Distribution Of Lyapunov Exponents Research Articles

The Parallel Chaotification Map and Its Application

A universal modular plus parallel (MPP) method is proposed to construct enhanced chaotic maps, including one-dimensional MPP chaotic map (1D-MPPCM) and high-dimensional MPPCM (HD-MPPCM). It is theoretically proved that 1D-MPPCM model can significantly increase the Lyapunov exponent (LE) and parameter range of seed chaotic maps. To further increase the system dimension, the HD-MPPCM model is established through the close-loop parallel coupling mechanism. Based on several typical seed chaotic maps, some new parallel chaotic maps are obtained by self-parallel and hybrid-parallel, and their dynamics are analyzed by phase diagram, LEs, permutation entropy (PE) complexity and statistic <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\chi^2$</tex-math> </inline-formula> . The simulation results show that the proposed maps have large maximum Lyapunov exponent (MLE), PE complexity, and uniform distribution. In particular, HD-MPPCMs have some interesting characteristics, such as full positive LEs, hyperchaotic behavior, global chaos, and full attractor distribution, which are the potential model for engineering applications. To further verify the practicability, the proposed maps are implemented on DSP platform, and applied to pseudo-random number generator (PRNG).

Quad Key-Secured 3D Gauss Encryption Compression System with Lyapunov Exponent Validation for Digital Images

High-dimensional systems are more secure than their lower-order counterparts. However, high security with these complex sets of equations and parameters reduces the transmission system’s processing speed, necessitating the development of an algorithm that secures and makes the system lightweight, ensuring that the processing speed is not compromised. This study provides a digital image compression–encryption technique based on the idea of a novel quad key-secured 3D Gauss chaotic map with singular value decomposition (SVD) and hybrid chaos, which employs SVD to compress the digital image and a four-key-protected encryption via a novel 3D Gauss map, logistic map, Arnold map, or sine map. The algorithm has three benefits: First, the compression method enables the user to select the appropriate compression level based on the application using a unique number. Second, it features a confusion method in which the image’s pixel coordinates are jumbled using four chaotic maps. The pixel position is randomized, resulting in a communication-safe cipher text image. Third, the four keys are produced using a novel 3D Gauss map, logistic map, Arnold map, or sine map, which are nonlinear and chaotic and, hence, very secure with greater key spaces (2498). Moreover, the novel 3D Gauss map satisfies the Lyapunov exponent distribution, which characterizes any chaotic system. As a result, the technique is extremely safe while simultaneously conserving storage space. The experimental findings demonstrate that the method provides reliable reconstruction with a good PSNR on various singular values. Moreover, the applied attacks demonstrated in the result section prove that the proposed method can firmly withstand the urge of attacks.

The regular dynamics through the finite‐time Lyapunov exponent distributions in 3D Hamiltonian systems

AbstractWe investigate the application as a technique of the so‐called finite‐time Lyapunov exponent, a scalar value that measures the average, or integrated, separation between trajectories in integrable dynamical systems to quantify the predictability limit of the asymptotic global behavior of old compact star systems (1–10 Gyr), for instance, neutron stars. For this purpose, the Hamiltonian systems with a three‐dimensional self‐gravitating axisymmetric potential are superimposed due to the influence of different birth and kick velocity distributions. We find the behavior of the density distributions P(x) for the finite‐time exponents, which follow a log‐normal distribution (transformed into a Gaussian distribution), with a given mean deviation centered around the global values, at larger Δt. It is shown that simple models of distributions reflect the underlying dynamics. We also analyze the predictability and the deviation vector ω(t), obtained from the distributions, by randomly selecting the initial deviation directions, and consequently the stellar dynamic stability of the periodic orbits, in three‐dimensional (3D) Hamiltonian systems over time. The behavior of the orbits is strongly influenced by the regular nature of the motion.

Information production in homogeneous isotropic turbulence.

We study the Reynolds number scaling of the Kolmogorov-Sinai entropy and attractor dimension for three-dimensional homogeneous isotropic turbulence through the use of direct numerical simulation. To do so, we obtain Lyapunov spectra for a range of different Reynolds numbers by following the divergence of a large number of orthogonal fluid trajectories. We find that the attractor dimension grows with the Reynolds number as Re^{2.35} with this exponent being larger than predicted by either dimensional arguments or intermittency models. The distribution of Lyapunov exponents is found to be finite around λ≈0 contrary to a possible divergence suggested by Ruelle. The relevance of the Kolmogorov-Sinai entropy and Lyapunov spectra in comparing complex physical systems is discussed.

