Positive Lyapunov Exponents Research Articles

N-dimensional hyperchaotic discrete map with desired positive Lyapunov exponents and application to UART secure communication

n-dimensional hyperchaotic discrete map with desired positive Lyapunov exponents and application to UART secure communication

Multi-winged Lorenz attractors due to bifurcations of a periodic orbit with multipliers (−1,i,−i)

Abstract We show that bifurcations of periodic orbits with multipliers ( − 1 , i , − i ) can lead to the birth of pseudohyperbolic (i.e. robustly chaotic) Lorenz-like attractors of three different types: one is a discrete analogue of the classical Lorenz attractor, and the other two are new. We call them two- and four-winged ‘Simó angels’. These three attractors exist in an orientation-reversing, three-dimensional, quadratic Hénon map. Our analysis is based on a numerical study of a normal form for this bifurcation, a three-dimensional system of differential equations with a Z4-symmetry. We investigate bifurcations in the normal form and describe those responsible for the emergence of the Lorenz attractor and the continuous-time version of the Simó angels. Both for the normal form and the 3D Hénon map, we have found open regions in the parameter space where the attractors are pseudohyperbolic, implying that for every parameter value from these regions every orbit in the attractor has positive top Lyapunov exponent.

Chaotic dynamics in a class of generalized memristive maps.

The memory effects of the memristors in nonlinear systems make the systems generate complicated dynamics, which inspires the development of the applications of memristors. In this article, the model of the discrete memristive systems with the generalized Ohm's law is introduced, where the classical Ohm's law is a linear relationship between voltage and current, and a generalized Ohm's law is a nonlinear relationship. To illustrate the rich dynamics of this model, the complicated dynamical behavior of three types of maps with three types of discrete memristances is investigated, where a cubic function representing a kind of generalized Ohm's law is used, and this cubic function is a simplified characteristic of the famous tunnel diode. The existence of attractors with one or two positive Lyapunov exponents (corresponding to chaotic or hyperchaotic dynamics) is obtained, and the coexistence of (infinitely) many attractors is observable. A hardware device is constructed to implement these maps and the analog voltage signals are experimentally acquired.

Dynamical complexity and multifractal analysis of geomagnetic activities at high temporal scales over three solar cycles

Activities in geospace occur at different time scales. Understanding geomagnetic activity at high temporal scales will give insight into fast dynamics in geospace. This study aims to investigate dynamical complexities in geomagnetic activities at a high temporal scale across three solar cycles. Geomagnetic activities, as represented by 5-min SYM-H data, were considered in this study over three solar cycles (22–24) from 1986 to 2019. Chaos analysis using sample entropy, Lyapunov exponent, and correlation dimension indicates that the geomagnetic activities are driven by intrinsic complex and chaotic processes. Positive Lyapunov exponent values between 0.13 and 0.18, 0.15–0.18, and 0.16–0.19 were obtained for solar cycles 22, 23, and 24 respectively. Furthermore, geomagnetic activities were also found to have multifractal structures driven by high fractal exponents with fine structures. A positive relationship was obtained between the annual mean values of SYM-H and the degree of complexity. It is concluded that geomagnetic activities have a short prediction horizon.

Rigidity of Lyapunov exponents for derived from Anosov diffeomorphisms

Abstract For a class of volume-preserving partially hyperbolic diffeomorphisms (or non-uniformly Anosov) $f\colon {\mathbb {T}}^d\rightarrow {\mathbb {T}}^d$ homotopic to linear Anosov automorphism, we show that the sum of the positive (negative) Lyapunov exponents of f is bounded above (respectively below) by the sum of the positive (respectively negative) Lyapunov exponents of its linearization. We show this for some classes of derived from Anosov (DA) and non-uniformly hyperbolic systems with dominated splitting, in particular for examples described by Bonatti and Viana [SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J. Math.115(1) (2000), 157–193]. The results in this paper address a flexibility program by Bochi, Katok and Rodriguez Hertz [Flexibility of Lyapunov exponents. Ergod. Th. & Dynam. Sys.42(2) (2022), 554–591].

