Smallest Lyapunov Exponent Research Articles

Border-collision bifurcation route to strange nonchaotic attractors in the piecewise linear normal form map

A new route to strange nonchaotic attractors (SNAs) is investigated in a quasiperiodically driven nonsmooth map. It is shown that the smooth quasiperiodic torus becomes nonsmooth (continuous and non-differentiable) due to the border-collision bifurcation of the torus. As the coefficients change, the nonsmooth torus gets gradually fractal, forming strange nonchaotic attractors. It is termed the border-collision bifurcation route to SNAs. A novel feature of this route is that a large number of SNAs are formed, and the parameter area of SNAs makes up about 40 % of the given parameter regions. These SNAs are identified by the largest Lyapunov exponents and the phase sensitivity exponents. They are also characterized by the distribution of finite-time Lyapunov exponents. Unlike other types of SNAs, the distribution of finite-time Lyapunov exponents has its maximum at a relatively small negative Lyapunov exponent, which contributes largely to lead to the abundance of SNAs.

Comparative Analysis of Correlation and Kaplan–Yorke Dimensions for Discrete-Time Fractional Systems

The aim of this paper is to investigate the discrete-time fractional systems from the following aspects. First, the discrete-time fractional unified system in Caputo sense is established with the help of Euler’s discretization method. Furthermore, the dynamic behaviors of the discrete-time fractional Lü system (DFLS) which is deemed as a representative for unified system are observed. Then, the correlation dimension ([Formula: see text]) and Kaplan–Yorke dimension ([Formula: see text]) of the DFLS are evaluated by the aid of Grassberger–Procaccia algorithm and the Lyapunov exponent spectrum, respectively. Finally, the intrinsic connections between [Formula: see text] and [Formula: see text] are analyzed by the statistical modeling idea when the DFLS is in chaotic vibrations. The main results show that [Formula: see text] shares a positive correlation with [Formula: see text] for the chaotic DFLS, while the differences between [Formula: see text] and [Formula: see text] are not only related to the ratio of the largest and smallest Lyapunov exponents, but also closely tied up with the fractional order [Formula: see text] itself.

Trunk Postural Control Strategies Among Persons With Lower-Limb Amputation While Walking and Performing a Concurrent Task.

Altered trunk movements during gait in persons with lower-limb amputation are often associated with an increased risk for secondary health conditions; however, the postural control strategies underlying such alterations remain unclear. In this secondary analysis, the authors employed nonlinear measures of triplanar trunk accelerations via short-term Lyapunov exponents to investigate trunk local stability as well as spatiotemporal gait parameters to describe gait mechanics. The authors also evaluated the influence of a concurrent task on trunk local stability and gait mechanics to explore if competition for neuromuscular processing resources can assist in identifying unique strategies to control kinematic variability. Sixteen males with amputation-8 transtibial and 8 transfemoral-and 8 uninjured males (controls) walked on a treadmill at their self-selected speed (mean = 1.2m/s ±10%) in 5 experimental conditions (8min each): 4 while performing a concurrent task (2 walking and 2 seated) and 1 with no concurrent task. Individuals with amputation demonstrated significantly smaller Lyapunov exponents than controls in all 3 planes of motion, regardless of concurrent task or level of amputation (P < .0001). Individuals with transfemoral amputation walked with wider strides compared with individuals with transtibial amputation and controls (P < .0001). Individuals with amputation demonstrated more trunk kinematic variability in the presence of wider strides compared with individuals without amputation, and it appears that performing a concurrent cognitive task while walking did not change trunk or gait mechanics.

A novel chaotic map constructed by geometric operations and its application

Chaotic maps are widely used in the designs of different security applications due to their series of attractive features, but many commonly used chaotic maps have several weaknesses such as limited chaotic range, small Lyapunov exponent and heavy computational cost. Although there have been many methods to overcome these shortcomings, most of them lack theoretical analysis and the improvement effect is quite limited. In order to better solve these problems, from the perspective of geometry, this paper constructs a novel chaotic map with complicated dynamic behavior, which is called piecewise cubic map. We mathematically derive its Lyapunov exponent and some associated theorems and corollaries, which reflects that the new map owns good chaotic performance. Meanwhile, it still retains the advantage of high iteration efficiency. These characteristics provide good theoretical guarantee for the security and efficiency of encryption applications based on it. Moreover, a novel pseudo-random number generator based on piecewise cubic map is designed to investigate its application in cryptography. Performance evaluation shows that the proposed generator can efficiently produce random sequences with high quality.

