Lyapunov Indicator Research Articles

Electromagnetic field and chaotic charged-particle motion around hairy black holes in Horndeski gravity

The Wald vector potential is an exact solution of the source-less Maxwell equations regarding an electromagnetic field of a vacuum uncharged black hole like the Kerr background black hole in an asymptotically uniform magnetic field. However, it is not if the black hole is a nonvacuum solution in a theory of modified gravity with extra fields or a charged Kerr–Newman spacetime. To satisfy the source-less Maxwell equations in this case, the Wald vector potential must be modified and generalized appropriately. Following this idea, we derive an expression for the vector potential of an electromagnetic field surrounding a hairy black hole in the Horndeski modified gravity theory. Explicit symplectic integrators with excellent long-term behaviour are used to simulate the motion of charged particles around the hairy black hole immersed in the external magnetic field. The recurrence plot method based on the recurrence quantification analysis uses diagonal structures parallel to the main diagonal to show regular dynamics, but adopts no diagonal structures to indicate chaotic dynamics. The method is efficient to detect chaos from order in the curved spacetime, as the Poincaré map and the fast Lyapunov indicator are.

A Kustaanheimo–Stiefel regularization of the elliptic restricted three-body problem and the detection of close encounters with fast Lyapunov indicators

We present the Kustaanheimo–Stiefel (KS) regularization of the elliptic restricted three-body problem (ER3BP) at the secondary body P2, and discuss its use to study a category of transits through its Hill’s sphere (fast close encounters). Starting from the Hamiltonian representation of the problem using the synodic rotating–pulsating reference frame and the true anomaly of P2 as independent variable, we perform the regularization at the secondary body analogous to the circular case by applying the classical KS transformation and the iso-energetic reduction in an extended 10-dimensional phase-space; this translates into an efficient algorithm that can be readily implemented to numerically integrate the equations of motion. Using such a regularized Hamiltonian we recover a definition of fast close encounters in the ER3BP for small values of the mass parameter μ (while we do not require a smallness condition on the eccentricity of the primaries), and we show that for these encounters the solutions of the variational equations are characterized by an exponential growth during the fast transits through the Hill’s sphere. Thus, for small μ, we justify the effectiveness of the regularized fast Lyapunov indicators (RFLIs) to detect orbits with multiple fast close encounters. Finally, we provide numerical demonstrations and show the benefits of the regularization in terms of the computational cost.

Re-examination of the transportation abilities of the 5:2 MMR with Jupiter

Resonances in the main asteroid belt play a significant role in the dynamical evolution of small bodies. They are capable of driving objects into the near-Earth object (NEO) region as well. This work re-examines the transportation abilities of the 5:2 mean motion resonance (MMR) with Jupiter. We focus on a greater portion of the resonance than the previous study that used a similar method. We are also interested in an elimination course along $q 0.26\,$au that was discovered previously. Moreover, we search for the orbits of potentially hazardous asteroids and for orbits that correspond to recent L chondrite meteorites. Firstly, short-term fast Lyapunov indicator maps of the 5:2 MMR were computed in order to distinguish between stable and unstable orbits. Then over $10\,000$ unstable particles were selected and integrated for a longer period of time, up to $10\,$Myr, to reveal the transportation abilities of the resonance. During our simulation, $99.45<!PCT!>$ of test particles became NEOs, $9.43<!PCT!>$ reached the orbit with a semi-major axis, $a<1\,$au, and over $27<!PCT!>$ of particles migrated to low perihelion distances, $q<0.005\,$au. In addition, $92.8<!PCT!>$ of the particles entered the Hill sphere of the Earth and over $97<!PCT!>$ reached an orbit at which we would classify them as potentially hazardous if they were sufficiently large. However, our simulation did not confirm ejections along $q 0.26\,$au. Our results suggest that there is some kind of discrepancy between using the MERCURIUS integrator (REBOUND package) and the ORBIT9 integrator (OrbFit package). This subject is worth additional examination.

