Lyapunov Characteristic Exponents Research Articles

Bifurcation analysis, quasi-periodic and chaotic behavior of generalized Pochhammer-Chree equation

We take into account the bifurcation analysis of the generalized Pochhammer-Chree (PC) equation that describes the dynamics of several systems in science and engineering. The considered model is changed into a planar dynamical system by applying the Galilean transformation. The phase portraits are plotted by considering suitable values of the bifurcation parameters. The considered model is solved using the RK method to compute the supernonlinear and nonlinear wave solutions. All phase portraits and wave solutions are depicted in the phase plane by simply fixing the relevant parameters values. The equilibrium points are obtained, and the same are classified. Moreover, sensitive analysis for different initial value problems is applied to analyze the quasiperiodic, chaotic behavior and time series after introducing an extrinsic periodic perturbation term. In addition, the Lyapunov characteristic exponents, Poincare section and bifurcation diagrams are also discussed to examine the chaotic pattern of the model. Numerical simulation results show that changing the frequencies and amplitude values impacts the dynamical features of the considered model.

Rich phenomenology of the solutions in a fractional Duffing equation

AbstractIn this paper, we characterize the chaos in the Duffing equation with negative linear stiffness and a fractional damping term given by a Caputo fractional derivative of order $$\alpha $$ α ranging from 0 to 2. We use two different numerical methods to compute the solutions, one of them new. We discriminate between regular and chaotic solutions by means of the attractor in the phase space and the values of the Lyapunov Characteristic Exponents. For this, we have extended a linear approximation method to this equation. The system is very rich with distinct behaviours. In the limits $$\alpha $$ α to 0 or $$\alpha $$ α to 2, the system tends to basically the same undamped system with a behaviour clearly different from the classical Duffing equation.

Rotorcraft stability analysis using Lyapunov characteristic exponents estimated from multibody dynamics

AbstractStability analysis of complex, nonlinear dynamical systems is a challenge. The use of Lyapunov Characteristic Exponents through a Jacobian-less method is proposed as a means to identify the Maximum Lyapunov Characteristic Exponent, namely the fundamental stability indicator of a generic problem, solely from time series obtained through general-purpose multibody dynamics simulations of complex rotorcraft aeromechanics models. The method is first applied to a relatively simple scenario concerning the identification of ground resonance. Then, its application to more complex models is addressed by studying the aeroelastic stability and identifying the whirl flutter of the XV-15 tiltrotor using a comprehensive aeroelastic model.

Confined chaos and the chaotic angular motion of Atlas, a Saturn’s inner satellite

ABSTRACT The dynamics of the Solar system exhibit inherent chaos and instability. Mathematical tools, such as the maximum Lyapunov characteristic exponent (LCE) and Lyapunov time (TL), play a crucial role in providing a qualitative understanding of chaos within celestial objects, such as asteroids and moonlets. Celestial bodies with relatively small Lyapunov times have garnered significant research interest due to their stable orbits, a phenomenon referred to as stable or confined chaos. Notable examples include Saturn’s satellites: Atlas, with a Lyapunov time on the order of 10 yr, Prometheus, and Pandora. This work aims to study the chaotic behaviour of the Atlas satellite and its relatively small TL. We present a three-dimensional model approach designed to isolate the radial contribution from the LCE and assess its influence within the LCE. Our investigation focuses on the Saturn system, comprising Saturn itself, along with its satellites Atlas, Prometheus, Pandora, and Mimas. To estimate the radial contribution of the LCE, we find the projection of the radial vector of a ghost Atlas (a slightly displaced Atlas) onto the Atlas radial vector, which allows us to calculate the difference between the radial vectors. This methodology enables us to estimate the radial contribution of the LCE and calculate the Lyapunov time. Remarkably, our results demonstrate that orbits remain confined even for integration times exceeding TL. Furthermore, we investigate the temporal behaviour of Atlas’ angular position in its orbit, potentially shedding light on chaotic angular dynamics.