Chaotic mixing in crystalline granular media

We study the Lagrangian kinematics of steady three-dimensional Stokes flow over simple cubic (SC) and body-centred cubic (BCC) lattices of close-packed spheres, and uncover the mechanisms governing chaotic mixing in these crystalline structures. Due to the cusp-shaped sphere contacts, the topology of the skin friction field is fundamentally different to that of continuous (non-granular) media, such as open pore networks, with significant implications for fluid mixing. Weak symmetry breaking of the flow orientation with respect to the lattice symmetries imparts a transition from regular to strong chaotic mixing in the BCC lattice, whereas the SC lattice only exhibits weak mixing. Whilst the SC and BCC lattices posses the same symmetry point group, these differences are explained in terms of their space groups. This insight is used to develop accurate predictions of the Lyapunov exponent distribution over the parameter space of mean flow orientation. These results point to a general theory of mixing and dispersion based upon the inherent symmetries of arbitrary crystalline structures.

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.

Infinite Number of Hidden Attractors in Memristor-Based Autonomous Duffing Oscillator

In this paper, the recent and emerging phenomenon of hidden oscillations is observed in a newly implemented memristor-based autonomous Duffing oscillator for the first time. The hidden oscillations are presented and quantified by various statistical measures. The system shows a large number of hidden attractors for a wide range of the system parameters. This study indicates that hidden oscillations can exist not only in piecewise-linear but also in smooth nonlinear circuits and systems. The distribution of Lyapunov exponents and the basin of attraction are explored to understand the nature of the hidden oscillations. We have also discussed the new phenomenon of periodic line invariant. An experimental demonstration is also presented using real time analog circuit.

Bursting behavior in degenerate optical parametric oscillator under noise

The distribution of Lyapunov exponents is calculated in the two-parameter region, and parameter is set carefully to reproduce bursting state from the optical parametric oscillator thus can be consistent with the previous experiments. The input field amplitude is changed to trigger different modes of outputs and then external noise is imposed to detect the transition of subharmonic mode and fundamental mode, it could be helpful to understand the potential mechanism for electromagnetic radiation-induced transition of electrical activities in neurons.

Delocalization in one-dimensional tight-binding models with fractal disorder II: existence of mobility edge

In the previous work, we investigated the correlation-induced localization-delocalization transition (LDT) of the wavefunction at band center ($E=0$) in the one-dimensional tight-binding model with fractal disorder [Yamada, EPJB (2015) 88, 264]. In the present work, we study the energy ($E \neq 0$) dependence of the normalized localization length and the delocalization of the wavefunction at the different energy in the same system. The mobility edges in the LDT arise when the fractal dimension of the potential landscape is larger than the critical value depending on the disorder strength, which is consistent with the previous result.In addition, we present the distribution of individual NLL and Lyapunov exponent in the system with LDT.

Realization of synchronization of nonlinear oscillators under intermittent coupling controlled by pulse signal

Based on the Lyapunov stability theory, an improved Lyapunov function scheme is used to understand the complete synchronization of hyperchaotic systems by imposing pulse linear coupling on the response system. According to this scheme, the controller begins to control the response system in a period when the output error variables are increasing; otherwise, the controller turns off. The distribution of conditional Lyapunov exponent versus coupling intensity, and the synchronization cost (averaged power consumption of controller) is calculated, respectively. By designing an exponential type of Lyapunov function, it is found that complete synchronization could be realized between two Chen hyperchaotic systems and two 4-dimensional LC hyperchaotic systems. Our numerical results are consistent with the previous theoretical discussion.

Nonlinear Speech Analysis and Modeling for Malayalam Vowel Recognition

This paper presents a novel method for the recognition of Malayalam vowel phonemes using nonlinear speech parameters such as maximal Lyapunov exponent and Phase Space Anti-diagonal Point Distribution. Reconstructed phase space is used as a base for extracting these features. The results show that the proposed nonlinear features have significant discriminatory power. The combined feature vectors of nonlinear feature and MFCC features yield increased recognition accuracy for five Malayalam vowel phonemes.

Statistical properties of the localization measure in a finite-dimensional model of the quantum kicked rotator.

We study the quantum kicked rotator in the classically fully chaotic regime K=10 and for various values of the quantum parameter k using Izrailev's N-dimensional model for various N≤3000, which in the limit N→∞ tends to the exact quantized kicked rotator. By numerically calculating the eigenfunctions in the basis of the angular momentum we find that the localization length L for fixed parameter values has a certain distribution; in fact, its inverse is Gaussian distributed, in analogy and in connection with the distribution of finite time Lyapunov exponents of Hamilton systems. However, unlike the case of the finite time Lyapunov exponents, this distribution is found to be independent of N and thus survives the limit N=∞. This is different from the tight-binding model of Anderson localization. The reason is that the finite bandwidth approximation of the underlying Hamilton dynamical system in the Shepelyansky picture [Phys. Rev. Lett. 56, 677 (1986)] does not apply rigorously. This observation explains the strong fluctuations in the scaling laws of the kicked rotator, such as the entropy localization measure as a function of the scaling parameter Λ=L/N, where L is the theoretical value of the localization length in the semiclassical approximation. These results call for a more refined theory of the localization length in the quantum kicked rotator and in similar Floquet systems, where we must predict not only the mean value of the inverse of the localization length L but also its (Gaussian) distribution, in particular the variance. In order to complete our studies we numerically analyze the related behavior of finite time Lyapunov exponents in the standard map and of the 2×2 transfer matrix formalism. This paper extends our recent work [Phys. Rev. E 87, 062905 (2013)].