Lyapunov exponent as a signature of dissipative many-body quantum chaos

A distinct feature of Hermitian quantum chaotic dynamics is the exponential increase of certain out-of-time-order correlation (OTOC) functions around the Ehrenfest time with a rate given by a Lyapunov exponent. Physically, the OTOCs describe the growth of quantum uncertainty that crucially depends on the nature of the quantum motion. Here, we employ the OTOC in order to provide a precise definition of dissipative quantum chaos. For this purpose, we compute analytically the Lyapunov exponent for the vectorized formulation of the large-q limit of a q-body Sachdev-Ye-Kitaev model coupled to a Markovian bath. These analytic results are confirmed by an explicit numerical calculation of the Lyapunov exponent for several values of q≥4 based on the solutions of the Schwinger-Dyson and Bethe-Salpeter equations. We show that the Lyapunov exponent decreases monotonically as the coupling to the bath increases and eventually becomes negative at a critical value of the coupling signaling a transition to a dynamics which is no longer quantum chaotic. Therefore, a positive Lyapunov exponent is a defining feature of dissipative many-body quantum chaos. The observation of the breaking of the exponential growth for sufficiently strong coupling suggests that dissipative quantum chaos may require in certain cases a sufficiently weak coupling to the environment. Published by the American Physical Society 2024

Constructing a Nondegenerate m-Dimensional Integer-Domain Chaotic Map Model over GF(2n) with Application in PRNG

To address the problem of dynamic degradation over a finite-precision platform of chaotic maps and the reversibility of linear chaotic maps, we propose an improved model over GF([Formula: see text]) that is called the nondegenerate m-Dimensional [Formula: see text] Integer-Domain Chaotic Maps (mD-IDCMs). This model incorporates modular exponentiation operation, and is capable of constructing nondegenerate IDCMs of any dimension. Moreover, we prove the irreversibility of mD-IDCM and analyze its chaotic behaviors in terms of positive Lyapunov Exponents (LEs). The results of theoretical analysis show that the proposed mD-IDCM model can obtain the desired positive LEs by appropriately configuring its coefficient matrix. Then, we present two instances, and analyze their LEs, Kolmogorov entropy, Sample entropy, Correlation dimension, and the dynamic analysis indicates that the chaotic map constructed by mD-IDCM has ergodicity within a sufficiently large chaotic range. Finally, we design a Pseudo-Random Number Generator (PRNG) with a key to verify the practicability of the mD-IDCM.

A Mañé-Manning formula for expanding measures for endomorphisms of ℙ^{𝕜}

Let k ≥ 1 k \ge 1 be an integer and f f a holomorphic endomorphism of P k ( C ) \mathbb {P}^k (\mathbb {C}) of algebraic degree d ≥ 2 d\geq 2 . We introduce a dynamical volume dimension for ergodic f f -invariant probability measures with strictly positive Lyapunov exponents. In particular, this class of measures includes all ergodic measures whose measure-theoretic entropy is strictly larger than ( k − 1 ) log ⁡ d (k-1)\log d , a natural generalization of the class of measures of positive measure-theoretic entropy in dimension 1. The volume dimension is equivalent to the Hausdorff dimension when k = 1 k=1 , but depends on the dynamics of f f to incorporate the absence of an analogue of Koebe’s theorem and the non-conformality of holomorphic endomorphisms for k ≥ 2 k\geq 2 . If ν \nu is an ergodic f f -invariant probability measure with strictly positive Lyapunov exponents, we prove a generalization of the Mañé-Manning formula relating the volume dimension, the measure-theoretic entropy, and the sum of the Lyapunov exponents of ν \nu . As a consequence, we give a characterization of the first zero of a natural pressure function for such expanding measures in terms of their volume dimensions. For hyperbolic maps, such zero also coincides with the volume dimension of the Julia set, and with the exponent of a natural (volume-)conformal measure. This generalizes results by Denker-Urbański [Nonlinearity 4 (1991), pp. 365–384; Trans. Amer. Math. Soc. 328 (1991), pp. 563–587] and McMullen [Comment. Math. Helv. 75 (2000), pp. 535–593] in dimension 1 to any dimension k ≥ 1 k\geq 1 . Our methods mainly rely on a theorem by Berteloot-Dupont-Molino [Ann. Inst. Fourier (Grenoble) 58 (2008), pp. 2137–2168], which gives a precise control on the distortion of inverse branches of endomorphisms along generic inverse orbits with respect to measures with strictly positive Lyapunov exponents.

