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The study examines the propagation of solitary waves in optical fibers controlled by the Radhakrishnan–Kundu–Lakshmanan model (RKL) equation, which includes a cubic–quintic non-Kerr refractive index. The first step in this study involves utilizing the extended algebraic method to derive solitonic solutions for the given equation. These solutions are subsequently graphically represented for a variety of parameter values. The proposed strategy allows for the discovery of a range of solutions.The controlling model is incorporated into a planar dynamical system via the Galilean transformation, resulting in the transformation of the governing model. The effects of both the quintic nonlinearity and self-phase modulation on the perturbed and unperturbed systems are explored using the effective potential and related phase portrait. The governing model is thus transformed into a planar dynamical system. Furthermore, this study harnessed a variety of numerical techniques. These methods encompassed time series analysis, phase portraits examination, and meticulous sensitivity inspections. Additionally, we employed a Poincaré section to assess the equation’s responsiveness to external perturbations. The outcomes reveal a fascinating observation: the perturbed system’s behavior shifts from a quasi-periodic state to a chaotic one, contingent upon the amplitude and frequency of the external perturbation. Moreover, we conducted multistability analysis for specific sets of physical parameters, uncovering the equation’s tendency to exhibit multistability in such scenarios.

Through an appropriate change of reference frame and rescalings of the variables and the parameters introduced, the Hamiltonian of the three-body problem is written as a perturbed Kepler problem. In this system, new Delaunay variables are defined and a suitable configuration of the phase space and the mass parameters is chosen. In such a system, wide regions of librational and rotational motions where orbits are regular and stable are found. Close to the separatrix of these regions, the existence of chaotic motions presenting a double rotational and librational dynamics is proved, numerically, through Poincaré sections and the use of FLI. [ABSTRACT FROM AUTHOR]

We present our recent contributions to the theory of Lagrangian descriptors for discriminating ordered and deterministic chaotic trajectories. The class of Lagrangian descriptors we are dealing with is based on the Euclidean length of the orbit over a finite time window. The framework is free of tangent vector dynamics and is valid for both discrete and continuous dynamical systems. We review its last advancements and touch on how it illuminated recently Dvorak's quantities based on maximal extent of trajectories' observables, as traditionally computed in planetary dynamics. [ABSTRACT FROM AUTHOR]

Machine learning (ML) is the branch of computer science that studies computer algorithms that can learn from data. It is mainly divided into supervised learning, where the computer is presented with examples of entries, and the goal is to learn a general rule that maps inputs to outputs, and unsupervised learning, where no label is provided to the learning algorithm, leaving it alone to find structures. Deep learning is a branch of machine learning based on numerous layers of artificial neural networks, which are computing systems inspired by the biological neural networks that constitute animal brains. In asteroid dynamics, machine learning methods have been recently used to identify members of asteroid families, small bodies images in astronomical fields, and to identify resonant arguments images of asteroids in three-body resonances, among other applications. Here, we will conduct a full review of available literature in the field and classify it in terms of metrics recently used by other authors to assess the state of the art of applications of machine learning in other astronomical subfields. For comparison, applications of machine learning to Solar System bodies, a larger area that includes imaging and spectrophotometry of small bodies, have already reached a state classified as progressing. Research communities and methodologies are more established, and the use of ML led to the discovery of new celestial objects or features, or new insights in the area. ML applied to asteroid dynamics, however, is still in the emerging phase, with smaller groups, methodologies still not well-established, and fewer papers producing discoveries or insights. Large observational surveys, like those conducted at the Zwicky Transient Facility or at the Vera C. Rubin Observatory, will produce in the next years very substantial datasets of orbital and physical properties for asteroids. Applications of ML for clustering, image identification, and anomaly detection, among others, are currently being developed and are expected of being of great help in the next few years. [ABSTRACT FROM AUTHOR]

Within this work, the radial Proportional Derivative (PD-) controller along with the eight-poles electro-magnetic actuator are introduced as a novel control strategy to suppress the lateral oscillations of a non-linear Jeffcott-rotor system. The proposed control strategy has been designed such that each pole of the magnetic actuator generates an attractive magnetic force proportional to the radial displacement and radial velocity of the rotating shaft in the direction of that pole. According to the proposed control mechanism, the mathematical model that governs the non-linear interactions between the Jeffcott system and the magnetic actuator has been established. Then, an analytical solution for the obtained non-linear dynamic model has been derived using perturbation analysis. Based on the extracted analytical solution, the motion bifurcation of the Jeffcott system has been investigated before and after control via plotting the different response curves. The obtained results illustrate that the uncontrolled Jeffcott-rotor behaves like a hard-spring duffing oscillator and responds with bi-stable periodic oscillation when the rotor angular speed is higher than the system's natural frequency. It is alsomfound that the system, before control, can exhibit stable symmetric motion with high vibration amplitudes in both the horizontal and vertical directions, regardless of the eccentricity magnitude. In addition, the acquired results demonstrate that the introduced control technique can eliminate catastrophic bifurcation behaviors and undesired vibration of the system when the control parameters are designed properly. However, it is reported that the improper design of the controller gains may destabilize the Jeffcott system and force it to perform either chaotic or quasi-periodic motions depending on the magnitudes of both the shaft eccentricity and the control parameters. Finally, to validate the accuracy of the obtained results, numerical simulations for all response curves have been introduced which have been in excellent agreement with the analytical investigations. [ABSTRACT FROM AUTHOR]