Multilayered bubbling route to SNA in a quasiperiodically forced electronic circuit with experimental and analytical confirmation

A new route to strange nonchaotic attractor (SNA), known as multilayered bubble route to SNA, has been identified in a quasiperiodically forced series LCR circuit with a simple nonlinear element. Upon increasing the system control parameter, the stable orbits of the torus become unstable, which induces formation of bubbles in the neighborhood of the resonating region of the torus. We have observed three tori with three smooth branches in the Poincaré map which gradually loose their smoothness and ultimately approach bubble formation, and then approach fractal behavior via SNAs before the onset of chaos. The bubbles gradually enlarge and subsequently another three layers of bubbles are formed as a function of the control parameter. The layers get increasingly wrinkled as a function of the control parameter, resulting in the creation of SNAs which are characterized by Poincaré maps. The multilayered bubble route to SNA is then confirmed by experimental Poincaré maps and explicit analytical solution is developed to further confirm it. Numerically observed bubbling route is characterized qualitatively in terms of phase portraits, power spectrum and further characterized quantitatively, by singular-continuous spectrum analysis, phase sensitivity measure, distribution of finite time Lyapunov exponents, largest Lyapunov exponent and its variance.

Multilayered bubbling route to SNA in a quasiperiodically forced electronic circuit with a simple nonlinear element

A new route to strange nonchaotic attractors (SNAs), known as multilayered bubble route to SNA, has been identified in a quasiperiodically forced series $$\textit{LCR}$$ circuit with a simple nonlinear element. Upon increasing the control parameter, the stable orbits of the torus become unstable, which induces formation of bubbles in the neighbourhood of the resonating region of the torus. We have observed three tori with three smooth branches in the Poincare map which gradually loose their smoothness and ultimately approach bubble formation, and then approach fractal behaviour via SNAs before the onset of chaos. The bubbles gradually enlarge and subsequently another three layers of bubbles are formed as a function of the control parameter. The layers get increasingly wrinkled as a function of the control parameter, resulting in the creation of SNAs which are charaterized by Poincare maps. Apart from the multilayered bubble route to SNA, Heagy–Hammel and fractalization routes to SNA have also been numerically observed and are characterized qualitatively interms of phase portraits, power spectrum. The above mentioned three routes are further characterized quantitatively, by singular-continuous spectrum analysis, phase sensitivity measure, distribution of finite time Lyapunov exponents, largest Lyapunov exponent and its variance.

Improving Chaotic Generator’s Speed Performance for Secure Information Transmitting

In this paper an improved Lozi system speed performance is achieved in application to chaotic switched model. Generally speaking, the main advantage of the switched encryption model is its robustness to the noise, while the main drawback is the slow processing speed. In this article we propose to gain model productivity by adjusting parameter in Lozi chaotic generator. Chaos generator is sensitive to any structure, therefore the solution shouldn’t influence the pseudo randomness which is required for encryption. We provide the results of switched chaotic model based on Lozi chaotic generator with changeable parameter studied for chaoticity and pseudo-randomness with NIST, largest Lyapunov exponent, auto-correlation, cross-correlation and cumulative distribution.

A statistical approach to estimate the Lyapunov spectrum in disc brake squeal

The estimation of squeal propensity of a brake system from the prediction of unstable vibration modes using the linear complex eigenvalue analysis (CEA) in the frequency domain has its fair share of successes and failures. While the CEA is almost standard practice for the automotive industry, time domain methods and the estimation of Lyapunov spectra have not received much attention in brake squeal analyses. One reason is the challenge in estimating the true Lyapunov exponents and their discrimination against spurious ones in experimental data. A novel method based on the application of the Eckmann-Ruelle matrices is proposed here to estimate Lyapunov exponents by using noise in a statistical procedure. It is validated with respect to parameter variations and dimension estimates. By counting the number of non-overlapping confidence intervals for Lyapunov exponent distributions obtained by moving a window of increasing size over bootstrapped same-length estimates of an observation function, a dispersion measure׳s width is calculated and fed into a Bayesian beta-binomial model. Results obtained using this method for benchmark models of white and pink noise as well as the classical Henon map indicate that true Lyapunov exponents can be isolated from spurious ones with high confidence. The method is then applied to accelerometer and microphone data obtained from brake squeal tests. Estimated Lyapunov exponents indicate that the pad׳s out-of-plane vibration behaves quasi-periodically on the brink to chaos while the microphone׳s squeal signal remains periodic.

