Chaotic Trajectories Research Articles

Nonlinear Forced Vibrations of Plates with Oscillating Inclusions

These studies focus on the forced vibrations of rectangular plates incorporating oscillating inclusions uniformly distributed in the carrier elastic medium. Plate’s equations of motion consist of the Lamé equation for a carrier material and the equation for oscillators representing the inclusions. The oscillators are acted upon by the nonlinear friction force, described by Coulomb’s modified law, and a cubically nonlinear elastic force. The simply supported forced plate is considered. Using the Galerkin approximation, the problem of plate vibrations is reduced to the study of a system of ODEs. According to numerical and qualitative analyses, the application of harmonic forcing causes the appearance of periodic, quasiperiodic, and chaotic regimes in the system. In particular, the analytical expression is derived for the periodic regimes of the forcing frequency. The scenarios of vibration’s development is identified at the variation of forcing amplitude. The quasiperiodic and chaotic modes are studied by means of Poincaré section technique, and Fourier and Lyapunov spectra analyses. The statistical properties of the sequences of temporal intervals between maxima of system’s solutions are considered in more detail. The sequence extracted from the chaotic trajectory is shown to possess long-term correlations and approximately obeys the Weibull distribution.

On extensive dynamics of a Cournot heterogeneous model with optimal response.

The objective of this paper is the study of the dynamical properties analysis of an original specification of the classical Cournot heterogeneous model with optimal response; specifically, a new approach that considers ordinal utility instead of cardinal monetary amounts is proposed where the classical decision of quantity is disentangled from the decision on imitation. The analysis is performed by means of bifurcation diagrams, the 0-1 test for chaos, power spectral density, histograms, and trajectory analysis. For this purpose, a new perturbation parameter ε of the initial condition is introduced, and together with the intensity of choice parameter β determining the share of responders vs imitators, the system is researched. Depending on ε and β, extreme reach dynamics, and coexisting attractors, periodic and chaotic trajectories are investigated through massive simulations. Those dynamics represent alternation between stability, cycles and chaos in the market. As the dynamics are completely endogenous, it means that swings in economy are intrinsic to the system and that they may persist unless controlled.

Exploring the dynamical evolution of minimoon 2020 CD3 in a circular restricted three-body problem framework

Asteroid 2020 CD3 was one of the most recently captured natural objects that became Earth’s transient satellite or minimoon. The asteroid is very likely residing in an Earth-like orbit (ELO) with 1:1 mean-motion resonance and occasionally captured by the Earth. In this study, the dynamical evolution of this object is conducted in a framework of Circular Restricted Three-Body Problem (CR3BP). Five thousand test objects are generated by distributing the orbital space with a random uniform scheme around the object’s nominal values comprising ten sigma-uncertainties of the orbital parameters and four Metonic cycles of the epoch. This numerical task has run for 1000 years with the IAS15 integrator known for its adaptive integration and high accuracy for millions of years of simulation. We found that almost all of the test particles are lying in a boundary region, hence follow chaotic and quasi-periodic trajectories and have become minimoon with an average of 15-16 times. The duration of captures of more than a year is very rare. Interestingly, there seems a preferred recapture time at ∼ 30 years. Most of them still stay in the ELO region, although a few have either been hit the Earth or expelled into vast space.

Multiple dynamics analysis of Lorenz-family systems and the application in signal detection

Inspired by the non-autonomous chaos system and the transient chaotic oscillation, based on the chaos systems of the Lorenz family, the system structure, parameters, and orders are adjusted, and the damping oscillation, sustained chaotic oscillation, transient chaotic oscillation, and chaotic oscillation with multiple chaotic trajectories are found. Firstly, the constrain conditions of the system parameters and fractional-orders are derived. Secondly, one external driving force signal and two control parameters are added to the system, different dynamic behaviors of the system under the excitement of different types, features of the external driving force signals are analyzed, simulated, and discussed. The phase diagram, bifurcation diagram, and Lyapunov exponents, etc are adopted for analysis. Finally, based on the adjusted fractional-order nonlinear systems, a kind of signal detection scheme that can be used under a strong noise environment is designed, some key information of the signal to be detected can be acquired. The simulation results show the effectiveness of the designed signal detection scheme.