Generalised Lyapunov stretching in a weakly chaotic open dynamical system

Abstract Chaotic dynamical systems may be characterised by a positive Lyapunov exponent, which measures the exponential rate of separation of nearby trajectories. However in a wide range of so-called weakly chaotic systems, the separation of nearby trajectories is sub-exponential, for example stretched exponential, in time; and therefore in such cases the Lyapunov exponent vanishes. When a hole is introduced in chaotic systems, the Lyapunov exponent on the system’s fractal repeller can be related to the generation of entropy and the escape rate from the system via the escape rate formalism, but no suitable generalisation exists to weakly chaotic systems. In this work we show that in a paradigmatic one-dimensional weakly chaotic iterated map, the Pomeau-Manneville map, the generation of generalised Lyapunov stretching is completely suppressed in the presence of a hole. These results are shown based on numerical evidence, and explained with a fully analytic stochastic model.

Chaotic dynamic analysis of electrical contact resistance measured in sliding current-carrying friction

This work explores the dynamic laws of electrical contact resistance (ECR) signals measured in the sliding current-carrying friction tests based on chaos theory. It is found that ECR time series have a chaotic nature, strongly verified by the positive largest Lyapunov exponent λ1. The chaos may originate from the combined friction effects on short and long timescales. Phase space analysis indicates that the ECR phase trajectory adheres to the “convergence–stability” evolution law over time, which pertains closely to energy dissipation. Hence, the chaotic attractor exists during current-carrying friction. Strikingly, wear surfaces confirm that the ECR phase trajectory evolution process corresponds to the stages of “formation–keeping” of the chaotic attractor, and also agrees with the electrical wear stages of “running-in–steady-state”. Further, the chaos quantifier variation is consistent with the phase space analysis results to identify wear transition quantitatively.

Designing five-dimensional non-degeneracy chaotic system and its application in reversible data hiding

The application of privacy protection techniques combining chaotic systems with data hiding has attracted considerable attention in recent years. However, the existing schemes have been found to be defective in terms of security and embedding capacity. This paper proposes a construction method for a 5-dimensional non-degeneracy chaotic system (5D-NDCS) to address the shortcomings of the previous approach. The 5D-NDCS possesses five positive Lyapunov exponents, and each dimension can generate chaotic sequences with strong randomness. Utilizing the pseudo-random sequence generated by the 5D-NDCS to encrypt the compressed image can enhance the security of the encrypted image. Subsequently, this paper introduces a joint bit-plane compression (JBPC) algorithm, which is based on the chaotic system and pixel value prediction technique. This algorithm effectively utilizes the pixel correlation of the original image to compress the bit sequence sufficiently, thereby reserving the hiding space. Finally, this paper proposes a reversible data hiding in encrypted images (RDHEI) scheme that supports multiple data hiders, based on the JBPC algorithm. The experimental results demonstrate the scheme’s advantages in terms of embedding rate, security, and fault tolerance.

Control and synchronisation of an extended hyperchaotic system based on robust switching control

This paper introduces a new class of hyperchaotic systems developed based on Chen's hyperchaotic system. The hyperchaotic nature of the developed system is proven through the calculation of the positive Lyapunov exponent and simulation. A robust feedback control method is then introduced to stabilise the system. To enhance the security of telecommunication channels for information exchange, a switching structure comprising extended hyperchaotic systems is presented. The synchronisation problem of the proposed structure is addressed using a robust switched linear control method. The simulation results show the effectiveness of the proposed controller in guaranteeing system stability and solving the synchronisation problem thus increasing security in telecommunication networks.

Effects of Two Quantum Correction Parameters on Chaotic Dynamics of Particles near Renormalized Group Improved Schwarzschild Black Holes