Dual-mode coupling resonance and dynamic stability of axially moving ferromagnetic thin plate strips in alternating magnetic field

This paper investigates the dual-mode coupling resonance and dynamic stability of axially moving ferromagnetic thin plate strips in alternating magnetic field. According to the basic electromagnetic theory and the magnetoelastic mechanics model, Lorentz electromagnetic force and torque generated by eddy current effect and the magnetizing force generated by magnetization are determined for the ferromagnetic plate strip. Introducing Von Karman geometric nonlinearity, the nonlinear governing equation of axially moving ferromagnetic plate strips under the combined action of mechanical load and magnetizing force is established by Hamilton's principle. For the simply supported plate strip, Galerkin method and the multi-scale method are employed to solve the modal coupling resonance dominated by electromagnetic excitation, obtaining the amplitude-frequency response equations. The stability of resonance responses is analyzed by the Lyapunov stability theory. Through parametric research, the amplitude-frequency characteristic curves, bifurcation diagrams and Lyapunov exponent spectra of each mode are plotted; the effect of different parameters on vibration characteristics and dynamic stability under the modal coupling is analyzed for the system. The results show that as one of the modal amplitudes enters the resonance region, the other modal amplitude decays into the "collapse" region due to energy exchange between modes; each modal amplitudes has more complex multi-valued and jumping phenomena for the nonlinear coupling effect. The increase in velocity will affect the stability of system; the enhancement of magnetic field will make the system change into chaos at high velocity.

ON THE CORRECT COMPUTATION OF ALL LYAPUNOV EXPONENTS IN HAMILTONIAN DYNAMICAL SYSTEMS

The Lyapunov characteristic exponents are a useful indicator of chaos in astronomical dynamical systems. They are usually computed through a standard, very effcient and neat algorithm published in 1980. However, for Hamiltonian systems the expected result of pairs of opposite exponents is not always obtained with enough precision. We find here why in these cases the initial order of the deviation vectors matters, and how to sort them in order to obtain a correct result.

Synchronization Analysis of a Master-Slave BEC System via Active Control

This paper will focus on theoretical treatment of the dynamic of the Bose-Einstein Condensate (BEC) systems contained different external trapping potentials. We construct the phase space diagrams and Lyapunov Characteristic Exponents (LCEs) for master and slave systems depended on the system parameters and propose a nonlinear control for the synchronization of systems in their chaotic states. The synchronization is obtained in master-slave scheme for different initial values. Numerical results are also given to show the efficiency of the used control technique. 

Quantitative Aeroelastic Stability Prediction of Wings Exhibiting Nonlinear Restoring Forces

In engineering practice, eigen-solution is used to assess the stability of linear dynamical systems. However, the linearity assumption in dynamical systems sometimes implies simplifications, particularly when strong nonlinearities exist. In this case, eigen-analysis requires linerisation of the problem and hence fails to provide a direct stability estimation. For this reason, a more reliable tool should be implemented to predict nonlinear phenomena such as chaos or limit cycle oscillations. One method to overcome this difficulty is the Lyapunov Characteristic Exponents (LCEs), which provide quantitative indications of the stability characteristics of dynamical systems governed by nonlinear time-dependent differential equations. Stability prediction using Lyapunov Characteristic Exponents is compatible with the eigen-solution when the problem is linear. Moreover, LCE estimations do not need a steady or equilibrium solution and they can be calculated as the system response evolves in time. Hence, they provide a generalization of traditional stability analysis using eigenvalues. These properties of Lyapunov Exponents are very useful in aeroelastic problems possessing nonlinear characteristics, which may significantly alter the aeroelastic characteristics, and result in chaotic and limit cycle behaviour. A very common nonlinearity in flexible systems is the nonlinear restoring force such as cubic stiffness, which would substantially benefit from using LCEs in stability assessment. This work presents the quantitative evaluation of aeroelastic stability indicators in the presence of nonlinear restoring force. The method is demonstrated on a two-dimensional aeroelastic problem by comparing the system behaviour and estimated Lyapunov Exponents.