Nostalgia for Futures Past

This article investigates the intellectual and political legacies of Maoist and other left Third Worldist thought of the 1970s through an analysis of Naeem Mohaiemen’s 2020 film, Afsan’s Long Day, The Young Man Was. The film presents a narrative of the history of the 1970s anti-imperialist left, and in Brechtian fashion, compels the viewer to take an emotionally distanced assessment of the left’s chaotic trajectory through a unique engagement with themes of time, history, and repetition. In the article, I explore several aspects of the film − its relationship to the ‘essay film’ as a genre, the historical context of the global 1970s as depicted in the film, and then conclude with a discussion of the theoretical concept, hauntology, in order to interrogate the political praxis of the film. The 1970s Global left represents a future that never came to be but yet continues to animate contemporary left thought.

Parameter Prediction of Marine Seawater Cooling System Based on Chaos-Elman Combined Model

To improve the accuracy of short-term prediction of marine seawater cooling system parameters and reduce the accuracy loss caused by chaotic data, a combined prediction model combining chaos theory and Elman feedback neural network was proposed in this paper. Firstly, the C-C algorithm and G-P algorithm were used to calculate the two reconstruction parameters respectively, and according to the properties of the maximum Lyapunov exponent, it could be determined that the original one-dimensional time series had chaotic characteristics. Secondly, phase space reconstruction technique was used to map the one-dimensional seawater outlet temperature time series of intercooler to the high-dimensional space to find its chaotic phase trajectory, and the number of adjacent points was determined by HQ (Hannan Quinn) information criterion. Finally, the Chaos-Elman combined prediction model was used to train and test 300 sets of seawater outlet temperature data of intercooler, and the accuracy of prediction was compared with that of single structure model. The experimental results were as follows: the average prediction accuracy based on chaos theory was 95.5%, the prediction accuracy of Elman neural network was 95.1%, and the prediction accuracy of Chaos-Elman combined model could reach 98.7%. In this study, the Chaos-Elman combined model has better prediction accuracy than the single model. It can be concluded that the Chaos-Elman combined model can be used for the short-term prediction of seawater cooling system parameters, and the prediction accuracy and reliability are high.

A comprehensive study on the characterization of chaotic advection flow and heat transfer in a twisted pipe with elliptical cross-section

Chaotic advection flow is one of the most useful methods to enhance mixing and heat transfer simultaneously in the laminar flow which is generated by spatially chaotic trajectories of the fluid particles. The present study aims to characterize the behavior of chaotic advection flow and heat transfer in a twisted pipe. The considered geometry consists of three 90° bends with an elliptical cross-section; the angle between curvature planes of consecutive bends is 90°. Numerical simulations are performed for both steady-state and pulsatile (unsteady) inlet velocity conditions with the Reynolds numbers in the range of 200 ≤ Re ≤ 450, wall temperatures 310 ≤ Tw(K) ≤ 340, velocity amplitude ratios (the ratio of peak oscillatory velocity component to the mean flow velocity) 1 ≤ β ≤ 3, and Womersley numbers 2.1118 ≤ α ≤ 7.9018. Mixing and heat transfer are analyzed both qualitatively and quantitatively for various considered conditions. The results revealed that the elliptical cross-section has a considerable impact besides the chaotic configuration on mixing and heat transfer enhancement. In the steady-state flow, the desirable system efficiency can be achieved either by increasing the wall temperature or decreasing the Reynolds number. Moreover, in the pulsatile flow, the Womersley numbers in the range of 2.1118 ≤ α ≤ 3.9509 provide a favorable condition to augment the mixing and heat transfer in the twisted pipe.

A revisit to the past plague epidemic (India) versus the present COVID-19 pandemic: fractional-order chaotic models and fuzzy logic control.

