Chaotic Trajectories Research Articles

Digital Communication Systems Based on Three-Dimensional Chaotic Attractors

In this paper, we propose a methodology to design chaos-based communication systems, which exploits the topological structure of 3-D chaotic attractors. The first step consists in defining a proper partition of a Poincaré section of the attractor and the subsequent encoding of the chaotic trajectories. Then, the evolution mechanism of the chaotic attractor, according to the dynamical restrictions imposed by the chaotic flow, is represented by a state diagram, where each state represents a region of the Poincaré section or a branch in the template of the chaotic attractor. The state transitions are associated with segments of chaotic trajectories that connect the corresponding regions of the Poincaré section. The chaotic signals are transmitted over both additive white Gaussian noise and Rayleigh flat fading channels, and a trellis structure derived from the state diagram is used at the decoder to estimate the transmitted information sequence. Finally, the bit error rate performance of the system is analyzed.

A Multi-Interval Homotopy Analysis Method Using Multi-Objective Optimization for Analytically Analyzing Chaotic Dynamics in Memristive Circuit

Memristive nonlinear system has drawn much attention in recent years, due to its rich and complex dynamical characteristics. However, there are few studies focus on the analytical analysis of this significant system. In this paper, a novel analytical method for analyzing the chaotic trajectories of memristive circuit is proposed. This method combines Homotopy Analysis Method (HAM) and Multi-objective Optimization (MO), i.e., the convergence control parameter of traditional HAM is divided into lots of subintervals in the time domain and respectively optimized by MO, for accurately solving the Ordinary Differential Equations describing memristive circuits. Hence, this method is named by MO-based multi-interval HAM (MO-MIHAM). By using MO-MIHAM, we accurately tracked the chaotic trajectories of the classical Memristor-Capacitor-Inductor (MCL) circuit and current memristive Band Pass Filter (BPF) chaotic circuit. Furthermore, based on the comparisons of errors between analytical approximate solutions derived from MO-MIHAM and solutions solved by traditional homotopy-based analytical methods and by Runge-Kutta-Fehlberg Method (RKF45) based numerical analysis, we found that, MO-MIHAM is characterized by higher approximation accuracy and computational performance (comprehensively considering the accuracy, computational complexity and execution time by a proposed Quality Factor) among the homotopy-based analytical methods, due to the optimized convergence control parameters in subintervals. Besides this major advantage, MO-MIHAM enables both qualitative and quantitative analyses and high freedom to choose homotopy-related terms for simplicity, and it is insensitive to convergence issues. Therefore, it is a powerful tool for exploratory studies for analytically analyzing chaotic dynamics in memristive circuit.

2D Logistic-Modulated-Sine-Coupling-Logistic Chaotic Map for Image Encryption

Due to the significant properties of unpredictability, ergodicity, and initial state sensitivity, chaotic system is widely used as a useful tool in image encryption. In this paper, we propose a 2-dimensional logistic-modulated-sine-coupling-logistic chaotic map (LSMCL), where we use the logistic map to modulate Sine map and couple the result of modulation and Sine map together. In terms of chaotic trajectory, Lyapunov exponent, and Kolmogorov entropy, comparing with other existing chaotic maps, we can observe that LSMCL has better chaotic performance. Furthermore, we propose an LSMCL-based image encryption algorithm with two rounds of permutation and diffusion operation. First, we provide a secret key generation procedure to generate the initial values and do permutation operation with the chaotic matrix by LSMCL. Furthermore, in diffusion procedure, we use two different chaotic matrices generated by LSMCL to change the pixel values in row and column. Finally, we provide some theoretical analyses and simulations to confirm the security and the validity of the proposed algorithm.

Chaos control of nonlinear aeroelastic pitch plunge model

This paper deals with the control of chaos in a nonlinear aeroelastic pitch-plunge model for aircraft wings. While design of nonlinear controllers to asymptotically stabilise the initial bifurcation exists in the current literature, the primary goal of this paper is to control the chaotic motion using feedback linearisation. For this purpose, a pitch-plunge model with two flaps is considered for the study. The approach proposed for the control of chaos is that of a tracking problem where the system is controlled to a defined limit cycle; thus the chaotic motion is replaced by orbital stability. Using Lie algebra, a feedback linearisation is performed by transforming the nonlinear space to a linear one. The error dynamics is established as the deviation of the chaotic trajectory from that of the desired path and tracking is achieved by reducing the error to zero. The ability of this approach to control chaos is demonstrated for a certain set of parameters of the pitch-plunge model, chosen from the literature. In order to validate the approach followed in this paper, the Rossler system undergoing chaotic motion is tracked to a LCO and results are compared with the literature.