Characterisation of chaos and mean-motion resonances in meteoroid streams

Context. Dynamically linking a meteor shower with its parent body is challenging, and chaos in the dynamics of meteoroid streams may contribute to this challenge. For a robust identification of parent bodies, it is therefore necessary to quantify the amount of chaos involved in the evolution of meteoroid streams. Aims. Characterising chaos in meteoroid streams through the aid of chaos maps is still a new field of study. Thus, we examine two very different meteoroid streams, the Draconids and the Leonids, in order to obtain a general view of this topic. Methods. We used the method developed in a previous paper dedicated to Geminids, drawing chaos maps with the orthogonal fast Lyapunov indicator. We chose four particle size ranges to investigate the effect of non-gravitational forces. As the dynamics are structured by mean-motion resonances with planets, we computed the locations and widths of the resonances at play. We used semi-analytical formulas valid for any eccentricity and inclination and an arbitrary number of planets. Results. We pinpoint which mean-motion resonances with Jupiter play a major role in the dynamics of each meteoroid stream. We show how those resonances tend to trap mostly large particles, preventing them from meeting with Jupiter. We also study particles that manage to escape those resonances, for example, due to the gravitational perturbation of Saturn. Finally, we explain why non-gravitational forces do not disturb the dynamics much, contrary to what is observed for the Geminids.

Circular string in a black p-brane leading to chaos

We consider a pulsating string near a non-extremal black p-brane (p=5 and p=6) and investigate the chaos in the corresponding string dynamics by examining the Fast Lyapunov indicator(FLI) and Poincare section. In our system, the energy and the charge play the role of control parameters. For generic values of these parameters, the numerical results show that the dynamics primarily fall into three modes: capture, escape to infinity, and quasiperiodic depending on the initial location (near to or far away from the black brane horizon) of the string. Finally, probing for different values of the winding number (n) the dynamics turns out to be sensitive to n. In particular, we observe the point particle (n = 0) scenario to be integrable whereas at higher n the dynamics seems to be chaotic.

Application of Symmetric Explicit Symplectic Integrators in Non-Rotating Konoplya and Zhidenko Black Hole Spacetime

In this study, we construct symmetric explicit symplectic schemes for the non-rotating Konoplya and Zhidenko black hole spacetime that effectively maintain the stability of energy errors and solve the tangent vectors from the equations of motion and the variational equations of the system. The fast Lyapunov indicators and Poincaré section are calculated to verify the effectiveness of the smaller alignment index. Meanwhile, different algorithms are used to separately calculate the equations of motion and variation equations, resulting in correspondingly smaller alignment indexes. The numerical results indicate that the smaller alignment index obtained by using a global symplectic algorithm is the fastest method for distinguishing between regular and chaotic cases. The smaller alignment index is used to study the effects of parameters on the dynamic transition from order to chaos. If initial conditions and other parameters are appropriately chosen, we observe that an increase in energy E or the deformation parameter η can easily lead to chaos. Similarly, chaos easily occurs when the angular momentum L is small enough or the magnetic parameter Q stays within a suitable range. By varying the initial conditions of the particles, a distribution plot of the smaller alignment in the X–Z plane of the black hole is obtained. It is found that the particle orbits exhibit a remarkably rich structure. Researching the motion of charged particles around a black hole contributes to our understanding of the mechanisms behind black hole accretion and provides valuable insights into the initial formation process of an accretion disk.

Applicability of the 0–1 test for chaos in magnetized Kerr–Newman spacetimes

The dynamics of electrically neutral or charged particles around a magnetized Kerr–Newman black hole immersed in an external electromagnetic field can be described by a dimensionless Hamiltonian system. This Hamiltonian is given an appropriate time transformation, which allows for construction of explicit symplectic integrators. Selecting one of the integrators with good accuracy, long-term stabilized Hamiltonian error behavior and less computational cost, we employ the 0–1 binary test correlation method to distinguish between regular and chaotic dynamics of electrically neutral or charged particles. The correlation method is almost the same as the techniques of Poincaré map and fast Lyapunov indicators in identifying the regular and chaotic two cases. It can well describe the dependence of the transition from regularity to chaos on varying one or two dynamical parameters. From a statistical viewpoint, chaos occurs easily under some circumstances with an increase of the external magnetic field strength and the particle electric charge and energy or a decrease of the black hole spin and the particle angular momentum. A small change of the black hole electric charge does not very sensitively affect the dynamics of neutral particles. With the black hole electric charge increasing, positively charged particles do not easily yield chaotic motions, but negatively charged particles do. On the other hand, the effect of a small change of the black hole magnetic charge on the dynamical transition from order to chaos has no universal rule.

Chaos Detection by Fast Dynamic Indicators in Reflecting Billiards.