Within this work, an advanced control algorithm was proposed to eliminate the non-linear vibrations of the rotor electro-magnetic suspension system. The suggested control algorithm is known as the Adaptive Linear Integral Positive Position Feedback controller (ALIPPF-controller). The ALIPPF-controller is a combination of first-order and second-order filters that are coupled linearly to the targeted non-linear system in order to absorb the excessive vibratory energy. According to the introduced control strategy, the dynamical model of the controlled rotor system was established as six non-linear differential equations that are coupled linearly. The obtained dynamical model was investigated analytically applying the asymptotic analysis, where the slow-flow equations were extracted. Based on the derived slow-flow equations, the bifurcation behaviors of the controlled system were explored by plotting the different bifurcation diagrams. In addition, the performance of the ALIPPF-controller in eliminating the rotor lateral vibrations was compared with the conventional Positive Position Feedback (PPF) controller. The acquired results illustrated that the ALIPPF-controller is the best control technique that can eliminate the considered system's lateral vibrations regardless of the angular speed and eccentricity of the rotating shaft. Finally, to demonstrate the accuracy of the obtained analytical results, numerical validation was performed for all obtained bifurcation diagrams that were in excellent agreement with the analytical solutions. [ABSTRACT FROM AUTHOR]

Close encounters or resonances overlaps can create chaotic motion in small bodies in the Solar System. Approaches that measure the separation rate of trajectories that start infinitesimally near, or changes in the frequency power spectrum of time series, among others, can discover chaotic motion. In this paper, we introduce the ACF index (ACFI), which is based on the auto-correlation function of time series. Auto-correlation coefficients measure the correlation of a time-series with a lagged duplicate of itself. By counting the number of auto-correlation coefficients that are larger than 5% after a certain amount of time has passed, we can assess how the time series auto-correlates with each other. This allows for the detection of chaotic time-series characterized by low ACFI values. [ABSTRACT FROM AUTHOR]

Chaotic motion affecting small bodies in the Solar system can be caused by close encounters or collisions or by resonance overlapping. Chaotic motion can be detected using approaches that measure the separation rate of trajectories that starts infinitesimally close or changes in the frequency power spectrum of time series, among others. In this work, we introduce an approach based on the autocorrelation function of time series, the ACF index (ACFI ). Autocorrelation coefficients measure the correlation of a time series with a lagged copy of itself. By measuring the fraction of autocorrelation coefficients obtained after a given time lag that are higher than the 5% null hypothesis threshold, we can determine how the time series autocorrelates with itself. This allows identifying unpredictable time series, characterized by low values of ACFI . Applications of ACFI to orbital regions affected by both types of chaos show that this method can correctly identify chaotic motion caused by resonance overlapping, but it is mostly blind to close encounters induced chaos. ACFI could be used in these regions to select the effects of resonance overlapping. [ABSTRACT FROM AUTHOR]

Within this work, the radial Proportional Derivative (PD-) controller along with the eight-poles electro-magnetic actuator are introduced as a novel control strategy to suppress the lateral oscillations of a non-linear Jeffcott-rotor system. The proposed control strategy has been designed such that each pole of the magnetic actuator generates an attractive magnetic force proportional to the radial displacement and radial velocity of the rotating shaft in the direction of that pole. According to the proposed control mechanism, the mathematical model that governs the non-linear interactions between the Jeffcott system and the magnetic actuator has been established. Then, an analytical solution for the obtained non-linear dynamic model has been derived using perturbation analysis. Based on the extracted analytical solution, the motion bifurcation of the Jeffcott system has been investigated before and after control via plotting the different response curves. The obtained results illustrate that the uncontrolled Jeffcott-rotor behaves like a hard-spring duffing oscillator and responds with bi-stable periodic oscillation when the rotor angular speed is higher than the system’s natural frequency. It is alsomfound that the system, before control, can exhibit stable symmetric motion with high vibration amplitudes in both the horizontal and vertical directions, regardless of the eccentricity magnitude. In addition, the acquired results demonstrate that the introduced control technique can eliminate catastrophic bifurcation behaviors and undesired vibration of the system when the control parameters are designed properly. However, it is reported that the improper design of the controller gains may destabilize the Jeffcott system and force it to perform either chaotic or quasi-periodic motions depending on the magnitudes of both the shaft eccentricity and the control parameters. Finally, to validate the accuracy of the obtained results, numerical simulations for all response curves have been introduced which have been in excellent agreement with the analytical investigations.

Within this work, an advanced control algorithm was proposed to eliminate the non-linear vibrations of the rotor electro-magnetic suspension system. The suggested control algorithm is known as the Adaptive Linear Integral Positive Position Feedback controller (ALIPPF-controller). The ALIPPF-controller is a combination of first-order and second-order filters that are coupled linearly to the targeted non-linear system in order to absorb the excessive vibratory energy. According to the introduced control strategy, the dynamical model of the controlled rotor system was established as six non-linear differential equations that are coupled linearly. The obtained dynamical model was investigated analytically applying the asymptotic analysis, where the slow-flow equations were extracted. Based on the derived slow-flow equations, the bifurcation behaviors of the controlled system were explored by plotting the different bifurcation diagrams. In addition, the performance of the ALIPPF-controller in eliminating the rotor lateral vibrations was compared with the conventional Positive Position Feedback (PPF) controller. The acquired results illustrated that the ALIPPF-controller is the best control technique that can eliminate the considered system’s lateral vibrations regardless of the angular speed and eccentricity of the rotating shaft. Finally, to demonstrate the accuracy of the obtained analytical results, numerical validation was performed for all obtained bifurcation diagrams that were in excellent agreement with the analytical solutions.