Stability Analysis of Nonlinear Rotating Systems Using Lyapunov Characteristic Exponents Estimated From Multibody Dynamics

Abstract The use of Lyapunov characteristic exponents to assess the stability of nonlinear, time-dependent mechanical systems is discussed. Specific attention is dedicated to methods capable of estimating the largest exponent without requiring the Jacobian matrix of the problem, which can be applied to time histories resulting from simulations performed with existing multibody solvers. Helicopter ground resonance is analyzed as the reference application. Improvements over the available literature are: the problem is formulated in physical coordinates, without eliminating periodicity through multiblade coordinates; the rotation of the blades is not linearized; the problem is modeled considering absolute positions and orientations of parts. The dynamic instability that arises at some angular velocities when the isotropy of the rotor is broken (e.g., caused by the failure of one lead-lag damper, a design test condition) is observed to evolve into a large amplitude limit cycle, where the usual Floquet–Lyapunov analysis of the linearized time-periodic simply predicts instability.

A theoretical framework for nonlinear dynamics in finance systems: Implications for stability, control and synchronization

It has been widely observed that most deterministic dynamical systems go into chaos for some values of their parameters. One of the most popular and widely used criteria is the conditional Lyapunov exponents, which constitute average measurements of expansion or shrinkage of small displacements along the synchronized trajectory. The Lyapunov characteristic exponents play a crucial role in the description of the behaviour of dynamical systems as they can be used to analyse the stability limit sets and to check sensitive dependence on initial conditions, that is, the presence of chaotic attractors. In this paper, Lyapunov stability theory and linear matrix inequalities (LMI) are employed to design control functions for the respective, control, and synchronization of the chaotic and hyperchaotic finance systems. The designed linear matrix inequalities (LMI) nonlinear controllers are capable of stabilizing the chaotic and hyperchaotic finance systems at any position as well as controlling it to track any trajectory that is a smooth function of time. The respective chaotic attractors were found to have a moderate value of the largest Lyapunov exponents (0.874959 s^(-1) and 0.650847 s^(-1)) with associated (Lyapunov) dimensions of 1.00 and 2.00 for the chaotic and hyperchaotic finance systems respectively. Based on Lyapunov stability theory and linear matrix inequalities (LMI), some necessary and sufficient criteria for stable synchronous behaviour are obtained and an exact analytic estimate of the threshold coupling, k_th, for complete chaos synchronization is derived. Finally, numerical simulation results are presented to validate the feasibility of the theoretical analysis.

PREDICTIVE ANALYSIS OF TIME SERIES WITH LONG-TERM MEMORY

Predictive analysis with the use of nonlinear dynamics methods, chaos theory of real time series, characterizing the prevalence of a certain class of skin pathologies in Ukraine was carried out. The basis for such studies is the Tackens' theorem. The considered time series does not conform to the normal distribution law, and the hypothesis of a trend is not confirmed for them. According to previous studies, the authors calculated the fractality index \mu, the value of which indicates the state of relative stability of the studied process. The assessment of the correlation confirmed the practical absence of the influence of the present on the future in the studied numerical series. The bifurcation of the attractor revealed during the construction of the time series phase portrait allows the system to have such changes in its state that can be interpreted as jump-like or close to them. The value of the Lyapunov characteristic exponent confirms that the trajectory of the studied time series is chaotic. In the presented research, the procedure of qualitative analysis of the time series was carried out. Using the R/S procedure of fractal analysis, the effect of the time series long-term memory was revealed, the «depth of memory of the time series starting point» and the Hurst index were estimated. According to the performed calculations, the behavior of the H trajectory and R/S trajectory is such that it gives grounds to assert that the time series has a long-term memory. When carrying out the R/S procedure of fractal analysis of the time series as a whole, the time series of the family Q(X) of the initial time series X(t) were analyzed. The distribution of the memory depth estimates was constructed, the fuzzy set L(Q(X)) describing «memory depth of the time series X(t)» as a whole was formed, which is received from a sequence of pairs {1;\mu(1)}, where \mu(1) is the value of the membership function «depth l» of the fuzzy set L(Q(X)). The presence in the time series of the long-term memory effect makes it possible to apply the method of cellular automata in predicting its values.

Chaotic dynamics from coupled magnetic monodomain and Josephson current.