The propagation of electromagnetic waves in a closed domain with a reflecting boundary amounts, in the eikonal approximation, to the propagation of rays in a billiard. If the inner medium is uniform, then the symplectic reflection map provides the polygonal rays' paths. The linear response theory is used to analyze the stability of any trajectory. The Lyapunov and reversibility error invariant indicators provide an estimate of the sensitivity to a small initial random deviation and to a small random deviation at any reflection, respectively. A family of chaotic billiards is considered to test the chaos detection effectiveness of the above indicators.

Theory and applications of fast Lyapunov indicators to model problems of celestial mechanics

Theory and applications of fast Lyapunov indicators to model problems of celestial mechanics

Application of Explicit Symplectic Integrators in the Magnetized Reissner–Nordström Spacetime

In recent works by Wu and Wang a class of explicit symplectic integrators in curved spacetimes was presented. Different splitting forms or appropriate choices of time-transformed Hamiltonians are determined based on specific Hamiltonian problems. As its application, we constructed a suitable explicit symplectic integrator for surveying the dynamics of test particles in a magnetized Reissner–Nordström spacetime. In addition to computational efficiency, the scheme exhibits good stability and high precision for long-term integration. From the global phase-space structure of Poincaré sections, the extent of chaos can be strengthened when energy E, magnetic parameter B, or the charge q become larger. On the contrary, the occurrence of chaoticity is weakened with an increase of electric parameter Q and angular momentum L. The conclusion can also be supported by fast Lyapunov indicators.

Characterisation of chaos in meteoroid streams

Context. Dynamically linking a meteor shower with its parent body can be challenging. This is in part due to the limits of the tools available today (such as D-criteria) but is also due to the complex dynamics of meteoroid streams. Aims. We choose a method to study chaos in meteoroid streams and apply it to the Geminid meteoroid stream. Methods. We decided to draw chaos maps. Amongst the chaos indicators we studied, we show that the orthogonal fast Lyapunov indicator is particularly well suited to our problem. The maps are drawn for three bin sizes, ranging from 10−1 to 10−4 m. Results. We show the influence of mean-motion resonances with the Earth and with Venus, which tend to trap the largest particles. The chaos maps present three distinct regimes in eccentricity, reflecting close encounters with the planets. We also study the effect of non-gravitational forces. We determine a first approximation of the particle size rlim needed to counterbalance the resonances with the diffusion due to the non-gravitational forces. We find that, for the Geminids, rlim lies in the range [3; 8] × 10−4 m. However, rlim depends on the orbital phase space.

Analytical and numerical estimates for solar radiation pressure semi-secular resonances

The aim of this work is to provide new insights on the dynamics associated to the resonances which arise as a consequence of the coupling of the effect due to the oblateness of the Earth and the Solar Radiation Pressure (SRP) effect for an uncontrolled object with moderate to high area-to-mass ratio. Analytical estimates for the location of the resulting resonant equilibrium points are provided, together with formulas to compute the maximum amplitude of the corresponding variation in the eccentricity, as a function of the initial conditions of the object and of its area-to-mass ratio. The period of the variations of the eccentricity and inclination due to such resonances is estimated using classical formulas. A classification based on the strength of the SRP resonances is provided. The estimates presented in the paper are validated using numerical tools, including the use of Fast Lyapunov Indicators to draw phase portraits and bifurcation diagrams. Many FLI maps depicting the location and overlapping of SRP resonances are presented. The results from this paper suggest that SRP resonances could be modeled in the context of either the Extended Fundamental Model by Breiter (1999) or the Second Fundamental Model by Henrard and Lemaitre (1983).

Surfing in the phase space of spin--orbit coupling in binary asteroid systems

ABSTRACTFor a satellite with an irregular shape, which is the common shape among asteroids, the well-known spin--orbit resonance problem could be changed to a spin--orbit coupling problem since a decoupled model does not accurately capture the dynamics of the system. In this paper, having provided a definition for close binary asteroid systems, we explore the structure of the phase space in a classical Hamiltonian model for spin--orbit coupling in a binary system. To map out the geography of resonances analytically and the cartography of resonances numerically, we reformulate a fourth-order gravitational potential function, in Poincare variables, via Stokes coefficients. For a binary system with a near-circular orbit, isolating the Hamiltonian near each resonance yields the pendulum model. Analysis of the results shows the geographical information, including the location and width of resonances, is modified due to the prominent role of the semimajor axis in the spin--orbit coupling model but not structurally altered. However, this resulted in modified Chirikov criterion to predict onset of large-scale chaos. For a binary system with arbitrary closed orbit, we thoroughly surf in the phase space via cartography of resonances created by fast Lyapunov indicator maps. The numerical study confirms the analytical results, provides insight into the spin--orbit coupling, and shows some bifurcations in the secondary resonances which can occur due to material transfer. Also, we take the (65803) Didymos binary asteroid as a case to show analytical and numerical results.