The chaotic motions were shown by the vibrations of a bulk permanent magnet (PM) supported by a cylindrical high-Tc superconductor (HTSC) under external disturbance in very early experiments. There are few theoretical studies on the nonlinear behaviors of the HTSC levitation system as the complex interaction between the HTSC and PM. In this paper, we developed a numerical program to analyze the nonlinear vibrations of the levitated PM coupled with the electromagnetic force, taking into account the heat diffusion on HTSC. When the levitation system displayed chaotic oscillations by vertical excitation, the electromagnetic and thermal characteristics of the HTSC will change. The dynamic responses are shown by comparing the calculation results in which the thermal effects are taken into account and not. The numerical results display the displacements of PM, magnetic levitation forces, the acceleration, and the maximum Lyapunov exponent. The results show that the thermal effect will influence the stability of the levitation system. [ABSTRACT FROM AUTHOR]

The nonlinear behaviors and energy harvesting of an unsymmetric cross-ply square composite laminated plate with a piezoelectric patch is presented. The unsymmetric cross-ply square composite laminated plate has two stable equilibrium positions by applying thermal stress, thus having snap-through with larger amplitude between the two stable equilibrium positions relative to the general laminated plate. Based on the von-Karman large deformation theory, the nonlinear electromechanical coupling equations of motion of the unsymmetric composite laminated plate with a piezoelectric patch are derived by using Hamilton’s principle. The influence of the base excitation amplitude on nonlinear behaviors and energy harvesting are investigated. For different base excitation amplitudes, the motions of the system demonstrate periodic motion, quasi-periodic motion, chaotic motion and snap-through, and two single-well chaotic attractors and a two-well chaos attractor coexist. Moreover, the power generation efficiency is optimal when the excitation amplitude is in a certain range due to its own unique nonlinear characteristics. The unsymmetric cross-ply square composite laminated plate subjected to thermal stress can actually be called a kind of bistable composite shell structure that has a broad application prospect in combination with morphing aircraft, large deployable antenna and solar panel, which are very likely to have nonlinear vibration.

This paper focuses on the nonlinear dynamics near internal resonance of a truncated FGM conical shell. The FGM conical shell is subjected to the aerodynamic load and the in-plane excitation along the meridian direction. Material properties depend on the temperature and the constituent phases of the truncated FGM conical shell. The volume fractions are modified in the thickness direction based on a power-law function continuously and smoothly. The first-order piston theory is applied for the supersonic aerodynamic pressure. Based on the first-order shear deformation theory, von-Karman type nonlinear geometric assumptions, Hamilton principle and Galerkin method, the nonlinear equations of motion for the truncated FGM conical shell are derived. The averaged equations of the truncated FGM conical shell are obtained under the situation of 1:1 internal resonance and 1/2 subharmonic resonance by using the method of multiple scales. The frequency-response curves, the force-response curves, the bifurcation diagrams, the phase portraits, the time history diagrams, and the Poincare maps are obtained by using numerical calculations. The influences of the Mach number, the exponent of volume fraction and the in-plane excitation on the nonlinear resonant behaviors of the truncated FGM conical shell are investigated. • Nonlinear dynamics near internal resonance of a truncated FGM conical shell are studied. • The FGM conical shell is subjected to the aerodynamic load and the in-plane excitation. • The first-order piston theory is applied for the supersonic aerodynamic pressure. • The nonlinear equations of motion for the truncated FGM conical shell are derived. • The influences of the Mach number and the in-plane excitation are investigated. [ABSTRACT FROM AUTHOR]