Large Deviation Estimates and Hölder Regularity of the Lyapunov Exponents for Quasi-periodic Schrödinger Cocycles

AbstractWe consider one-dimensional quasi-periodic Schrödinger operators with analytic potentials. In the positive Lyapunov exponent regime, we prove large deviation estimates, which lead to refined Hölder continuity of the Lyapunov exponents and the integrated density of states, in both small Lyapunov exponent and large coupling regimes. Our results cover all the Diophantine frequencies and some Liouville frequencies.

Research on cascading high-dimensional isomorphic chaotic maps

In order to overcome the security weakness of the discrete chaotic sequence caused by small Lyapunov exponent and keyspace, a general chaotic construction method by cascading multiple high-dimensional isomorphic maps is presented in this paper. Compared with the original map, the parameter space of the resulting chaotic map is enlarged many times. Moreover, the cascaded system has larger chaotic domain and bigger Lyapunov exponents with proper parameters. In order to evaluate the effectiveness of the presented method, the generalized 3-D Hénon map is utilized as an example to analyze the dynamical behaviors under various cascade modes. Diverse maps are obtained by cascading 3-D Hénon maps with different parameters or different permutations. It is worth noting that some new dynamical behaviors, such as coexisting attractors and hyperchaotic attractors are also discovered in cascaded systems. Finally, an application of image encryption is delivered to demonstrate the excellent performance of the obtained chaotic sequences.

Decay of the distance autocorrelation and Lyapunov exponents

This work presents numerical evidence that for discrete dynamical systems with one positive Lyapunov exponent the decay of the distance autocorrelation is always related to the Lyapunov exponent. Distinct decay laws for the distance autocorrelation are observed for different systems, namely, exponential decays for the quadratic map, logarithmic for the Hénon map, and power-law for the conservative standard map. In all these cases the decay exponent is close to the positive Lyapunov exponent. For hyperchaotic conservative systems the power-law decay of the distance autocorrelation is not directly related to any Lyapunov exponent.

Dynamical analogues of rank distributions

We present an equivalence between stochastic and deterministic variable approaches to represent ranked data and find the expressions obtained to be suggestive of statistical-mechanical meanings. We first reproduce size-rank distributions N(k) from real data sets by straightforward considerations based on the assumed knowledge of the background probability distribution P(N) that generates samples of random variable values similar to real data. The choice of different functional expressions for P(N): power law, exponential, Gaussian, etc., leads to different classes of distributions N(k) for which we find examples in nature. Then we show that all of these types of functions can be alternatively obtained from deterministic dynamical systems. These correspond to one-dimensional nonlinear iterated maps near a tangent bifurcation whose trajectories are proved to be precise analogues of the N(k). We provide explicit expressions for the maps and their trajectories and find they operate under conditions of vanishing or small Lyapunov exponent, therefore at or near a transition to or out of chaos. We give explicit examples ranging from exponential to logarithmic behavior, including Zipf’s law. Adoption of the nonlinear map as the formalism central character is a useful viewpoint, as variation of its few parameters, that modify its tangency property, translate into the different classes for N(k).

Dynamics and Optimization Control of a Robust Chaotic Map

Robust chaos in the discrete system is suggested to have practical as well as theoretical importance since it can obtain reliable operation in the chaotic mode. However, it receives only moderate attention and only focuses on a finite chaotic parameter space and small Lyapunov exponents. This paper introduces a two-dimensional smooth map and studies its robustness of chaos in the infinite parameter space. Then, a compound operation-based optimization control method is introduced to increase the map complexity in the measure of Lyapunov exponent. The introduced method is simple and provides a new pathway for exploring the robustness and complexity of discrete chaotic system. Finally, we design a chaos-based pseudo-random number generator (CPNG) based on the optimized robust chaotic map, and the careful analysis shows that the proposed CPNG has high quality of randomness and has passed the rigorous National Institute of Standards and Technology (NIST) test.

Perubahan Panjang Lokalisasi dan Kerapatan Keadaan Elektron pada Molekul DNA Poli(dA)-Poli(dT) Karena Medan Magnet

The localization length and the Density Of State (DOS) of electron in a Poly(dA)-Poly(dT) DNA molecule at two temperatures have been calculated for several values of magnetic field. The calculation are carried out on a DNA molecule model that consists of adenine (A) and Thymine (T) nucleobases and sugar-phosphate backbone . The DNA molecule is modeled in tight binding Hamiltonian approach with semi empiri cal Slater-Koster theory and Peierls phase factor for introducing the effect of temperature and magnetic field, respectively . In the model, electron is allowed to move between nucleobase sites, backbode sites, and between backbone site and nucleobase site. The localization length is obtained from the smallest Lyapunov exponent which is calculated using transfer matri x method along with Gram-Schmidt ort h onormali zation procedure . The DOS is calculated using Green ’s function method by taking into account the presence of metallic electrode at both ends of the DNA molecule . The localization length and the DOS change as a result of the change in electron wave phase due to magnetic field. The change is observed at both temperature used in the study, but the change at lower temperature is larger than the one at higher temperature. Thermally agitated vibrational twisting motion perturbs electron motion in the DNA molecule such that the influence of magnetic field on the localization length and the DOS of electron at higher temperature becomes smaller.