Chaotic simulation for nanofluidic particles

The analysis of ferro-nanofluidic flow is carried out for magnetic and temperature variability. Indicators such as Small Alignment Index (SALI), Fast Lyapunov Indicator (FLI) and Dynamic Lyapunov Indicator (DLI) compute regular as well as chaotic motions. LCEs & Bifurcations also carried out for Ha as well as Ra variations. Entropy quantifies probability of haphazardness in the dynamism. It is observed that for case of rectangular cavity this chaotic state begins through Ra = 26.2 indicating that at extremely high temperatures this rectangular cavity becomes randomized proving that whenever these blood vessels have non-uniform deposition even if there is regular flow, patient has a better chance at survival as randomness happens and due to non-uniformity of deposition there exist narrow in between passages that provide stable blood- flow.

A detailed dynamical model for inclination-only dependent lunisolar resonances. Effect on the “eccentricity growth” mechanism

The focus of this paper is on inclination-only dependent lunisolar resonances, which shape the dynamics of a MEO (Medium Earth Orbit) object over secular time scales (i.e. several decades). Following the formalism of Daquin et al. (2022), we discuss an analytical model yielding the correct form of the separatrices of each one of the major lunisolar resonances in the “action” space (i,e) (inclination, eccentricity) for any given semi-major axis a. We then highlight how our method is able to predict and explain the main structures found numerically in Fast Lyapunov Indicator (FLI) cartography. We focus on explaining the dependence of the FLI maps from the initial phase of the argument of perigee ω and of the longitude of the ascending node Ω of the object and of the moon ΩL. In addition, on the basis of our model, we discuss the role played by the Ω-ΩL and the 2Ω-ΩL resonances, which overlap with the inclination-only dependent ones as they sweep the region for increasing values of a, generating large domains of chaotic motion. Our results provide a framework useful in designing low-cost satellite deployment or space debris mitigation strategies, exploiting the natural dynamics of lunisolar resonances that increase an object’s eccentricity up until it reaches a domain where friction leads to atmospheric re-entry.

Chaos in a Magnetized Modified Gravity Schwarzschild Spacetime

Based on the scalar–tensor–vector modified gravitational theory, a modified gravity Schwarzschild black hole solution has been given in the existing literature. Such a black hole spacetime is obtained through the inclusion of a modified gravity coupling parameter, which corresponds to the modified gravitational constant and the black hole charge. In this sense, the modified gravity parameter acts as not only an enhanced gravitational effect but also a gravitational repulsive force contribution to a test particle moving around the black hole. Because the modified Schwarzschild spacetime is static spherical symmetric, it is integrable. However, the spherical symmetry and the integrability are destroyed when the black hole is immersed in an external asymptotic uniform magnetic field and the particle is charged. Although the magnetized modified Schwarzschild spacetime is nonintegrable and inseparable, it allows for the application of explicit symplectic integrators when its Hamiltonian is split into five explicitly integrable parts. Taking one of the proposed explicit symplectic integrators and the techniques of Poincaré sections and fast Lyapunov indicators as numerical tools, we show that the charged particle can have chaotic motions under some circumstances. Chaos is strengthened with an increase of the modified gravity parameter from the global phase space structures. There are similar results when the magnetic field parameter and the particle energy increase. However, an increase of the particle angular momentum weakens the strength of chaos.