The internal resonances and the nonlinear breathing vibrations of an eccentric rotating composite laminated circular cylindrical shell rounding a parallel axis are studied for the first time, which is subjected to the lateral excitation and the temperature excitation. Based on Love thin shear deformation theory and von Kármán-type nonlinear relation, the nonlinear partial differential governing equations of motion for the eccentric rotating composite laminated circular cylindrical shell are established by Hamilton's principle. The nonlinear partial differential govern equations are discretized into a set of coupled nonlinear ordinary differential equations by Galerkin approach. Considering the effects of the eccentricity ratio, the geometric parameters and the excitation conditions, the internal resonance and nonlinear breathing vibrations of the eccentric rotating composite laminated circular cylindrical shell rounding a parallel axis are studied. The method of multiple scales is employed to obtain four-dimensional nonlinear averaged equations. Corresponding to several selected parameters, the frequency-response curves are obtained and the nonlinear dynamic behaviors and the jump phenomena are exhibited. The effects of different parameters on the nonlinear resonant responses are also studied. The periodic and chaotic motions of the eccentric rotating composite laminated circular cylindrical shell rounding a parallel axis are found when the rotating speed corresponds to the internal resonant point at the certain excitations. • Nonlinear vibrations of rotating laminated circular cylindrical shell are studied. • Nonlinear governing equations of motion are established. • Internal resonances are studied considering the eccentricity ratio and different parameters. • For several selected parameters, the frequency-response curves are obtained. • There exist periodic and chaotic motions of rotating laminated circular cylindrical shell. In this paper, the nonlinear breathing vibrations of an eccentric rotating composite laminated circular cylindrical shell are studied for the first time, which is subjected to the lateral and temperature excitations. Based on Donnell thin shear deformation theory, von Kármán-type nonlinear relation and Hamilton's principle, the nonlinear partial differential governing equations of motion are established for the eccentric rotating composite laminated circular cylindrical shell. The nonlinear partial differential governing equations of motion are discretized into a set of coupled nonlinear ordinary differential equations of motion by Galerkin approach. Based on the case of 1:2 internal resonance and principal parametric resonance-1/2 subharmonic resonance, the method of multiple scales is employed to obtain four-dimensional nonlinear averaged equations. Considering the effects of different parameters, for example, the eccentricity ratio, the geometric parameters and the excitations, the nonlinear dynamic behaviors and the jumping phenomena are exhibited for the frequency-response curves and the amplitude-response curves. The periodic and chaotic motions of the eccentric rotating composite laminated circular cylindrical shell are found when the rotating speed corresponds to the internal resonant point at the certain conditions for different lateral and temperature excitations. [ABSTRACT FROM AUTHOR]

Abstract Perturbed nonlinear Schrödinger (NLS) equation with the power-law nonlinearity in a nano optical fiber is studied with the help of its equivalent two-dimensional planar dynamic system and Hamiltonian. Via the bifurcation theory and qualitative theory, equilibrium points for the two-dimensional planar dynamic system are obtained. With the external perturbation taken into consideration, chaotic motions for the perturbed NLS equation with the power-law nonlinearity are derived based on the equilibrium points. [ABSTRACT FROM AUTHOR]

Based on the theory of non-linear dynamic systems, the influence of the perturbation of the wheel rotation speed on the quasi-steady curved motions of a two-axle railway bogie system with a realistic wheel/rail contact relation is investigated in this paper. Since the wheel/rail contact relation is non-linear, it is tabulated in a wheel/rail contact table. The bifurcation diagram of the bogie system is constructed with gradually increasing and decreasing speed in the speed range . A supercritical Hopf bifurcation, where the stable non-trivial stationary solution loses its stability, is found in both models with and without perturbation of the wheel rotation speed. In the model without perturbation of the wheel rotation speed, the first chaotic motions develop at the speed where the wheel flange contact starts. A period-doubling cascade of the bogie system through pitchfork bifurcations, which explains the transition from periodic solutions to chaotic motions of the bogie system, is found. Several jumps happen at higher speeds because of the coexistence of multiple attracting solutions, which should be avoided. A comprehensive investigation of the hysteresis phenomena is made. However, in the model with a perturbation of the wheel rotation speed, no chaotic motions are found. [ABSTRACT FROM AUTHOR]

The study examines the propagation of solitary waves in optical fibers controlled by the Radhakrishnan–Kundu–Lakshmanan model (RKL) equation, which includes a cubic–quintic non-Kerr refractive index. The first step in this study involves utilizing the extended algebraic method to derive solitonic solutions for the given equation. These solutions are subsequently graphically represented for a variety of parameter values. The proposed strategy allows for the discovery of a range of solutions. The controlling model is incorporated into a planar dynamical system via the Galilean transformation, resulting in the transformation of the governing model. The effects of both the quintic nonlinearity and self-phase modulation on the perturbed and unperturbed systems are explored using the effective potential and related phase portrait. The governing model is thus transformed into a planar dynamical system. Furthermore, this study harnessed a variety of numerical techniques. These methods encompassed time series analysis, phase portraits examination, and meticulous sensitivity inspections. Additionally, we employed a Poincaré section to assess the equation's responsiveness to external perturbations. The outcomes reveal a fascinating observation: the perturbed system's behavior shifts from a quasi-periodic state to a chaotic one, contingent upon the amplitude and frequency of the external perturbation. Moreover, we conducted multistability analysis for specific sets of physical parameters, uncovering the equation's tendency to exhibit multistability in such scenarios. • The propagation of solitary waves in optical fibers by Radhakrishnan–Kundu–Lakshmanan model (RKL) equation are discussed. • All possible phase portraits are presented. • Sensitive analysis is applied to discuss the sensitivity of the model. • Chaotic and Quasi-periodic behaviors are observed. • Multi-stability is reported in detail. [ABSTRACT FROM AUTHOR]

