Sum Of Lyapunov Exponents Research Articles

Generating rotationally hidden attractive sea via a new chaotic system with two mixed memristors

In this work, a novel 3D memristive chaotic system which has an exponential function is proposed. Especially, the sum of Lyapunov exponents in the proposed system is 0. It indicates that the system can generate attractive sea not attractor. In comparison with some other 3D chaotic systems, this type of chaotic system is relatively rare. In particular, the proposed system has non-equilibrium point, and it can produce hidden sea. Furthermore, the perpetual point of the proposed system is caculated. It is considered to be potentially related to the generation of hidden dynamics. By using the dynamic analysis tool such as 0–1 test and 2D dynamical map, the dynamic behaviors with different control parameters are analyzed. And then, based on the proposed 3D chaotic system, two new system models are reconstructed. The new model can produce the rotational hidden attractive sea with different angles. DSP implementation shows the feasibility of the system for industrial applications.

Partially hyperbolic random dynamics on Grassmannians

A sequence of invertible matrices given by a small random perturbation around a fixed diagonal partially hyperbolic matrix induces a random dynamics on the Grassmann manifolds. Under suitable weak conditions, it is known to have a unique invariant (Furstenberg) measure. The main result gives concentration bounds on this measure, showing that with high probability, the random dynamics stays in the vicinity of stable fixed points of the unperturbed matrix, in a regime where the strength of the random perturbation dominates the local hyperbolicity of the diagonal matrix. As an application, bounds on sums of Lyapunov exponents are obtained.

On fibrations approaching the Arakelov equality

The sum of Lyapunov exponents L_f of a semi-stable fibration is the ratio of the degree of the Hodge bundle by the Euler characteristic of the base. This ratio is bounded from above by the Arakelov inequality. Sheng-Li Tan showed that for fiber genus gge 2 the Arakelov equality is never attained. We investigate whether there are sequences of fibrations approaching asymptotically the Arakelov bound. The answer turns out to be no, if the fibration is smooth, or non-hyperelliptic, or has a small base genus. Moreover, we construct examples of semi-stable fibrations showing that Teichmüller curves are not attaining the maximal possible value of L_f.

A New Four-Dimensional Non-Hamiltonian Conservative Hyperchaotic System

This paper presents a new four-dimensional non-Hamiltonian conservative hyperchaotic system. In the absence of equilibrium points in the system, the phase trajectories generated by the system have hidden features. The conservative features that vary with the parameter have been analyzed in detail by Lyapunov exponent spectrum, bifurcation diagram, the sum of Lyapunov exponents, and the fractional dimensions, and during the analysis, multiple quasi-periodic four-dimensional tori as well as hyperchaotic attractors have been observed. The Poincaré sections confirm these dynamic behaviors. Amidst the process of studying the dynamical behavior of the system with initial values, the hidden extreme multistability, and the initial offset boosting behavior, the results have been witnessed for the very first time in a conservative chaotic system. The phase diagram and attraction basin also confirm this assertion, while two complex transient transition behaviors have been observed. Moreover, through the introduction of a spectral entropy algorithm, the complexity analysis of the time sequences generated by the system have been performed and compared with the existing literature. The results show that the system has a high degree of complexity. The design and construction of the analog circuit of the system for simulation, the circuit experimental results are consistent with the numerical simulation, further verifying the physical realizability of the newly proposed system. This lays a good foundation for its practical application in engineering.

The third type of chaos in a system of two adaptively coupled phase oscillators.

We study a new type of attractor, the so-called reversible core, which is a mathematical image of mixed dynamics, in a strongly dissipative time-irreversible system of two adaptively coupled phase oscillators. The existence of mixed dynamics in this system was proved in our previous article [A. A. Emelianova and V. I. Nekorkin, Chaos 29, 111102 (2019)]. In this paper, we attempt to identify the dynamic mechanisms underlying the existence of mixed dynamics. We give the region of the existence of mixed dynamics on the parameter plane and demonstrate in what way, when a type of attractor changes, its main characteristics, such as its fractal dimension and the sum of Lyapunov exponents, transform. We demonstrate that when mixed dynamics appear in the system, the average frequencies of the oscillations in forward and reverse time begin to almost coincide, and its spectra gradually approach each other with an increase in the parameter responsible for the presence of mixed dynamics.

Entropy variation and Lyapunov exponents of a nonequilibrium system

A macroscopic state space is introduced to describe macro-states of a nonequilibrium system. The space is spanned by averages of basic variables of the system. The entropy variation rate of the system is derived as average trace of an entropy source matrix which is a term in a memory matrix of the system. The entropy variation and Lyapunov exponents are shown determined by the non-uniformity in time and in the macro-state space of fluctuation force associated with each variable of the system. This indicates a fundamental relationship between fluctuations and dissipation. It is shown that for an ergodic system, the sum of Lyapunov exponents is equal to the time-rate of information entropy or average trace of the entropy source matrix. The study is illustrated by a simple fluid.