Using chaotic maps for 3D boundary surveillance by quadrotor robot

Chaotic maps have been shown to be suitable for applications that require unpredictable behavior. In this paper, the chaotic maps are used for motion planning and control of a quadrotor for boundary surveillance purposes. The chaotic motion would prevent anticipation of the future movement of the quadrotor, especially in hostile situations where an intruding opponent exists. The chaotic trajectories are constructed based on a closed three-dimensional curve representing the boundary. It is demonstrated that the proposed algorithm for 3D planning generates chaotic paths by investigating the Lyapunov exponents. The desired unpredicted trajectory is used as input to an under-actuated control system. The performance of the algorithm is shown by simulation of the control system applied to the dynamic model.

Single Crystal-Lattice-Shaped Chaotic and Quasi-Periodic Flows with Time-Reversible Symmetry

In the literature, there are few conservative chaotic systems which are not obviously conservative according to their equations. This paper reports a 3D time-reversible symmetric chaotic system without equilibrium. The matrix form of the new system shows that there exists a Hamiltonian, which can exhibit interesting spatial structures (isosurfaces) controlled by different initial conditions. Numerical results shows that different initial conditions lead to different dynamical behaviors, such as quasi-periodic motion and conservative chaos. Moreover, the chaotic trajectories, visually, entwine around a isosurface and form a complicated topological structure like a single crystal lattice.

Characterization of magnetic field lines in connection with their transport in Beltrami magnetic fields

Single and double curl Beltrami magnetic fields are studied numerically in connection with their transport properties, and the results are compared and analyzed by characterizing the magnetic field lines. In the phase space of the single curl Beltrami field, islands of regular field lines are embedded into the chaotic sea, whereas the field lines for certain solutions of the double curl Beltrami field are chaotic over the entire space. Due to the presence of regular islands in phase space, the chaotic trajectories show stickiness phenomena which are characterized by the distribution of a chaotic field line. The dynamical traps in chaotic orbits due to stickiness phenomena are also characterized by the distribution of finite distance Lyapunov exponents. Finally, the recurrence length distribution of chaotic trajectories is plotted to understand their global behaviour.

Stickiness in double-curl Beltrami magnetic fields.

The double-curl Beltrami magnetic field in the presence of a uniform mean field is considered for investigating the nonlinear dynamical behavior of magnetic field lines. The solutions of the double-curl Beltrami equation being non-force-free in nature belong to a large class of physically interesting magnetic fields. A particular choice of solution for the double-curl equation in three dimensions leads to a wholly chaotic phase space. In the presence of a strong mean field, the phase space is a combination of closed magnetic surfaces and weakly chaotic regions that slowly tends to global randomness with a decreasing mean field. Stickiness is an important feature of the mixed phase space that describes the dynamical trapping of a chaotic trajectory at the border of regular regions. The global behavior of such trajectories is understood by computing the recurrence length statistics showing a long-tail distribution in contrast to a wholly chaotic phase space that supports a distribution which decays rapidly. Also, the transport characteristics of the field lines are analyzed in connection with their nonlinear dynamical properties.

On the Stability of a Cournot Dynamic Game Under the Influence of Information

In this paper we study the impact of information on the stability of a dynamic Cournot – type duopoly game. We suppose that one player searches for information about his rival before he makes his decision. We study how the amount of information acquired by player, influences the stability of Nash equilibrium. The game is modeled with a system of two difference equations. Existence and stability of equilibrium of this system are studied. We show numerically that the model gives chaotic and unpredictable trajectories as a consequence of change in the information parameter, but in our case there are also stable trajectories for each value of the information parameter. To provide some numerical evidence for the chaotic behavior of the system we present various numerical results including bifurcation diagrams, strange attractors, Lyapunov numbers and sensitive dependence on initial conditions.
 
 
 Keywords: Cournot dynamic game, Nash equilibrium, Stability, Chaos.

Clausius–Mossotti relation fractal modification

The microstructure characteristic and dielectric properties of doped BaTiO3-ceramics were investigated in the light of the Clausius–Mossotti relation (CMR) which incorporates both the Curie and Curie–Weiss Laws. The samples of doped BaTiO3 ceramics were prepared using conventional solid state procedure and sintered from 1290 °C to 1320 °C. Microstructural and compositional studies were performed by SEM equipped with EDS system. The Clausius–Mossotti relation is used to clarify the influence of dopant on the dielectric properties and BaTiO3 phase transformation. Curie parameters (C, Tc) were calculated by using a Curie–Weiss law. Also, a new approach on correlation between microstructure and dielectric properties of doped BaTiO3 based on fractal nature analysis (grains shapes, pores and intergranular contacts) have been developed, by using fractals in microstructure configurations reconstruction. This includes reconstruction of grain surface i.e. pore surface as tensor product of appropriate fractal curves. Particles chaotic motion trajectories is considered as Brownian motion which is then linearized in fractal manner. Fractality of grain surface, pore surface and particle motion introduce three fractal factors. Our hypothesis is that working temperature of BaTiO3-ceramics must be influenced by ceramics material fractality; these three fractality factors, appear as a correction of theoretic temperature appearing in Clausius–Mossotti relation i.e. Curie–Weiss law.