The nonlinear vibrations and chaotic motions of a simply supported symmetric cross-ply composite laminated piezoelectric rectangular plate subjected to the transverse and in-plane excitations are analyzed in the case of primary parametric resonance and 1:3 internal resonance. It is assumed that different layers of the symmetric cross-ply composite laminated piezoelectric rectangular plate are perfectly bonded to each other and with piezoelectric actuator layers embedded in the plate. Based on the Reddy’s third-order shear deformation plate theory, the nonlinear governing equation of motion for the composite laminated piezoelectric rectangular plate is derived by using the Hamilton’s principle. The Galerkin’s approach is employed to discretize the partial differential governing equation to a two-degree-of-freedom nonlinear system under combined the parametric and external excitations. The method of multiple scales is utilized to obtain the four-dimensional averaged equation. Numerical method is used to find the bifurcation diagram, the periodic and chaotic motions of the composite laminated piezoelectric rectangular plate. The numerical results illustrate the existence of the periodic and chaotic motions in the averaged equation. It is found that the chaotic responses are especially sensitive to the forcing and the parametric excitations. The influences of the transverse, in-plane and piezoelectric excitations on the bifurcations and chaotic behaviors of the composite laminated piezoelectric rectangular plate are investigated numerically. [ABSTRACT FROM AUTHOR]

In this paper, the nonlinear oscillations of a composite laminated circular cylindrical shell clamped along a generatrix and with the radial pre-stretched membranes at both ends are studied for the first time. The dynamic effect of membranes on the circular cylindrical shell is replaced by a nonlinear elastic excitation with the damping. Meanwhile, the parametric excitation of the changing temperature is also considered. Based on Reddy's third-order shear deformation theory and von Kármán-type nonlinear kinematics, the nonlinear partial differential equations of motion for the composite laminated circular cylindrical shell clamped along a generatrix are established by Hamilton's principle, which are derived into a set of coupled nonlinear ordinary differential equations by the Galerkin discretization. The asymptotic perturbation method is applied to obtain the four-dimensional nonlinear averaged equations in the case of 1:2 internal resonance and principal parametric resonance-1/2 subharmonic resonance. Corresponding to several selected values of the parameters, the frequency-response curves are obtained by numerical method. It is found that the static bifurcations, the jump phenomena as well as the hardening-spring-type nonlinearity behaviors are exhibited and that different parameters change the frequency-response curve shape. The numerical results based on the averaged equations are obtained to exhibit some intrinsically nonlinear dynamic behaviors of the composite laminated circular cylindrical shell clamped along a generatrix using the bifurcation diagram, waveform, phase plots and Poincaré maps. It is also found that there exist alternately the periodic and chaotic motions of the circular cylindrical shell clamped along a generatrix with the parameter excitation of temperature increases in a certain range. [ABSTRACT FROM AUTHOR]

Nonlinear dynamics of flexible multibeam structures modeled as an L-shaped beam are investigated systematically considering the modal interactions. Taking into account nonlinear coupling and nonlinear inertia, Hamilton's principle is employed to derive the partial differential governing equations of the structure. Exact mode functions are obtained by the coupled linear equations governing the horizontal and vertical beams and the results are verified by the finite element method. Then the exact modes are adopted to truncate the partial differential governing equations into two coupled nonlinear ordinary differential equations by using Galerkin method. The undamped free oscillations are studied in terms of Jacobi elliptic functions and results indicate that the energy exchanges are continual between the two modes. The saturation and jumping phenomena are then observed for the forced damped multibeam structure. Further, a higher-dimensional, Melnikov-type perturbation method is used to explore the physical mechanism leading to chaotic behaviors for such an autoparametric system. Numerical simulations are performed to validate the theoretical predictions. [ABSTRACT FROM AUTHOR]

The ( 2 + 1 ) -dimensional Zakharov equations arising from the propagation of a laser beam in a plasma are studied in this paper. Analytic soliton solutions are obtained by means of the symbolic computation, based on which we find that | E | is inversely related to ω p e , but positively related to m i and c s , while n is inversely related to ω p e and ω L , but positively related to n 0 , with E as the envelope of the high-frequency electric field, n as the plasma density, while ω p e , ω L , n 0 , m i and c s as the plasma electronic frequency, frequency of the laser beam, mean density of the plasma, mass of an ion and ion-sound velocity in the plasma, respectively. Head-on interaction is found to be transformed into an overtaking one with ω p e increasing or n 0 decreasing. Also, period of the bound-state interaction decreases with ω L decreasing. Considering the driving forces in the laser-induced plasma, we explore the associated chaotic motions as well as the effects of ω L , ω p e , k L , n 0 , m i , c s , ω F 1 and ω F 2 , where k L is the wave number of the laser beam, ω F 1 and ω F 1 represent the frequencies of driving forces, respectively. It is found that the chaotic motions can be weakened with ω p e , c s and ω F 1 increasing, or with n 0 , m i and ω F 2 decreasing, and the periodic motion can occur when ω F 1 reaches the critical value 2 π , while the chaotic motions are independent of ω L and k L . [ABSTRACT FROM AUTHOR]

Consider an autonomous ordinary differential equation in R n that has a homoclinic solution asymptotic to a hyperbolic equilibrium. The homoclinic solution is degenerate in the sense that the linear variational equation has 2 bounded, linearly independent solutions. We study bifurcation of the homoclinic solution under periodic perturbations. Using exponential dichotomies and Lyapunov–Schmidt reduction, we obtain general conditions under which the perturbed system can have transverse homoclinic solutions and nearby periodic or chaotic solutions. [ABSTRACT FROM AUTHOR]

