization, typically characterize agents’ investment decisions about ways to upgrade technology and enhance productivity. The prototypical model endogenously splits the productivity distribution in two: the right-hand side of the distribution is populated by innovators; the left-hand side is occupied by agents who follow a strategy of adoption or imitation. Given the assumptions of the model, the productivity of innovators grows at a constant rate (which directly depends on a constant probability of innovation). The evolution of the productivity of adopters may, in turn, implicate complex dynamics. Because the pace of productivity growth for adopters depends on the shape of the productivity distribution, different distributions might induce distinct growth paths, some of them potentially leading to the emergence of nonlinearities, such as limit cycles and chaos. This study investigates the presence of nonlinearities in technology adoption, for different configurations of the productivity distribution. Under reasonable parameterizations, endogenous fluctuations emerge as a plausible long-term equilibrium.

Spin Chaos of Exciton Polaritons in a Magnetic Field

The spin properties of exciton polaritons in a micropillar cavity placed in a static magnetic field and excited by a resonant light wave are studied theoretically. Owing to the Zeeman effect, a nonlinear polariton system has two branches of optical response that are characterized by opposite circular polarizations. An indirect mechanism of polarization reversal is predicted, according to which the current state of the system undergoes a transition to dynamical chaos, and then the alternative spin state is established spontaneously. Such spin switches, mediated by a chaotic stage, proceed in both directions near the same critical excitation amplitude, so that the sign of the circular polarization of the cavity radiation is directly determined by the intensity of the optical pump.

On polylinear differential realization of the determined dynamic chaos in the class of higher order equations with delay

The investigation has defined the characteristic criterion (and its modification) of solvability of the problem of differential realization of the bundle of controlled trajectory curves of determined chaotic dynamic processes in the class of bilinear non-autonomous ordinary second- and higher-order differential equations (with and without delay) in the separable Hilbert space. The problem statement under consideration belongs to the type of converse problems for the additive combination of nonstationary linear and bilinear operators of the evolution equation in the Hilbert space. The constructions of tensor products of the Hilbert spaces, structures of lattices with an orthocomplement, the theory of extension of M2 -operators and the functional apparatus of the entropy Relay Ritz operator represent the basis of this theory. It has been shown that in the case of the finite bundle of the controlled trajectory curves the existence of the property of sub-linearity of the given operator allows one to obtain sufficient conditions of existence of such realizations. Side by side with solving the main problems, grounded are topological-group conditions of continuity of projectivization of the Relay Ritz operator with computing the fundamental group (Poincare group) of its compact image. The results obtained give incentives for the development of the quantitative theory of converse problems of higher-order multilinear evolution equations with the operators of generalized delay describing, for example, differential modeling of nonlinear Van der Pol oscillators or Lorentz strange attractors.

Can the observable solar activity spectrum be reproduced by a simple dynamo model?

The temporal spectrum of the solar activity is more than just the main cycle. It contains different timescales, which can be considered as continuous components of the activity spectrum. The possibility of finding a realistic spectrum of the solar magnetic activity variation is analysed for several versions of a simple model for solar activity based on the original idea of E. Parker. In particular, we study the original set of partial differential equations with two versions of suppression of the dynamo action and the fourth-order dynamical system obtained by truncating the Parker equations. We show that the effects included in the models, i.e. the nonlinear dynamo suppression and the dynamical chaos, as well as random fluctuations of the dynamo drivers, are quite sufficient to obtain the main solar cycle and the continuous components of the spectrum similar to the observed ones. However, the capabilities of the approach under consideration substantially vary from one model to another. Each model reproduces a continuous component of the spectrum in a specific parameter range. This study has confirmed the view that the examination of various solar dynamo models with the aim to find a reasonable combination of main activity cycle and continuous spectrum of solar activity can be used as an additional test of their validity.

Dynamic Modeling and Chaos Suppression of the Permanent Magnet Synchronous Motor Drive with Sliding Mode Control

The permanent magnet synchronous motor (PMSM) is mathematically modeled and simulated in this study using MATLAB. PMSM is a nonlinear, multi-variable and time-varying system and due to nonlinearities and its strong coupling between its variables, the dynamical behaviour of the PMSM is complex. Therefore, under specified parameters and conditions, chaotic undesirable phenomenon arise in the PMSM. For a chaotic PMSM drive system, this paper proposes a sliding mode control (SMC) approach based on the Lyapunov stability theory to control and suppress the chaotic motion emergence. Firstly, the dynamic characteristics of the state equations of the PMSM drive system is analysed and demonstrated that it will appear chaos phenomenon at some certain parameters. Finally, the SMC strategy and Lyapunov stability theory are combined to introduce a sliding surface and produce a control rule. Simulation results are presented to verify that the proposed strategy can be successfully employed to control a chaotic PMSM and make the system asymptotically stable to the equilibrium point.