The ordinary (superconductor-insulator-superconductor) Josephson junction cannot exhibit chaos in the absence of an external ac drive, whereas in the superconductor-ferromagnet-superconductor Josephson junction, known as the φ_{0} junction, the magnetic layer effectively provides two extra degrees of freedom that can facilitate chaotic dynamics in the resulting four-dimensional autonomous system. In this work, we use the Landau-Lifshitz-Gilbert model for the magnetic moment of the ferromagnetic weak link, while the Josephson junction is described by the resistively capacitively shunted-junction model. We study the chaotic dynamics of the system for parameters surrounding the ferromagnetic resonance region, i.e., for which the Josephson frequency is reasonably close to the ferromagnetic frequency. We show that, due to the conservation of magnetic moment magnitude, two of the numerically computed full spectrum Lyapunov characteristic exponents are trivially zero. One-parameter bifurcation diagrams are used to investigate various transitions that occur between quasiperiodic, chaotic, and regular regions as the dc-bias current through the junction, I, is varied. We also compute two-dimensional bifurcation diagrams, which are similar to traditional isospike diagrams, to display the different periodicities and synchronization properties in the I-G parameter space, where G is the ratio between the Josephson energy and the magnetic anisotropy energy. We find that as I is reduced the onset of chaos occurs shortly before the transition to the superconducting state. This onset of chaos is signaled by a rapid rise in supercurrent (I_{S}⟶I) which corresponds, dynamically, to increasing anharmonicity in phase rotations of the junction.

Об орбитальной устойчивости спутников экзопланет

By calculating the Lyapunov characteristic exponents, an analysis was made of the stability of the secular orbital dynamics of satellites for a number of exoplanets in which the presence of an exomoon is possible. Three planets have been identified in which the existence of satellites with stable dynamics is possible: KOI-268, Kepler-1000b and Kepler-1442b. Estimates of the possible values of the exomoon orbital parameters are obtained. The dynamics of the satellites of the planets Kepler-1625b and Kepler-1708b already discovered from the analysis of observations is considered in detail. It is established that the semi-major axis of the orbit of the planet’s satellite Kepler-1625b probably has values from 10 a.u. up to 25 a.u. The solution available for the satellite of the planet Kepler-1708b (Kipping et al., 2022) corresponds to a stable satellite orbit.

Semi-analytical investigations on the dynamics of BeiDou inclined geosynchronous satellite orbit

A semi-analytical investigation on the dynamics of BeiDou IGSO is conducted in this paper. First, a semi-analytical propagator is constructed based on the mean element method. The comparison of results of semi-analytical propagation with reference orbits shows that all orbital elements except the semi-major axis agree well within six centuries. For some example orbits, the semi-analytical propagation of the semi-major axis is accurate only within several decades, which is related to the chaotic motion of the one-to-one tesseral resonance. Compared with the numerical ephemeris of Moon, an improvement of accuracy in the semi-analytical propagation is observed when a simple analytical ephemeris of Moon is adopted. This phenomenon agrees with the doubly averaging process of the third-body perturbation. Second, the dynamics of BeiDou IGSO are investigated using the Hamiltonian approach with simplified models. The transition between different resonance regions of one-to-one tesseral resonance is investigated by the computation of one-dimensional maximal Lyapunov characteristic exponent (1-mLCE). The secular resonance from the lunisolar perturbation modeled by a 2-DOF system is systematically investigated using the surface of section. The secondary resonance related to the precession of lunar node is identified by the computation of 1-mLCE. In the end, the reentry disposal for BeiDou IGSO is investigated as an optimization problem. With IGSO3 as an example of reentry orbit, the sensitivity analysis is conducted to study its stability and predictability.

A new way to compute the Lyapunov characteristic exponents for non-smooth and discontinues dynamical systems

One of the most important problems of nonlinear dynamics is related to the development of methods concerning the identification of the dynamical modes of the corresponding systems. The classical method is related to the calculation of the Lyapunov characteristic exponents (LCEs). Usually, to implement the classical algorithms for the LCEs calculation, the smoothness of the right-hand sides of the corresponding equations is required. In this work, we propose a new algorithm for the LCEs computation in systems with strong nonlinearities (these nonlinearities cannot be linearized) including the hysteresis. This algorithm uses the values of the Jacobi matrix in the vicinity of singularities of the right-hand sides of the corresponding equations. The proposed modification of the algorithm is also can be used for systems containing such design hysteresis nonlinearity as the Preisach operator. (Thus, this modification can be used for all members of the hysteresis family.) Moreover, the proposed algorithm can be successfully applied to the well-known chaotic systems with smooth nonlinearities. Examples of dynamical systems with hysteresis nonlinearities, such as the Lorentz system with hysteresis friction and the van der Pol oscillator with hysteresis block, are considered. These examples illustrate the efficiency of the proposed algorithm.