Nonlinear geometrically exact rod dynamics is of great interest in many areas of engineering. In recent years, the research was focused towards Timoshenko-type rod theories where shearing is of importance. However, in many general model of mechanisms and spatial deformations, it is desirable to have a displacement-only formulation, which brings us back to the classical Bernoulli beam. While it is well known for linear analysis, the Bernoulli beam is not as common in geometrically exact models of dynamics, especially when we want to incorporate the rotational inertia into the model. This paper is about the development of an energy-momentum integration scheme for the geometrically exact Bernoulli-type rod. We will show that the task is achievable and devise a general framework to do so. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We consider the $$N$$ -body problem with interaction potential $$U_\alpha =\frac{1}{\vert x_i-x_j\vert ^\alpha }$$ for $$\alpha >1$$ . We assume that the particles have all the same mass and that $$N$$ is the order $$\vert \mathcal {R}\vert $$ of the rotation group $$\mathcal {R}$$ of one of the five Platonic polyhedra. We study motions that, up to a relabeling of the $$N$$ particles, are invariant under $$\mathcal {R}$$ . By variational techniques we prove the existence of periodic and chaotic motions. [ABSTRACT FROM AUTHOR]

In this paper, a Fourier expansion-based differential quadrature (FDQ) method is developed to analyze numerically the transverse nonlinear vibrations of an axially accelerating viscoelastic beam. The partial differential nonlinear governing equation is discretized in space region and in time domain using FDQ and Runge-Kutta-Fehlberg methods, respectively. The accuracy of the proposed method is represented by two numerical examples. The nonlinear dynamical behaviors, such as the bifurcations and chaotic motions of the axially accelerating viscoelastic beam, are investigated using the bifurcation diagrams, Lyapunov exponents, Poincare maps, and three-dimensional phase portraits. The bifurcation diagrams for the in-plane responses to the mean axial velocity, the amplitude of velocity fluctuation, and the frequency of velocity fluctuation are, respectively, presented when other parameters are fixed. The Lyapunov exponents are calculated to further identify the existence of the periodic and chaotic motions in the transverse nonlinear vibrations of the axially accelerating viscoelastic beam. The conclusion is drawn from numerical simulation results that the FDQ method is a simple and efficient method for the analysis of the nonlinear dynamics of the axially accelerating viscoelastic beam. [ABSTRACT FROM AUTHOR]

The modified Zakharov-Kuznetsov (mZK) equation in the electrical transmission line is investigated in this paper. Different expressions on the parameters in the mZK equation are given. By means of the Hirota method, bilinear forms and soliton solutions of the mZK equation are obtained. Linear-stability analysis yields the instability condition for such soliton solutions. We find that the soliton amplitude becomes larger when the inductance L and capacitance C 0 decrease. Phase-plane analysis is conducted on the mZK equation for the properties at equilibrium points. Then, we investigate the perturbed mZK equation, which can be proposed when the external periodic force is considered. Both the weak and developed chaotic motions are observed. Our results indicate that the two chaotic motions can be manipulated with certain relation between the absolute values of nonlinear terms and the perturbed one. We also find that the chaotic motions can be weakened with the absolute values of L and C 0 decreased. [ABSTRACT FROM AUTHOR]

By using an extended Melnikov method on multi-degree-of-freedom Hamiltonian systems with perturbations, the global bifurcations and chaotic dynamics are investigated for a parametrically excited, simply supported rectangular buckled thin plate. The formulas of the rectangular buckled thin plate are derived by using the von Karman type equation. The two cases of the buckling for the rectangular thin plate are considered. With the aid of Galerkin's approach, a two-degree-of-freedom non-autonomous nonlinear system is obtained for the non-autonomous rectangular buckled thin plate. The high-dimensional Melnikov method developed by Yagasaki is directly employed to the non-autonomous ordinary differential equation of motion to analyze the global bifurcations and chaotic dynamics of the rectangular buckled thin plate. Numerical method is used to find the chaotic responses of the non-autonomous rectangular buckled thin plate. The results obtained here indicate that the chaotic motions can occur in the parametrically excited, simply supported rectangular buckled thin plate. [ABSTRACT FROM AUTHOR]

We use probability density functions (pdfs) of sums of orbit coordinates, over time intervals of the order of one Hubble time, to distinguish weakly from strongly chaotic orbits in a barred galaxy model. We find that, in the weakly chaotic case, quasi-stationary states arise, whose pdfs are well approximated by q-Gaussian functions (with 1 < q < 3), while strong chaos is identified by pdfs which quickly tend to Gaussians ( q = 1). Typical examples of weakly chaotic orbits are those that 'stick' to islands of ordered motion. Their presence in rotating galaxy models has been investigated thoroughly in recent years due to their ability to support galaxy structures for relatively long time scales. In this paper, we demonstrate, on specific orbits of 2 and 3 degree of freedom barred galaxy models, that the proposed statistical approach can distinguish weakly from strongly chaotic motion accurately and efficiently, especially in cases where Lyapunov exponents and other local dynamic indicators appear to be inconclusive. [ABSTRACT FROM AUTHOR]

Orbits in the principal planes of triaxial potentials are known to be prone to unstable motion normal to those planes, so that three dimensional investigations of those orbits are needed even though they are two dimensional. We present here an investigation of such orbits in the well known logarithmic potential which shows that the third dimension must be taken into account when studying them and that the instability worsens for lower values of the forces normal to the plane. Partially chaotic orbits are present around resonances, but also in other regions. The action normal to the plane seems to be related to the isolating integral that distinguishes regular from partially chaotic orbits, but not to the integral that distinguishes partially from fully chaotic orbits. [ABSTRACT FROM AUTHOR]