Polylinear Differential Realization of Deterministic Dynamic Chaos in the Class of Higher Order Equations with Delay

Polylinear Differential Realization of Deterministic Dynamic Chaos in the Class of Higher Order Equations with Delay

A Novel Eighth-Order Hyperchaotic System and Its Application in Image Encryption

With the advancement in information and communication technologies (ICTs), the widespread dissemination and sharing of digital images has raised concerns regarding privacy and security. Traditional methods of encrypting images often suffer from limitations such as a small key space and vulnerability to brute-force attacks. To address these issues, this paper proposes a novel eighth-order hyperchaotic system. This hyperchaotic system exhibits various dynamic behaviors, including hyperchaos, sub-hyperchaos, and chaos. The encryption scheme based on this system offers a key space larger than 22338. Through a comprehensive analysis involving histogram analysis, key space analysis, correlation analysis, entropy analysis, key sensitivity analysis, differential attack analysis, and cropping attack analysis, it is demonstrated that the proposed system is capable of resisting statistical attacks, brute force attacks, differential attacks, and cropping attacks, thereby providing excellent security performance.

Dynamic chaos of imaging measurements for characterizing gas–liquid nonlinear flow behaviour in a metallurgical reactor stirred by top‐blown air

AbstractA modern chaos detection strategy for imaging measurements is proposed in this work to quantify the thermal mixing state quality in a metallurgical reactor stirred by top‐blown air. Specifically, the improved C‐C algorithm is proposed to reconstruct the phase space of the nonlinear time series of the bubble characteristics under thermal conditions. Moreover, a chaos decision tree algorithm is introduced to extract the chaotic mixing characteristics of the thermal two‐phase mixing system for the first time. Experimental and calculated results show that all the possible mixing states of the mixing system are visualized by reconstructing the phase space with the help of the visualization technique of the thermal gas–liquid two‐phase flow. It is found that the attractor of the nonlinear time series of bubbles exhibited more serious variations while the chosen working condition was optimal. Furthermore, the obtained parameter represents the chaotic characteristics of the thermal gas–liquid two‐phase mixing system which has chaotic characteristics under various experimental conditions. Hence, the new strategy would be helpful and effective in beneficial exploration for understanding the nonlinear intensification mechanism of metallurgical thermal processes.

THE ROLE OF HIGHER MOMENTS ON THE DISTRIBUTION OF PARTICLES IN THE SPACE OF IMPULSES AT CYCLOTRON RESONANCES

The results of the analysis of the dynamics of charged particles under conditions of cyclotron resonances in the field of an intense electromagnetic wave are presented. Particular attention is paid to regimes with dynamic chaos. It is shown that there are two qualitatively different regimes. The appearance of the first one is due to the overlap of nonlinear cyclotron resonances. The second mode is related to intermittency. The moments and spectra of each of these regimes are determined. It is shown that with an increase in the intensity of an external electromagnetic wave, the first regime appears at the beginning and only then the second regime appears. A characteristic feature of the second regime is intermittency. Steps appear on the time dynamics of pulses in the second mode. It is shown that the spectra in the second mode are narrower than in the first mode. A characteristic feature of the second regime (the regime with intermittency) is the fact that the higher moments turn out to be larger than the lower moments. In the first regime, the highest moments decrease rapidly. To find the particle momentum distribution function, the generalized Fokker-Planck equation was used. Solutions of this equation are written out for some important cases.

Electronic-state chaos, intramolecular electronic energy redistribution, and chemical bonding in persisting multidimensional nonadiabatic systems.