A Nonlinearly Modulated Logistic Map with Delay for Image Encryption

Considering that a majority of the traditional one-dimensional discrete chaotic maps have disadvantages including a relatively narrow chaotic range, smaller Lyapunov exponents, and excessive periodic windows, a new nonlinearly modulated Logistic map with delay model (NMLD) is proposed. Accordingly, a chaotic map called a first-order Feigenbaum-Logistic NMLD (FL-NMLD) is proposed. Simulation results demonstrate that FL-NMLD has a considerably wider chaotic range, larger Lyapunov exponents, and superior ergodicity compared with existing chaotic maps. Based on FL-NMLD, we propose a new image encryption algorithm that joins the pixel plane and bit-plane shuffle (JPB). The simulation and test results confirm that JPB has higher security than simple pixel-plane encryption and is faster than simple bit-plane encryption. Moreover, it can resist the majority of attacks including statistical and differential attacks.

Second-Order Chaos Indicators MEGNO2 and OMEGNO2: Theory

Modifications of the Mean Exponential Growth factor of Nearby Orbits (MEGNO) linear variational method called MEGNO2 and OMEGNO2 indicators are introduced. The modifications are based on taking into account not only the linear, but also the nonlinear part of the increment of the phase flow in the divergence among nearby trajectories according to the second-order formulas. The new indicators allow one to determine more quickly the nature of the orbits under study in dynamical systems with zero or small Lyapunov exponents in comparison with the first-order variational indicators. They improve the analysis of regular regions and, in particular, periodic orbits as well as prevent the appearance of spurious structures in the resulting mappings.

Sinks with relatively large immediate basins and a refinement of Mañé’s C1 generic dichotomy

A C1 perturbation theorem creating sinks with relatively large immediate basins of attraction under the existence of a non-atomic ergodic measure admitting at most small positive Lyapunov exponents is provided. As an application, we prove a refinement of Mañé’s C1 generic dichotomy for surface diffeomorphisms, that is, C1 generically they have either (I) hyperbolicity; or (II) (by taking the inverse if necessary) infinitely many attracting periodic orbits each of which has either (a) relatively large immediate basins of attraction or (b) a pathological feature (i.e. the maximal and minimum norms of the derivatives along the periodic orbits increase and decrease exponentially, respectively, at the periods by a uniform rate). Moreover, the property (a) of (II) is considered from numerical viewpoints in the context of ‘observability’.

On nodes and modes in resting state fMRI

This paper examines intrinsic brain networks in light of recent developments in the characterisation of resting state fMRI timeseries — and simulations of neuronal fluctuations based upon the connectome. Its particular focus is on patterns or modes of distributed activity that underlie functional connectivity. We first demonstrate that the eigenmodes of functional connectivity – or covariance among regions or nodes – are the same as the eigenmodes of the underlying effective connectivity, provided we limit ourselves to symmetrical connections. This symmetry constraint is motivated by appealing to proximity graphs based upon multidimensional scaling. Crucially, the principal modes of functional connectivity correspond to the dynamically unstable modes of effective connectivity that decay slowly and show long term memory. Technically, these modes have small negative Lyapunov exponents that approach zero from below. Interestingly, the superposition of modes – whose exponents are sampled from a power law distribution – produces classical 1/f (scale free) spectra. We conjecture that the emergence of dynamical instability – that underlies intrinsic brain networks – is inevitable in any system that is separated from external states by a Markov blanket. This conjecture appeals to a free energy formulation of nonequilibrium steady-state dynamics. The common theme that emerges from these theoretical considerations is that endogenous fluctuations are dominated by a small number of dynamically unstable modes. We use this as the basis of a dynamic causal model (DCM) of resting state fluctuations — as measured in terms of their complex cross spectra. In this model, effective connectivity is parameterised in terms of eigenmodes and their Lyapunov exponents — that can also be interpreted as locations in a multidimensional scaling space. Model inversion provides not only estimates of edges or connectivity but also the topography and dimensionality of the underlying scaling space. Here, we focus on conceptual issues with simulated fMRI data and provide an illustrative application using an empirical multi-region timeseries.