A renormalized group improved Schwarzschild black hole spacetime contains two quantum correction parameters. One parameter γ represents the identification of cutoff of the distance scale, and another parameter Ω stems from nonperturbative renormalization group theory. The two parameters are constrained by the data from the shadow of M87* central black hole. The dynamics of electrically charged test particles around the black hole are integrable. However, when the black hole is immersed in an external asymptotically uniform magnetic field, the dynamics are not integrable and may allow for the occurrence of chaos. Employing an explicit symplectic integrator, we survey the contributions of the two parameters to the chaotic dynamical behavior. It is found that a small change of the parameter γ constrained by the shadow of M87* black hole has an almost negligible effect on the dynamical transition of particles from order to chaos. However, a small decrease in the parameter Ω leads to an enhancement in the strength of chaos from the global phase space structure. A theoretical interpretation is given to the different contributions. The term with the parameter Ω dominates the term with the parameter γ, even if the two parameters have same values. In particular, the parameter Ω acts as a repulsive force, and its decrease means a weakening of the repulsive force or equivalently enhancing the attractive force from the black hole. On the other hand, there is a positive Lyapunov exponent that is universally given by the surface gravity of the black hole when Ω≥0 is small and the external magnetic field vanishes. In this case, the horizon would influence chaotic behavior in the motion of charged particles around the black hole surrounded by the external magnetic field. This point can explain why a smaller value of the renormalization group parameter would much easily induce chaos than a larger value.

Sinusoidal oscillator parametrically forced to robust hyperchaotic states: the lumpkin case

The objective of this paper is to showcase the capability of the conventional circuit structure known as the Lumpkin oscillator, widely employed in practical applications, to operate in robust chaotic or hyperchaotic steady states. Through numerical analysis, we demonstrate that the generated signals exhibit a significant level of unpredictability and randomness, as evidenced by positive Lyapunov exponents, approximate entropy, recurrence plots, and other indicators of complex dynamics. We establish the structural stability of strange attractors through design and practical construction of a flow-equivalent fourth-order chaotic oscillator, followed by experimental measurements. The oscilloscope screenshots captured align well with the plane projections of the approximate solutions derived from the underlying mathematical models.

Lyapunov analysis of data-driven models of high dimensional dynamics using reservoir computing: Lorenz-96 system and fluid flow

We computed the Lyapunov spectrum and finite-time Lyapunov exponents of a data-driven model constructed using reservoir computing. This analysis was performed for two dynamics that exhibit a highly dimensionally unstable structure. We focused on the reconstruction of heterochaotic dynamics, which are characterized by the coexistence of different numbers of unstable dimensions. This was achieved by computing fluctuations in the number of positive finite-time Lyapunov exponents.

Анализ регулярности/хаотичности динамики шаровых скоплений в центральной области Млечного Пути

We analyzed the regularity/chaoticity of the orbits of 45 globular clusters in the central region of the Galaxy with a radius of 3.5 kpc, which are subject to the greatest influence from an elongated rotating bar. Various analysis methods were used, namely, methods for calculating the maximum characteristic Lyapunov exponents, the method of Poincaré sections, the frequency method based on the calculation of fundamental frequencies, as well as the visual assessment method. Bimodality was discovered in the histogram of the distribution of positive Lyapunov exponents, calculated in the classical version, without renormalization of the shadow orbit, which makes it possible to implement a probabilistic method of GC classification. To construct the orbits of globular clusters, we used a gravitational potential model with a bar in the form of a triaxial ellipsoid, described in detail in the work of Bajkova et al., Izvestiya GAO in Pulkovo, 2023, 228, 1. The following bar parameters were adopted: mass 1010M⊙, semimajor axis length 5 kpc, bar viewing angle 25o, rotation speed 40 km s−1 kpc−1. To form the 6D-phase space required for orbit integration, the most accurate astrometric data to date from the Gaia satellite (EDR3) (Vasiliev, Baumgardt, 2021), as well as new refined average distances to globular clusters (Baumgardt, Vasiliev, 2021) were used. A classification is made of globular clusters with regular and chaotic dynamics. As the analysis showed, globular clusters with small pericentric distances and large eccentricities are most susceptible to the influence of the bar and demonstrate the greatest chaos. It is shown that the results of classifying globular clusters by the nature of their orbital dynamics, obtained using the various methods of analysis considered in the work, correlate well with each other.