We study the chaotic, huge fluctuation of electronic state, resultant intramolecular energy redistribution, and strong chemical bonding surviving the fluctuation with exceedingly long lifetimes of highly excited boron clusters. Those excited states constitute densely quasi-degenerate state manifolds. The huge fluctuation is induced by persisting multidimensional nonadiabatic transitions among the states in the manifold. We clarify the mechanism of their coexistence and its physical significance. In doing so, we concentrate on two theoretical aspects. One is quantum chaos and energy randomization, which are to be directly extracted from the properties of the total electronic wavefunctions. The present dynamical chaos takes place through frequent transitions from adiabatic states to others, thereby making it very rare for the system to find dissociation channels. This phenomenon leads to the concept of what we call intramolecular nonadiabatic electronic-energy redistribution, which is an electronic-state generaliztion of the notion of intramolecular vibrational energy redistribution. The other aspect is about the peculiar chemical bonding. We investigate it with the energy natural orbitals (ENOs) to see what kind of theoretical structures lie behind the huge fluctuation. The ENO energy levels representing the highly excited states under study appear to have four robust layers. We show that the energy layers responsible for chaotic dynamics and those for chemical bonding are widely separated from each other, and only when an event of what we call "inter-layer crossing" happens to burst can the destruction of these robust energy layers occur, resulting in molecular dissociation. This crossing event happens only rarely because of the large energy gaps between the ENO layers. It is shown that the layers of high energy composed of complex-valued ENOs induce the turbulent flow of electrons and electronic-energy in the cluster. In addition, the random and fast time-oscillations of those high energy ENOs serve as a random force on the nuclear dynamics, which can work to prevent a concentration of high nuclear kinetic energy in the dissociation channels.

Kato Chaos in Linear Dynamics

This paper introduces the concept of Kato chaos to linear dynamics and its induced dynamics. This paper investigates some properties of Kato chaos for a continuous linear operator T and its induced operators T¯. The main conclusions are as follows: (1) If a linear operator is accessible, then the collection of vectors whose orbit has a subsequence converging to zero is a residual set. (2) For a continuous linear operator defined on Fréchet space, Kato chaos is equivalent to dense Li–Yorke chaos. (3) Kato chaos is preserved under the iteration of linear operators. (4) A sufficient condition is obtained under which the Kato chaos for linear operator T and its induced operators T¯ are equivalent. (5) A continuous linear operator is sensitive if and only if its inducing operator T¯ is sensitive. It should be noted that this equivalence does not hold for nonlinear dynamics.

Bounded Confidence and Cohesion-Moderated Pressure: A General Model for the Large-Scale Dynamics of Ordered Opinion.

Due to the development of social media, the mechanisms underlying consensus and chaos in opinion dynamics have become open questions and have been extensively researched in disciplines such as sociology, statistical physics, and nonlinear mathematics. In this regard, our paper establishes a general model of opinion evolution based on micro-mechanisms such as bounded confidence, out-group pressure, and in-group cohesion. Several core conclusions are derived through theorems and simulation results in the model: (1) assimilation and high reachability in social networks lead to global consensus; (2) assimilation and low reachability result in local consensus; (3) exclusion and high reachability cause chaos; and (4) a strong "cocoon room effect" can sustain the existence of local consensus. These conclusions collectively form the "ideal synchronization theory", which also includes findings related to convergence rates, consensus bifurcation, and other exploratory conclusions. Additionally, to address questions about consensus and chaos, we develop a series of mathematical and statistical methods, including the "energy decrease method", the "cross-d search method", and the statistical test method for the dynamical models, contributing to a broader understanding of stochastic dynamics.

New traveling solutions, phase portrait and chaotic pattern for the generalized (2+1)-dimensional nonlinear conformable fractional stochastic Schrödinger equations forced by multiplicative Brownian motion

The paper mainly studies the traveling wave solutions, phase portrait and chaotic pattern of the generalized fractional stochastic Schrödinger equations forced by multiplicative Brownian motion. By using the polynomial complete discrimination method, the hyperbolic functions, rational functions and Jacobi elliptic function solutions are obtained. Moreover, the physics properties of the obtained solutions are shown through three-dimensional, two-dimensional and contour graphs. In order to help better analyze dynamic behavior, phase diagrams, chaos behavior and sensitivity are plotted by using Maple software.

Punctuated chaos and indeterminism in self-gravitating many-body systems

Dynamical chaos is a fundamental manifestation of gravity in astrophysical, many-body systems. The spectrum of Lyapunov exponents quantifies the associated exponential response to small perturbations. Analytical derivations of these exponents are critical for understanding the stability and predictability of observed systems. This paper presents a new model for chaos in systems with eccentric and crossing orbits. Here, exponential divergence is not a continuous process but rather the cumulative effect of an ever-increasing linear response driven by discrete events at regular intervals, i.e. punctuated chaos. We show that long-lived systems with punctuated chaos can magnify Planck length perturbations to astronomical scales within their lifetime, rendering them fundamentally indeterministic.