Synchronization of chaotic systems: A microscopic description

The synchronization of coupled chaotic systems represents a fundamental example of self organization and collective behavior. This well-studied phenomenon is classically characterized in terms of macroscopic parameters, such as Lyapunov exponents, that help predict the systems transitions into globally organized states. However, the local, microscopic, description of this emergent process continues to elude us. Here we show that at the microscopic level, synchronization is captured through a gradual process of topological adjustment in phase space, in which the strange attractors of the two coupled systems continuously converge, taking similar form, until complete topological synchronization ensues. We observe the local nucleation of topological synchronization in specific regions of the systems attractor, providing early signals of synchrony, that appear significantly before the onset of complete synchronization. This local synchronization initiates at the regions of the attractor characterized by lower expansion rates, in which the chaotic trajectories are least sensitive to slight changes in initial conditions. Our findings offer an alternative description of synchronization in chaotic systems, exposing its local embryonic stages that are overlooked by the currently established global analysis. Such local topological synchronization enables the identification of configurations where prediction of the state of one system is possible from measurements on that of the other, even in the absence of global synchronization.

Probing analytical and numerical integrability: the curious case of (AdS5 \xd7 S5)\u03b7

Motivated by recent studies related to integrability of string motion in various backgrounds via analytical and numerical procedures, we discuss these procedures for a well known integrable string background (AdS5 × S5)η. We start by revisiting conclusions from earlier studies on string motion in (ℝ × S3)η and (AdS3)η and then move on to more complex problems of (ℝ × S5)η and (AdS5)η. Discussing both analytically and numerically, we deduce that while (AdS5)η strings do not encounter any irregular trajectories, string motion in the deformed five-sphere can indeed, quite surprisingly, run into chaotic trajectories. We discuss the implications of these results both on the procedures used and the background itself.

Clean numerical simulation: a new strategy to obtain reliable solutions of chaotic dynamic systems

It is well known that chaotic dynamic systems (such as three-body system, turbulent flow and so on) have the sensitive dependence on initial conditions (SDIC). Unfortunately, numerical noises (such as truncation error and round-off error) always exist in practice. Thus, due to the SDIC, long-term accurate prediction of chaotic dynamic systems is practically impossible. In this paper, a new strategy for chaotic dynamic systems, i.e. the Clean Numerical Simulation (CNS), is briefly described, together with its applications to a few Hamiltonian chaotic systems. With negligible numerical noises, the CNS can provide convergent (reliable) chaotic trajectories in a long enough interval of time. This is very important for Hamiltonian systems such as three-body problem, and thus should have many applications in various fields. We find that the traditional numerical methods in double precision cannot give not only reliable trajectories but also reliable Fourier power spectra and autocorrelation functions. In addition, it is found that even statistic properties of chaotic systems can not be correctly obtained by means of traditional numerical algorithms in double precision, as long as these statistics are time-dependent. Thus, our CNS results strongly suggest that one had better to be very careful on DNS results of statistically unsteady turbulent flows, although DNS results often agree well with experimental data when turbulent flows are in a statistical stationary state.

Experimental Observation of a Time-Driven Phase Transition in Quantum Chaos.

We report the first experimental observation of the time-driven phase transition in a canonical quantum chaotic system, the quantum kicked rotor. The transition bears a firm analogy to a thermodynamic phase transition, with the time mimicking the temperature and the quantum expectation of the rotor's kinetic energy mimicking the free energy. The transition signals a sudden change in the system's memory behavior: before the critical time, the system undergoes chaotic motion in phase space and its memory of initial states is erased in the course of time; after the critical time, quantum interference enhances the probability for a chaotic trajectory to return to the initial state, and thus the system's memory is recovered.

Exploring orbital dynamics and trapping with a generalized pilot-wave framework.

We explore the effects of an imposed potential with both oscillatory and quadratic components on the dynamics of walking droplets. We first conduct an experimental investigation of droplets walking on a bath with a central circular well. The well acts as a source of Faraday waves, which may trap walking droplets on circular orbits. The observed orbits are stable and quantized, with preferred radii aligning with the extrema of the well-induced Faraday wave pattern. We use the stroboscopic model of Oza et al. [J. Fluid Mech. 737, 552-570 (2013)] with an added potential to examine the interaction of the droplet with the underlying well-induced wavefield. We show that all quantized orbits are stable for low vibrational accelerations. Smaller orbits may become unstable at higher forcing accelerations and transition to chaos through a path reminiscent of the Ruelle-Takens-Newhouse scenario. We proceed by considering a generalized pilot-wave system in which the relative magnitudes of the pilot-wave force and drop inertia may be tuned. When the drop inertia is dominated by the pilot-wave force, all circular orbits may become unstable, with the drop chaotically switching between them. In this chaotic regime, the statistically stationary probability distribution of the drop's position reflects the relative instability of the unstable circular orbits. We compute the mean wavefield from a chaotic trajectory and confirm its predicted relationship with the particle's probability density function.