Covers of the projective line and the moduli space of quadratic differentials

Consider the Hurwitz space parameterizing covers of \({\mathbb{P}^1}\) branched at four points. We study its intersection with divisor classes on the moduli space of curves. As applications, we calculate the slope of Teichmuller curves parameterizing square-tiled cyclic covers. In addition, we come up with a relation among the slope of Teichmuller curves, the sum of Lyapunov exponents and the Siegel–Veech constant for the moduli space of quadratic differentials, which yields information for the effective cone of the moduli space of curves.

Structure-preserving algorithms for the Duffing equation

In this paper, the dissipative and the forced terms of the Duffing equation are considered as the perturbations of nonlinear Hamiltonian equations and the perturbational effect is indicated by parameter ε. Firstly, based on the gradient-Hamiltonian decomposition theory of vector fields, by using splitting methods, this paper constructs structure-preserving algorithms (SPAs) for the Duffing equation. Then, according to the Liouville formula, it proves that the Jacobian matrix determinants of the SPAs are equal to that of the exact flow of the Duffing equation. However, considering the explicit Runge–Kutta methods, this paper finds that there is an error term of order p+1 for the Jacobian matrix determinants. The volume evolution law of a given region in phase space is discussed for different algorithms, respectively. As a result, the sum of Lyapunov exponents is exactly invariable for the SPAs proposed in this paper. Finally, through numerical experiments, relative norm errors and absolute energy errors of phase trajectories of the SPAs and the Heun method (a second-order Runge–Kutta method) are compared. Computational results illustrate that the SPAs are evidently better than the Heun method when ε is small or equal to zero.

Construction of measures with expansion

We give a construction of measures with partial sum of Lyapunov exponents bounded from below.

Non-absolutely continuous foliations

We consider a partially hyperbolic diffeomorphism of a compact smooth manifold preserving a smooth measure. Assuming that the central distribution is integrable to a foliation with compact smooth leaves we show that this foliation fails to have the absolute continuity property provided that the sum of Lyapunov exponents in the central direction is not zero on a set of positive measure. We also establish a more general version of this result for general foliations with compact leaves.

UNIFORM LYAPUNOV EXPONENTS ARE REALISED BY ERGODIC INVARIANT MEASURES

Given a differentiable random dynamical system on a finite-dimensional differentiable manifold with a compact random invariant set, we show that there exists an ergodic invariant measure, supported by the invariant set, such that the leading Lyapunov exponent associated with this invariant measure equals the uniform Lyapunov exponent with respect to the invariant set. This is extended to sums of Lyapunov exponents and to the Lyapunov dimension of the set.

THE INSTANTANEOUS LYAPUNOV EXPONENT AND ITS APPLICATION TO CHAOTIC DYNAMICAL SYSTEMS

Any system containing at least one positive Lyapunov exponent is defined to be chaotic and the system dynamics become unpredictable. For a mechanical system, the sum of Lyapunov exponents is negative and related to the damping, and so can be utilised to monitor any changes of the damping mechanism. However, in order to track any changes the data segment used to compute the Lyapunov exponents must be short. This leads to problems since Lyapunov exponents are calculated from a long term averaged divergence rate. To overcome this problem, it is described how the sum of Instantaneous Lyapunov exponents is related to the generalised divergence of the flow and the damping of a mechanical system. Computer simulations using differential equations are implemented to demonstrate the significance of the sum of Instantaneous Lyapunov exponents. In practice, it may be difficult to obtain accurate Instantaneous Lyapunov exponents are introduced to overcome this. The sum of short term averaged Lyapunov exponents from a time series (from experimental data), and so short term averaged Lyapunov exponents is applied to experimental data to detect evolutionary changes in damping from high to low.

Sums of Lyapunov exponents for some polynomial maps of ${\bf C}^2$

We give a formula for the sum of the Lyapunov exponents of a nondegenerate polynomial map $f$ of ${\bf C}^{2}$ close to $(z,w)\to (z^d,w^d)$. The formula only involves the behavior of $f$ at infinity. In particular, it follows that the sum only depends on the homogeneous part of $f$ of degree $d$.

Lyapunov Instability of the Boundary-Driven Chernov–Lebowitz Model for Stationary Shear Flow

We report on the computation of full Lyapunov spectra of the boundary-driven Chernov–Lebowitz model for stationary planar shear flow. The Lyapunov exponents are calculated with a recently developed formalism for systems with elastic hard collisions. Although the Chernov–Lebowitz model is strictly energy conserving, any phase-space volume is subjected to a contraction due to the reflection rules of the hard disks colliding with the walls. Consequently, the sum of Lyapunov exponents is negative. As expected for an inhomogeneously driven system, the Lyapunov spectra do not obey the conjugate pairing rule. The external driving makes the system less chaotic, which is reflected in a decrease of the Kolmogorov–Sinai entropy if the driving is increased.