In this paper, the dynamical response of a coupled oscillator is investigated, taking in consideration the nonlinear behavior of a SMA spring coupling the two oscillators. Due to the nonlinear coupling terms, the system exhibits both regular and chaotic motions. The Poincaré sections for different sets of coupling parameters are verified. [ABSTRACT FROM AUTHOR]

As a result of resonance overlap, planetary systems can exhibit chaotic motion. Planetary chaos has been studied extensively in the Hamiltonian framework, however, the presence of chaotic motion in systems where dissipative effects are important, has not been thoroughly investigated. Here, we study the onset of stochastic motion in presence of dissipation, in the context of classical perturbation theory, and show that planetary systems approach chaos via a period-doubling route as dissipation is gradually reduced. Furthermore, we demonstrate that chaotic strange attractors can exist in mildly damped systems. The results presented here are of interest for understanding the early dynamical evolution of chaotic planetary systems, as they may have transitioned to chaos from a quasi-periodic state, dominated by dissipative interactions with the birth nebula. [ABSTRACT FROM AUTHOR]

Saturn's moon, Hyperion, is subject to strongly-varying solid body torques from its primary and lacks a stable spin state resonant with its orbital frequency. In fact, its rotation is chaotic, with a Lyapunov timescale on the order of 100 days. In 2005, Cassini made three close passes of Hyperion at intervals of 40 and 67 days, when the moon was imaged extensively and the spin state could be measured. Curiously, the spin axis was observed at the same location within the body, within errors, during all three fly-bys-~ 30° from the long axis of the moon and rotating between 4.2 and 4.5 times faster than the synchronous rate. Our dynamical modeling predicts that the rotation axis should be precessing within the body, with a period of ~ 16 days. If the spin axis retains its orientation during all three fly-bys, then this puts a strong constraint on the in-body precessional period, and thus the moments of inertia. However, the location of the principal axes in our model are derived from the shape model of Hyperion, assuming a uniform composition. This may not be a valid assumption, as Hyperion has significant void space, as shown by its density of 544± 50 kg m (Thomas et al. in Nature 448:50, ). This paper will examine both a rotation model with principal axes fixed by the shape model, and one with offsets from the shape model. We favor the latter interpretation, which produces a best-fit with principal axes offset of ~ 30° from the shape model, placing the A axis at the spin axis in 2005, but returns a lower reduced χ than the best-fit fixed-axes model. [ABSTRACT FROM AUTHOR]

The nonlinear dynamic responses of a string-beam coupled system subjected to harmonic external and parametric excitations are studied in this work in the case of 1:2 internal resonance between the modes of the beam and string. First, the nonlinear governing equations of motion for the string-beam coupled system are established. Then, the Galerkin's method is used to simplify the nonlinear governing equations to a set of ordinary differential equations with four-degrees-of-freedom. Utilizing the method of multiple scales, the eight-dimensional averaged equation is obtained. The case of 1:2 internal resonance between the modes of the beam and string — principal parametric resonance-1/2 subharmonic resonance for the beam and primary resonance for the string — is considered. Finally, nonlinear dynamic characteristics of the string-beam coupled system are studied through a numerical method based on the averaged equation. The phase portrait, Poincare map and power spectrum are plotted to demonstrate that the periodic and chaotic motions exist in the string-beam coupled system under certain conditions. [ABSTRACT FROM AUTHOR]

In this paper, a special nonsmooth transformation of time is combined with the shooting algorithm for visualization of the manifolds of periodic solutions and their bifurcations. The general class of nonlinear oscillators under smooth, nonsmooth, and impulsive loadings is considered. The corresponding boundary value problems with no singularities are obtained by introducing the periodic piecewise-linear (sawtooth) temporal argument. The Ueda circuit, that is Duffing's oscillator with no linear stiffness, is considered for illustration. It is shown that the temporal mode shape of the input can be responsible for qualitative features of the dynamics, such as transitions between the regular and random motions. The important role of unstable periodic orbits and their links with strange attractors are discussed. [ABSTRACT FROM AUTHOR]

Abstract: The dynamics of a harmonically excited single degree-of-freedom linear system with a feedback control, in which the actuator is subjected to dead zone and saturation constraints, is investigated in detail. The controlled system is mathematically modeled by a set of three piecewise linear equations. It is found that the system may exhibit nine types of symmetric and asymmetric period-one motions, which are characterized by a different number of crossing dead zone and saturation region per cycle. A solution for the symmetric period-one motion with a doubly crossing dead zone and saturation region is analytically constructed and its stability characteristics is examined. Other types of dynamic response such as sub-harmonic periodic motions and chaotic motions, found through numerical simulations, are also presented. [Copyright &y& Elsevier]

A technique based on the power cepstrum has been developed to analyze and characterize data of various nonlinear and chaotic motions. For repeatability and ready availability, nonlinear response data of a Duffing oscillator and van der Pol oscillator, generated numerically by the fourth order Runge-Kutta algorithm, were used in the investigation. Results obtained by the proposed technique using spectrum, cepstrum, and specepstrum which is defined as the spectrum of the logarithmic cepstrum, indicate that it is superior to methods previously available in the literature. [ABSTRACT FROM AUTHOR]