ENTROPIC EVALUATION OF INNOVATIVE SYSTEMS ENTERPRISE DEVELOPMENT

The model of the production complex always includes the object of management and the subject of management in the form of appropriate systems. The method of realization of relations between them depends on the level of development of industrial relations. At the present stage of development of machinery and technology to the method of implementation of these relations requires their integration into a single corporate governance system. In recent years, several approaches have been used to identify the mechanisms underlying the development and operation of complex systems. Particularly useful results were obtained in their study by methods of random matrix theory, mono- and multifractal analysis, chaos theory with reconstruction of the system trajectory in phase space and determination of its parameters such as correlation dimension, spectrum of Lyapunov indicators, recurrent maps. However, the use of some of the methods requires the stationary nature of the studied data, requires long time series and a comprehensive calculation of several parameters. Another approach to considering the features of complex systems is to calculate the characteristics of entropy. The paper considers problematic issues related to the innovative development of machine-building enterprises. An algorithm for increasing the innovation potential of an industrial enterprise is proposed. The authors present a variant of assessing the systemic nature of innovative development of machine-building enterprises. The indicators of evaluation and increase of the level of innovative development of industrial enterprises are systematized. The integral index and the system entropy index are determined.

Application of Explicit Symplectic Integrators in a Magnetized Deformed Schwarzschild Black Spacetime

Abstract Following the latest work of Wu et al., we construct time-transformed explicit symplectic schemes for a Hamiltonian system on the description of charged particles moving around a deformed Schwarzschild black hole with an external magnetic field. Numerical tests show that such schemes have good performance in stabilizing energy errors without secular drift. Meantime, tangent vectors are solved from the variational equations of the system with the aid of an explicit symplectic integrator. The obtained tangent vectors are used to calculate several chaos indicators, including Lyapunov characteristic exponents, fast Lyapunov indicators, a smaller alignment index, and a generalized alignment index. It is found that the smaller alignment index and generalized alignment index are the fastest indicators for distinguishing between regular and chaotic cases. The smaller alignment index is applied to explore the effects of the parameters on the dynamical transition from order to chaos. When the positive deformation factor and angular momentum decrease, or when the energy, positive magnetic parameter, and the magnitude of the negative deformation parameter increase, chaos easily occurs for the appropriate choices of initial conditions and the other parameters.

A deep dive into the $$2g+h$$ resonance: separatrices, manifolds and phase space structure of navigation satellites

Despite extended past studies, several questions regarding the resonant structure of the medium-Earth orbit (MEO) region remain hitherto unanswered. This work describes in depth the effects of the $2g+h$ lunisolar resonance. In particular, (i) we compute the correct forms of the separatrices of the resonance in the inclination-eccentricity space for fixed semi-major axis. This allows to compute the change in the width of the $2g+h$ resonance as the altitude increases. (ii) We discuss the crucial role played by the value of the inclination of the Laplace plane, $i_{L}$. Since $i_L$ is comparable to the resonance's separatrix width, the parametrization of all resonance bifurcations has to be done in terms of the proper inclination $i_{p}$, instead of the mean one. (iii) The subset of circular orbits constitutes an invariant subspace embedded in the full phase space, the center manifold $\mathcal{C}$. Using $i_p$ as a label, we compute its range of values for which $\mathcal{C}$ becomes a normally hyperbolic invariant manifold (NHIM). The structure of invariant tori in $\mathcal{C}$ allows to explain the role of the initial phase $h$ noticed in several works. (iv) Through Fast Lyapunov Indicator (FLI) cartography, we portray the stable and unstable manifolds of the NHIM as the altitude increases. Manifold oscillations dominate in phase space between $a=24,000$ km and $a=30,000$ km as a result of the sweeping of the $2g+h$ resonance by the $h-\Omega_{\rm{Moon}}$ and $2h-\Omega_{\rm{Moon}}$ resonances. The noticeable effects of the latter are explained as a consequence of the relative inclination of the Moon's orbit with respect to the ecliptic. The role of the phases $(h,\Omega_{\rm{Moon}})$ in the structures observed in the FLI maps is also clarified. Finally,(v) we discuss how the understanding of the manifold dynamics could inspire end-of-life disposal strategies.

SIGNAL PROCESSING AND CYBERSECURITY IN SOME CHAOTIC OPTICAL COMMUNICTAION SYSTEMS

A chaos –geometric approach to investigation of complex chaotic dynamical systems is applied to an analysis, modeling and processing the time series of emission intensities of chaotic transmitter/receiver systems (two unidirectionally coupled semiconductor laser systems in the all-optical scheme) suited for encoding at rates of GBit/s. the problem of a signal processing is directly connected with the corresponding cybersecurity in some optical chaos communictaion systems. The estimated values for the dynamic and topologic invariants such as the correlation and Kaplan-York dimensions, Lyapunov indicators, Kolmogorov entropy etc for investigated chaotic signal time series of two unidirectionally coupled semiconductor laser systems in the all-optical scheme.