India is one of the worst hit regions by the second wave of COVID-19 pandemic and ‘Black fungus’ epidemic. This paper revisits the Bombay Plague epidemic of India and presents six fractional-order models (FOMs) of the epidemic based on observational data. The models reveal chaotic dispersion and interactive coupling between multiple species of rodents. Suitable controllers based on fuzzy logic concept are designed to stabilise chaos to an infection-free equilibrium as well as to synchronise a chaotic trajectory with a regular non-chaotic one so that the unpredictability dies out. An FOM of COVID-19 is also proposed that displays chaotic propagation similar to the plague models. The index of memory and heredity that characterise FOMs are found to be crucial parameters in understanding the progression of the epidemics, capture the behaviour of transmission more accurately and reveal enriched complex dynamics of periodic to chaotic evolution, which otherwise remain unobserved in the integral models. The theoretical analyses successfully validated by numerical simulations signify that the results of the past Plague epidemic can be a pathway to identify infected regions with the closest scenarios for the present second wave of Covid-19, forecast the course of the outbreak, and adopt necessary control measures to eliminate chaotic transmission of the pandemic.

Dissolution identitaire dans Sérotonine de Michel Houellebecq

"Sérotonine, Michel Houellebecq’s latest novel, can be read as the account of a case of identity dissolution, at both an individual and a collective level, the two levels intersecting each other. The author proposes the reader a chaotic trajectory through the despondency-ridden universe of the narrating character, who travels across a given geographic, cultural, social, professional space representative of today’s France, in keeping with the free will of a whimsical memory and a type of writing that follows a continual chronological and emotional disruption. Our analytic approach lies at the crossroads of self hermeneutics and group psychology, while intending to show how the field of literary imagination and imagery capitalizes on the present-day world’s insecurity identity-wise."

On a Bertrand Duopoly Game With Heterogeneous Expectations and Emissions Costs

This study was based on the dynamics of a nonlinear Bertrand-type duopoly game with differentiated goods, linear demand and a cost function that included emissions costs. This duopoly game was modeled with a system of two difference equations. Existence and stability of equilibria of this system were studied. It was shown that the model gave more complex chaotic and unpredictable trajectories as a consequence of changes in the speed of adjustment parameter and horizontal product differentiation parameter. Numerical simulations showed that a higher value of the speed of adjustment and a higher or lower (negative) value of product differentiation (weaker or fiercer competition) can destabilize the economy. The chaotic features were justified numerically via computing Lyapunov numbers and sensitive dependence on initial conditions. Also, it was shown that in this case of a duopoly game, there were stable trajectories, and a higher (lower) degree of product differentiation did not tend to destabilize the economy.
 Keywords: Bertrand duopoly game, discrete dynamical system, heterogeneous expectations, stability, chaotic behavior

Ergodicity and Born's rule in an entangled three-qubit Bohmian system.

We study in detail the interplay between chaos and entanglement in the Bohmian trajectories of three entangled qubits, made of coherent states of the quantum harmonic oscillator. We find that all the three-dimensional (3D) chaotic trajectories are ergodic; namely, they have a common long time distribution of points regardless of the initial conditions, and for any nonzero entanglement, their number is much larger than in the corresponding two-qubit system. Furthermore, the range of entanglements for which practically all the trajectories are chaotic and ergodic is much larger than in the two-qubit case. Thus, as the dimensionality of the system increases, Born's rule becomes accessible to a wider range of arbitrary initial distributions than in the 2D case. Our numerical results lead to the conjecture that, for multiqubit systems, Born's rule is the limit of almost all initial distributions of particles.

Avoiding Dynamical Degradation in Computer Simulation of Chaotic Systems Using Semi-Explicit Integration: Rössler Oscillator Case

Dynamical degradation is a known problem in the computer simulation of chaotic systems. Data type limitations, sampling, and rounding errors give rise to the periodic behavior. In applications of chaotic systems in secure communication and cryptography systems, such effects can reduce data storage security and operation. In this study, we considered a possible solution to this problem by using semi-explicit integration. The key idea is to perturb the chaotic trajectory by switching between two integrators, which possess close but still different numerical solutions. Compared with the traditional approach based on the perturbation of the bifurcation parameter, this technique does not significantly change the nonlinear properties of the system. We verify the efficiency of the proposed perturbation method through several numerical experiments using the well-known Rössler oscillator.