The nonlinear dynamics of a clamped-clamped/sliding inextensional elastic beam subject to a harmonic axial load is investigated. The Galerkin method is used on the coupled bending-bending-torsional nonlinear equations with inertial and geometric nonlinearities and the resulting two second order ordinary differential equations are studied by the method of multiple time seales and by direct numerical integration. The amplitude equations are analyzed for steady and Hopf bifurcations. Depending on the amplitude of excitation, the damping and the ratio of principal flexural rigidities, various qualitatively distinct frequency response diagrams are uncovered and limit cycles and chaotic motions are found. In the truncated two-degree-of-freedom system the transition from periodic to chaotic amplitude-modulated motions is via the process of torus doubling and subsequent destruction of the torus. [ABSTRACT FROM AUTHOR]

The nonlinear behaviors and energy harvesting of an unsymmetric cross-ply square composite laminated plate with a piezoelectric patch is presented. The unsymmetric cross-ply square composite laminated plate has two stable equilibrium positions by applying thermal stress, thus having snap-through with larger amplitude between the two stable equilibrium positions relative to the general laminated plate. Based on the von-Karman large deformation theory, the nonlinear electromechanical coupling equations of motion of the unsymmetric composite laminated plate with a piezoelectric patch are derived by using Hamilton's principle. The influence of the base excitation amplitude on nonlinear behaviors and energy harvesting are investigated. For different base excitation amplitudes, the motions of the system demonstrate periodic motion, quasi-periodic motion, chaotic motion and snap-through, and two single-well chaotic attractors and a two-well chaos attractor coexist. Moreover, the power generation efficiency is optimal when the excitation amplitude is in a certain range due to its own unique nonlinear characteristics. The unsymmetric cross-ply square composite laminated plate subjected to thermal stress can actually be called a kind of bistable composite shell structure that has a broad application prospect in combination with morphing aircraft, large deployable antenna and solar panel, which are very likely to have nonlinear vibration. [ABSTRACT FROM AUTHOR]

This paper aims at the bifurcations and chaotic motions of a harmonically driven rectangular plate subjected to a uniform incompressible subsonic airflow. The plate equation of motion is derived by considering the von Karman's large deflection and Kelvin's type damping of material. A Galerkin-type solution is applied for the plate stress function and the aerodynamic force. The governing partial differential equation of the system is transformed into ordinary differential equations using the Galerkin method. The divergence instability and the pitchfork-like bifurcation of the plate are explored by theoretical and numerical analysis. The bifurcations of fixed points and periodic motions are thoroughly analyzed. The periodic motions can experience symmetry breaking/restoring bifurcations, period-doubling bifurcations, and saddle–node-like bifurcations, which are vital to the transition between different types of motions. Two typical bifurcation processes feature the bifurcation structure. The first one describes the change between the small and the large periodic orbits; the second one refers to the change between various large periodic orbits. Two criteria are used to predict the chaotic motions, which play a significant role in the transition between the small-orbit and the large-orbit periodic motions. The first one is the classical Holmes–Melnikov's criterion, and the second one is an approximated criterion that is newly developed from the resonant-response analysis of a reduced system. Results show that the current criterion brings some noticeable improvements compared with Holmes–Melnikov's criterion. • A complete modeling of a periodically driven three dimensional plate with large deflection in subsonic flow is developed. • The plate loses its stability by divergence and experiences various bifurcations of both fixed points and periodic motions. • The bifurcation structure of system is featured by two typical changing processes of different types of periodic motions. • An approximated criterion for prediction of chaos in the transition between the small-orbit and the large-orbit periodic motions is developed. [ABSTRACT FROM AUTHOR]

The complex coupling characteristics of a dual-rotor system make the mechanism of nonlinear vibrations different from that of a single rotor system. Research of nonlinear vibration of the dual-rotor is beneficial to improve the stability of an aero-engine. Based on nonlinear spring characteristics of rotors, a novel nonlinear dual-rotor dynamics model coupled by the inter-shaft bearing is proposed, and internal resonances on the dual-rotor system are systematically studied. Considering nonlinearities, internal resonance phenomena occurring in the vicinity of the coupling critical rotational speed are systematically analyzed by numerical simulation results such as time histories, Poincare maps, bifurcation maps and largest Lyapunov exponents. Theoretical solutions are calculated based on the improved shooting method and stabilities of theoretical solutions are investigated by using the Floquet theory. The theoretical solutions are consistent with numerical simulation results well. The concepts of the energy track shifting, the energy-Poincare map, stabilities of energy tracks and the energy supplying function are introduced to analyze nonlinear vibration characteristics for the first time. The internal resonance phenomena of the low pressure rotor and the complex nonlinear vibrations of the high pressure rotor are qualitatively analyzed based on the energy viewpoint, and the energy tracks are verified by experiments. Research shows that energy tracks can characterize vibration characteristics well and the energy supplying function provide a possible way to quantify the energy transfer between rotors in the multi-rotor system. • Novel dual-rotor model considering nonlinear spring characteristics of rotors. • Vibration analysis based on the energy tracks shifting and energy-Poincare maps. • Internal resonances induced by nonlinear coupling relationship between rotors. • Analysis of energy transfer between rotors based on energy supplying functions. [ABSTRACT FROM AUTHOR]