Mathematical Analysis of Peculiar Behavior by Chaotic, Fractional and Strange Multiwing Attractors

Strange attractors are fascinating not only for the fractal structure most of them display, but also for the freakish chaotic and multiscroll trajectories they portray. In this paper, Haar wavelet numerical method is used to address the solvability of the 3D Lu–Chen model combined with the classical Caputo fractional differentiation. The full model, named Caputo–Lu–Chen model is solved analytically and a complete error analysis is performed to test the convergence of the method. It reveals that the error made using such a technique is negligible. Numerical simulations, performed in pure fractional and standard integer cases, show that the Caputo–Lu–Chen system maintains its status of strange, chaotic and multiscroll attractor. The graphics in both cases present similar features characterized by attractors with many scrolls. Hence, the Caputo derivative order when introduced as parameter in the Lu–Chen system does not remove the multiscroll property of generated attractors; it even becomes an important control parameter able to change the dynamics of the whole system. This result can be very important in control theory.

Three-dimensional billiards: Visualization of regular structures and trapping of chaotic trajectories.

The dynamics in three-dimensional (3D) billiards leads, using a Poincaré section, to a four-dimensional map, which is challenging to visualize. By means of the recently introduced 3D phase-space slices, an intuitive representation of the organization of the mixed phase space with regular and chaotic dynamics is obtained. Of particular interest for applications are constraints to classical transport between different regions of phase space which manifest in the statistics of Poincaré recurrence times. For a 3D paraboloid billiard we observe a slow power-law decay caused by long-trapped trajectories, which we analyze in phase space and in frequency space. Consistent with previous results for 4D maps, we find that (i) trapping takes place close to regular structures outside the Arnold web, (ii) trapping is not due to a generalized island-around-island hierarchy, and (iii) the dynamics of sticky orbits is governed by resonance channels which extend far into the chaotic sea. We find clear signatures of partial transport barriers. Moreover, we visualize the geometry of stochastic layers in resonance channels explored by sticky orbits.

Distance correlation detecting Lyapunov instabilities, noise-induced escape times and mixing

The properties of the statistical method of distance correlation between multivariate data are analysed in the context of nonlinear dynamical systems. The distance correlation between the noisy and the noiseless quadratic maps are studied in periodic and chaotic regimes. Results are compared to the classical method of Pearson’s correlation. While distance and Pearson’s correlations are affected by the Lyapunov exponent from the noiseless orbits, only the distance correlation is able to recognize the correct qualitative behaviour of noise-induced escape times decays and the mixing of chaotic trajectories. In addition, the distance correlation is capable of detecting distinct size of attractors. The main goal of this work is to establish the validity of the distance correlation as an method of correlation between multivariate data in dynamical systems.

Influence of quadrupole–quadrupole-type interaction on the chaotic dynamics of $$\alpha $$ α -helical proteins

By proposing a model Hamiltonian in the first quantised form we investigate the chaotic dynamics of \(\alpha \)-helical proteins by taking into account the quadrupole–quadrupole-type interaction. The dynamics is studied by deriving Hamilton’s equations of motion and by plotting the time-series evolution and phase-space trajectories. Chaotic trajectories are observed in the phase-space plots. The effect of the interaction parameters on the stability of proteins is also discussed.

A study of chaotic searching paths for their application in an ultrasonic scanner

Most non-destructive ultrasonic automatic scanning tests are programmed with a complete systematic trajectory. However, when the location of a discontinuity (disturbance source) and the searching space are unknown, an exhaustive search can be time and computational inefficient. In this paper, the efficiency of chaotic trajectories to search for a discontinuity in an area with limited information is studied. The probability of detection and its relationship with topological properties of the proposed chaotic dynamic model is analyzed. The Pearson’s random walk model is modified to substitute random for chaotic behavior by transferring the solutions of the chaotic dynamical model to the phase and modulus of the discrete path trajectories. This provides a simple description of the trajectories in a vector space where only magnitude and phase are required. Following this approach, the chaotic trajectories were generated in the working plane and then transferred to a Cartesian robotic ultrasonic system. The system was tested without human supervision on unknown environments for ultrasonic detection of hidden defects. The results show that the studied chaotic walk model provides efficient ultrasonic scanning trajectories for defect detection and can outperform systematic scanning when the geometry of the workspace is complex and unknown.