Universal distribution of Lyapunov exponents for products of Ginibre matrices

Starting from exact analytical results on singular values and complex eigenvalues of products of independent Gaussian complex random N × N matrices, also called the Ginibre ensemble, we rederive the Lyapunov exponents for an infinite product. We show that for a large number t of product matrices, the distribution of each Lyapunov exponent is normal, and we compute its t-dependent variance as well as corrections in a large-t expansion. Originally Lyapunov exponents are defined for the singular values of the product matrix that represents a linear time evolution. Surprisingly a similar construction for the moduli of the complex eigenvalues yields the very same exponents and normal distributions to leading order. We discuss a general mechanism for 2 × 2 matrices showing why the singular values and the radii of complex eigenvalues collapse onto the same value in the large-t limit. We thereby rederive Newmanʼs triangular law which has a simple interpretation as the radial density of complex eigenvalues in the circular law, and study the commutativity of the two limits and on the global scale and on the local scale. We show as a mathematical byproduct that a particular asymptotic expansion of a Meijer G-function with large index leads to a Gaussian.

Numerical investigation in chaos characteristics of macro-mixing efficiency of two-dimensional supersonic mixing layer

Coherent structures of two-dimensional supersonic turbulent mixing layers with the engineering background have been investigated numerically with large eddy simulation method. The chaos characteristics of transverse sections of mixing layer flowfield have been obtained through the analyses of two chaos characteristic parameters, Lyapunov exponent and correlation dimension. The results show that under the conditions of the present numerical simulations and physical models, the distribution of correlation dimension is roughly consistent with that of streamwise turbulent intensity at transverse sections of mixing layers through analysis of nonlinear time series, as may be used to measure the macro-mixing efficiency of mixing layers quantitatively. The distribution of maximal Lyapunov exponent could be used to validate the quantitative results. Therefore, a potential experimental method measuring the thickness of mixing layers with low costs on the basis of chaos characteristic parameters has been proposed firstly.

Anomalous diffusion of field lines and charged particles in Arnold-Beltrami-Childress force-free magnetic fields

The cosmic magnetic fields in regions of low plasma pressure and large currents, such as in interstellar space and gaseous nebulae, are force-free in the sense that the Lorentz force vanishes. The three-dimensional Arnold-Beltrami-Childress (ABC) field is an example of a force-free, helical magnetic field. In fluid dynamics, ABC flows are steady state solutions of the Euler equation. The ABC magnetic field lines exhibit a complex and varied structure that is a mix of regular and chaotic trajectories in phase space. The characteristic features of field line trajectories are illustrated through the phase space distribution of finite-distance and asymptotic-distance Lyapunov exponents. In regions of chaotic trajectories, an ensemble-averaged variance of the distance between field lines reveals anomalous diffusion—in fact, superdiffusion—of the field lines. The motion of charged particles in the force-free ABC magnetic fields is different from the flow of passive scalars in ABC flows. The particles do not necessarily follow the field lines and display a variety of dynamical behavior depending on their energy, and their initial pitch-angle. There is an overlap, in space, of the regions in which the field lines and the particle orbits are chaotic. The time evolution of an ensemble of particles, in such regions, can be divided into three categories. For short times, the motion of the particles is essentially ballistic; the ensemble-averaged, mean square displacement is approximately proportional to t2, where t is the time of evolution. The intermediate time region is defined by a decay of the velocity autocorrelation function—this being a measure of the time after which the collective dynamics is independent of the initial conditions. For longer times, the particles undergo superdiffusion—the mean square displacement is proportional to tα, where α &amp;gt; 1, and is weakly dependent on the energy of the particles. These super-diffusive characteristics, both of magnetic field lines and of particles moving in these fields, strongly suggest that theories of transport in three-dimensional chaotic magnetic fields need a shift from the usual paradigm of quasilinear diffusion.

Evidence for a fluctuation theorem in an atmospheric circulation model

An investigation of the distribution of finite time trajectory divergence is performed on an atmospheric global circulation model. The distribution of the largest local Lyapunov exponent shows a significant probability for negative values over time spans up to 10 days. This effect is present for resolutions up to wave numbers ℓ=42 (≈250 km). The probability for a negative local largest Lyapunov exponent decreases over time, similarly to the predictions of the fluctuation theorem for entropy production. The model used is hydrostatic with variable numbers of vertical levels and different horizontal resolutions.