Numerical analysis and circuit realization of the modified LÜ chaotic system

A novel three-dimensional autonomous chaotic system from the LÜ chaotic system is given. By using the theoretical analysis and numerical simulation, we provide an insight into the dynamic properties and characterizations of this system, such as Hopf bifurcation. In particular, we are interested in focusing on the dependence of varying parameters on chaos with the help of some chaos indicators including the fast Lyapunov indicator, small alignment indexes and Lyapunov exponent. It is shown that growing the parameter c leads to the extent of chaos. Finally, a chaotic electronic circuit is designed for the realization of the chaotic attractor with aim of Multisim software, and it gives almost the same rules of types of orbits as numerical ones by an alternating value of a circuital resistor.

Circuit Simulation of the Modified Lorenz System

We propose a novel three-dimensional autonomous chaotic system originating from the Lorenz chaotic system. With help of the theoretical analysis and the numerical simulation, the dynamic properties and characterization of this system are studied such as Hopf bifurcation etc.. In particular, fast Lyapunov indicator, small alignment indexes and Lyapunov exponent are used to estimate the effects on the dynamics, of varying parameter that correspond to ordered or chaotic orbits. It is found that increasing value of the parameter r always leads to the extent of chaos. Finally, by Multisim software, a chaotic electronic circuit is designed for the realization of the chaotic attractor, and along with the change of circuital resistor value, it gives almost the same rules of types of orbits as numerical ones.

“Weak quantum chaos” and its resistor network modeling

Weakly chaotic or weakly interacting systems have a wide regime where the common random matrix theory modeling does not apply. As an example we consider cold atoms in a nearly integrable optical billiard with a displaceable wall (piston). The motion is completely chaotic but with a small Lyapunov exponent. The Hamiltonian matrix does not look like one taken from a Gaussian ensemble, but rather it is very sparse and textured. This can be characterized by parameters s and g which reflect the percentage of large elements and their connectivity, respectively. For g we use a resistor network calculation that has a direct relation to the semilinear response characteristics of the system, hence leading to a prediction regarding the energy absorption rate of cold atoms in optical billiards with vibrating walls.

Exponentially small splitting of separatrices in the perturbed McMillan map

The McMillan map is a one-parameter family of integrable sym- plectic maps of the plane, for which the origin is a hyperbolic xed point with a homoclinic loop, with small Lyapunov exponent when the parameter is small. We consider a perturbation of the McMillan map for which we show that the loop breaks in two invariant curves which are exponentially close one to the other and which intersect transversely along two primary homoclinic orbits. We compute the asymptotic expansion of several quantities related to the splitting, namely the Lazutkin invariant and the area of the lobe between two consecutive primary homoclinic points. Complex matching techniques are in the core of this work. The coecients involved in the expansion have a resurgent origin, as shown in (14).

On a new hyperchaotic system

A new hyperchaotic system is presented, which has two large positive and one small negative Lyapunov exponent over a large range of parameters. As a consequence, system orbits strongly expand in some directions but rapidly shrink in some other directions. These strong expansions and shrinking lead the system orbits to be more disordered and random. Bifurcation and Poincaré-map analyses further show that the system has very rich bifurcations in different directions and extremely complicated dynamics overall. Spectral analysis shows that the system in the hyperchaotic mode has an extremely broad frequency bandwidth of high magnitudes, verifying its unusual random nature and indicating its great potential for some relevant engineering applications such as secure communications.

Lyapunov spectra of billiards with cylindrical scatterers: Comparison with many-particle systems

The dynamics of a system consisting of many spherical hard particles can be described as a single point particle moving in a high-dimensional space with fixed hypercylindrical scatterers with specific orientations and positions. In this paper, the similarities in the Lyapunov exponents are investigated between systems of many particles and high-dimensional billiards. The billiards contain cylindrical scatterers which have isotropically distributed orientations and homogeneously distributed positions. The dynamics of the isotropic billiards are calculated using a Monte Carlo simulation, and a reorthogonalization process is used to find the Lyapunov exponents. The results are compared to numerical results for systems of many hard particles as well as the analytical results for the high-dimensional Lorentz gas. The smallest three-quarters of the positive exponents behave more like the exponents of hard-disk systems than the exponents of the Lorentz gas. This similarity shows that the hard-disk systems may be approximated by a spatially homogeneous and isotropic system of scatterers for a calculation of the smaller Lyapunov exponents, apart from the exponent associated with localization. The method of the partial stretching factor is used to calculate these exponents analytically, with results that compare well with simulation results of hard disks and hard spheres.