Individual chaotic behaviour of the S-stars in the Galactic centre

Located at the core of the Galactic centre, the S-star cluster serves as a remarkable illustration of chaos in dynamical systems. The long-term chaotic behaviour of this system can be studied with gravitational N-body simulations. By applying a small perturbation to the initial position of star S5, we can compare the evolution of this system to its unperturbed evolution. This results in two solutions that diverge exponentially, defined by the separation in position space δr, with an average Lyapunov timescale of ∼420 yr, corresponding to the largest positive Lyapunov exponent. Even though the general trend of the chaotic evolution is governed in part by the supermassive black hole Sagittarius A∗ (Sgr A∗), individual differences between the stars can be noted in the behaviour of their phase-space curves. We present an analysis of the individual behaviour of the stars in this Newtonian chaotic dynamical system. The individuality of their behaviour is evident from offsets in the position space separation curves of the S-stars and the black hole. We propose that the offsets originate from the initial orbital elements of the S-stars, where Sgr A∗ is considered in one of the focal points of the Keplerian orbits. Methods were considered to find a relation between these elements and the separation in position space. Symbolic regression provides the clearest diagnostics for finding an interpretable expression for the problem. Our symbolic regression model indicates that ⟨δr⟩ ∝ e2.3, implying that the time-averaged individual separation in position space increases rapidly with the initial eccentricity of the S-stars.

A non-degenerate n-dimensional integer domain chaotic map model with application to PRNG

To address the limitations of existing chaotic maps, we proposed a non-degenerate n-dimensional (n ≥ 2) integer domain chaotic map (nD-IDCM) model that can construct any non-degenerate n-dimensional integer domain chaotic maps. Moreover, we analyzed its chaotic behavior through Lyapunov exponent, and found that the nD-IDCM generates chaotic sequences in the integer domain, which effectively resolves the issue of finite precision effect when implementing existing chaotic maps on computers or digital devices. To verify the effectiveness of nD-IDCM, we presented two instances to demonstrate how the positive Lyapunov exponents can be regulated by manipulating the parameter matrix. Subsequently, we have scrutinized their dynamical behavior using Kolmogorov entropy, sample entropy, correlation dimension and randomness testing via TestU01. Finally, to assess the feasibility of nD-IDCM, we devised a keyed pseudo random number generator (PRNG) based on a 3D-IDCM that can ensure superior randomness and unpredictability. Experimental results indicated that integer domain chaotic maps constructed using nD-IDCM have desirable Lyapunov exponents and exhibit ergodicity within a sufficient larger chaotic range.

Strong Probabilistic Stability in Holomorphic Families of Endomorphisms of ℙk (ℂ) and Polynomial-Like Maps

Abstract We prove that, in stable families of endomorphisms of $\mathbb{P}^{k} (\mathbb C)$, the measurable holomorphic motion of the Julia sets introduced by Berteloot, Dupont, and the first author is unbranched at almost every point with respect to all measures on the Julia set with strictly positive Lyapunov exponents and not charging the post-critical set. This provides a parallel in this setting to the probabilistic stability of Hénon maps by Berger–Dujardin–Lyubich. An analogous result holds in families of polynomial-like maps of large topological degree. In this case, we also give a sufficient condition for the positivity of the Lyapunov exponents of an ergodic measure in terms of its measure-theoretic entropy, generalizing to this setting an analogous result by de Thélin and Dupont valid on $\mathbb{P}^{k} (\mathbb C)$.

Analysis of total electron content over the African low-latitude region during the maximum phase of solar cycle 24 (2012–2014)

This study examined the ionospheric variation over the African low-latitude region using Total Electron Content (TEC) and its wavelet power spectrum (WPS). TEC data were obtained from five Global Positioning System (GPS) stations located in the trough and crests of the equatorial ionization anomaly (EIA) in both hemispheres during quiet periods of the maximum phase of solar cycle 24 (2012–2014). Our results revealed a striking equinoctial asymmetry (EA) with stronger TEC and higher solar flux in September equinox than March equinox of 2012–2013, with the reverse being the case in 2014. As such, the difference in solar flux between both equinoxes might have played a dominant role in driving the EA. Furthermore, we found a hemispheric asymmetry in TEC with higher value towards the crest in the northern hemisphere (southern hemisphere) in September equinox (March equinox) of 2012–2013. In 2014, nevertheless, TEC was remarkably higher towards the crest located in the southern hemisphere in September equinox only. The chaotic time series methods used to examine the TEC variations showed positive Lyapunov exponents (LE) at all stations. This indicated the presence of chaoticity in the African EIA. Nevertheless, stations located towards the southern crest (Eldoret and Dodoma) had the maximum LE values while those at the magnetic equator (Adis Ababa) and northern crest (Nama and Al Wajh) had the minimum values. Consequently, the changes in the internal dynamics of the equatorial ionosphere have impacts on its chaotic behavior.