Enhanced synchronization due to intermittent noise

We propose a novel scheme to regulate noise infusion into the chaotic trajectories of uncoupled complex systems to achieve complete synchronization. So far the noise-induced synchronization utilize the uncontrolled noise that can be applied in the entire state space. Here, we consider the controlled (intermittent) noise which is infused in the restricted state space to realize enhanced synchronization. We find that the intermittent noise, which is applied only to a fraction of the state space, restricts the trajectories to evolve within the contraction region for a longer period of time. The basin stability of the synchronized states (SS) is found to be significantly enhanced compared to uncontrolled noise. Additionally, we uncover that the SS prevail for an extended range of noise intensity. We elucidate the results numerically in the Lorenz chaotic system, the Pikovski–Rabinovich circuit model and the Hindmarsh–Rose neuron model.

Generation mechanism and evolution of five-state meshing behavior of a spur gear system considering gear-tooth time-varying contact characteristics

Teeth disengagement or back-side teeth engagement induced by backlash reduces the transmission quality and dynamic performance of gear systems, and the accurate interpretation of multi-state meshing behavior can provide guidance for structural optimization and performance evaluation. Therefore, the multi-state meshing behavior of the gear system is elaborated. A new nonlinear dynamic model of a spur gear system with five-state meshing behavior is established based on the time-varying backlash and contact ratio. The time-varying meshing stiffness and time-varying backlash considering the elastic contact of gear teeth, gear temperature rise and lubrication are included in the model. The five-state meshing behavior is clearly characterized by constructing five Poincaré maps, and its generation mechanism is revealed using dynamic meshing force time history, teeth relative displacement time history and phase portrait. The bifurcation and evolution of five-state meshing behavior are analyzed under the effects of load factor, meshing frequency and error coefficient. The results show that the mutation in the direction of dynamic meshing force leads to teeth disengaging and back-side single- or double-tooth contact, forming multi-state meshing behavior. Bifurcation caused by parameter changes greatly affects the evolution of five-state meshing behavior, particularly grazing bifurcation can decrease the number of teeth disengagement. Chaotic behavior or trajectory expansion inspires multi-state meshing vibration of the system. Previous gear system models could not reveal these phenomena due to ignoring the multi-state meshing behavior.

Hardware Efficient Pseudo-Random Number Generator Using Chen Chaotic System on FPGA

This paper introduces an FPGA implementation of a pseudo-random number generator (PRNG) using Chen’s chaotic system. This paper mainly focuses on the development of an efficient VLSI architecture of PRNG in terms of bit rate, area resources, latency, maximum length sequence, and randomness. First, we analyze the dynamic behavior of the chaotic trajectories of Chen’s system and set the parameter’s value to maintain low hardware design complexity. A circuit realization of the proposed PRNG is presented using hardwired shifting, additions, subtractions, and multiplexing schemes. The benefit of this architecture, all the binary multiplications (except [Formula: see text] and [Formula: see text] operations are performed using hardwired shifting. Moreover, the generated sequences pass all the 15 statistical tests of NIST, while it generates pseudo-random numbers at a uniform clock rate with minimum hardware complexity. The proposed architecture of PRNG is realized using Verilog HDL, prototyped on the Virtex-5 FPGA (XC5VLX50T) device, and its analysis has been done using the Matlab tool. Performance analysis confirms that the proposed Chen chaotic attractor-based PRNG scheme is simple, secure, and hardware efficient, with high potential to be adopted in cryptography applications.

Dynamics of an unbalanced circular foil and point vortices in an ideal fluid

This paper addresses the problem of the motion of an unbalanced circular foil and point vortices in an ideal incompressible fluid. Using Bernoulli's theorem for unsteady potential flow, the force due to the pressure from the fluid on the foil is obtained for an arbitrary vortex motion. A detailed analysis is made of the case of free vortex motion in which a Hamiltonian reduction by symmetries is performed. For the resulting system, relative equilibria corresponding to the motion of an unbalanced foil and a vortex in a circle or in a straight line are found and their stability is investigated. New examples of stationary configurations of a vortex and a foil are given. Using a Poincaré map, it is also shown that in the general case of an unbalanced circular foil the reduced system exhibits chaotic trajectories.