Orbits of charged particles trapped in a dipole magnetic field.

Motion of a test charged particle in a dipole magnetic field can be reduced to a two degree-of-freedom Hamiltonian system due to the axisymmetry of the dipole field. We carried out a systematic study of orbits of low-energy trapped charged particles in the dipole field via calculation of their Lyapunov characteristic exponents (LCEs) with random initial conditions in the four-dimensional phase space. Since there is at most one positive LCE, these orbits are classified as chaotic ones with one positive LCE and quasi-periodic ones with vanishing LCEs. The dependence of the fraction of quasi-periodic orbits in the phase space on the particle energy is given, which reveals a discrete spectrum, confirming the results of earlier studies. It is also found that most quasi-periodic orbits are confined near the equatorial plane and away from the dipole except for some at very low energies, while chaotic ones are ergodic. The distribution of the maximum LCE (mLCE) appears to vary gradually in the phase space and chaotic orbits with very low values of the mLCE wander near quasi-periodic orbits for a significant amount of time before merging into the sea of chaos.

Rocard’s 1941 Chaotic Relaxation Econometric Oscillator

In the beginning of the Second World War, the French physicist, Yves Rocard, published a book entitled Théorie des Oscillateurs (Theory of Oscillators). In Chapter V, he designed a mathematical model consisting of a set of three nonlinear differential equations and allowing to account for economic cycles. Numerical integration of his model has highlighted a chaotic attractor. Its analysis with classical tools such as bifurcation diagram and Lyapunov Characteristic Exponents has confirmed the chaotic features of its solution. It follows that Rocard’s 1941 chaotic econometric model has thus most likely preceded Lorenz’ butterfly of twenty-two years. Moreover, apart from this historical discovery which upsets historiography, it is also established that this “new old” three-dimensional autonomous dynamical system is a new jerk system whose solution exhibits a chaotic attractor the topology of which varies, from a double scroll attractor to a Möbius-strip and then to a toroidal attractor, according to the values of a control parameter.

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.

Investigation of chaos behavior and integral sliding mode control on financial risk model

<abstract> <p>This paper reports the finding of a new financial chaotic system. A new control law for completely synchronizing the new financial chaotic system with itself has been established using adaptive integral sliding mode control. We also find that the new financial chaotic system has fascinating traits including symmetry, equilibrium points, multistability, Lyapunov exponents and bifurcation diagrams. We illustrate all the main results of this research work using MATLAB phase plots. The Lyapunov characteristic exponents and analysis using bifurcation diagrams have resulted in a new financial chaos system showing chaos phenomena in the intervals of parameters 0 < <italic>a</italic> < 15, and parameters 0 < <italic>b</italic> < 0.25. The results of this study can be used to predict if there is chaos in financial risk. Chaotic systems have many applications in engineering like cryptosystems and secure communication systems.</p> </abstract>

Dynamics and Hopf Bifurcation of a Chaotic Chemical Reaction Model

Taking into account an ideal mixture and a well-stirred reactor, some dynamical aspects for a 3-dimensional chaotic system are carried out. Positivity and boundedness of solutions are discussed. Equilibria are investigated and method of linearization is implemented for asymptotic behavior of system about these equilibria. Lyapunov function is constructed to prove the global stability of positive equilibrium point. Moreover, it is proved that system undergoes Hopf bifurcation about its interior (positive) equilibrium. An explicit criterion of Hopf bifurcation without finding the eigenvalues is used for the existence of Hopf bifurcation. Numerical simulation is presented for the illustration of theoretical discussion. Lyapunov dimension is approximated and maximum Lyapunov characteristic exponents are plotted to ensure the chaotic behavior of the model.