Asymmetric and chaotic responses of dry friction oscillators with different static and kinetic coefficients of friction

The responses of a simple harmonically excited dry friction oscillator are analysed in the case when the coefficients of static and kinetic coefficients of friction are different. One- and two-parameter bifurcation curves are determined at suitable parameters by continuation method and the largest Lyapunov exponents of the obtained solutions are estimated. It is shown that chaotic solutions can occur in broad parameter domains—even at realistic friction parameters—that are tightly enclosed by well-defined two-parameter bifurcation curves. The performed analysis also reveals that chaotic trajectories are bifurcating from special asymmetric solutions. To check the robustness of the qualitative results, characteristic bifurcation branches of two slightly modified oscillators are also determined: one with a higher harmonic in the excitation, and another one where Coulomb friction is exchanged by a corresponding LuGre friction model. The qualitative agreement of the diagrams supports the validity of the results.

External-field-induced dynamics of a charged particle on a closed helix.

We investigate the dynamics of a charged particle confined to move on a toroidal helix while being driven by an external time-dependent electric field. The underlying phase space is analyzed for linearly and circularly polarized fields. For small driving amplitudes and a linearly polarized field, we find a split up of the chaotic part of the phase space, which prevents the particle from inverting its direction of motion. This allows for a nonzero average velocity of chaotic trajectories without breaking the well-known symmetries commonly responsible for directed transport. Within our chosen normalized units, the resulting average transport velocity is constant and does not change significantly with the driving amplitude. A very similar effect is found in case of the circularly polarized field and low driving amplitudes. Furthermore, when driving with a circularly polarized field, we unravel a second mechanism of the split up of the chaotic phase space region for very large driving amplitudes. There exists a wide range of parameter values for which trajectories may travel between the two chaotic regions by crossing a permeable cantorus. The limitations of these phenomena, as well as their implication on manipulating directed transport in helical geometries are discussed.

Multistability and dynamical properties of quantum ion-acoustic flow

Multistability and dynamical properties of ion-acoustic flow are studied in a quantum plasma containing positive beam ions, positive ions and electrons. A four dimensional conservative dynamical system has been proposed for the considered plasma system and is analyzed by considering effects of Mach number and quantum diffraction parameter. Coexistences of multiple chaotic trajectories, chaotic with quasiperiodic and multiperiodic trajectories and chaotic with quasi-periodic and periodic trajectories for ion-acoustic waves are established. The results are suitable for application in comprehending the beam-plasma interaction and studying dynamics of coexisting features in extreme astrophysical plasmas, such as, neutron stars.

A secure image protection algorithm by steganography and encryption using the 2D-TSCC* *Project supported by the National Natural Science Foundation of China (Grant No. 61672124), the Password Theory Project of the 13th Five-Year Plan National Cryptography Development Fund (Grant No. MMJJ20170203), Liaoning Provincial Science and Technology Innovation Leading Talents Program Project, China (Grant No. XLYC1802013), Key Research and Development Projects

Based on the two-dimensional (2D) tan-sin-cos-coupling (2D-TSCC), a new image protection method is designed, this method includes steganography and encryption. First, a 2D-TSCC system is designed. The 2D-TSCC has a large parameter space in a hyperchaotic state. The chaotic trajectory fills the entire window. The chaotic sequence generated by the 2D-TSCC has a good pseudorandomness, so it can be used in steganography and encryption. Then, the amount of information contained in each bit of the cover image is analyzed, and the three bits which carry the least amount of information are selected. The secret image is hidden in these three bits base on the 2D-TSCC. Finally, the carrier image is scrambled and diffused by the 2D-TSCC. The ciphertext is generated in this way. Send the ciphertext to the recipient through channel transmission, and the recipient obtains the secret image by decrypting twice.