Melnikov Approach Research Articles

Dynamics of Some Perturbed Morse-Type Oscillators: Simulations and Applications

The purpose of this paper is to investigate some Morse-type oscillators. In its original form, it is a model for describing the vibrations of a diatomic molecule. The Morse potential generalizes the harmonic oscillator by introducing deviations from the classical theoretical model. In the present study, we perturbed the Morse differential equation by several periodic terms based on the cosine function and by a damping term. The frequency is driven by different coefficients. The size of the deviations is controlled by another constant. We provide two modifications w.r.t. the damping term. The Melnikov approach is applied as an indicator of the possible chaotic opportunities. We also propose a novel approach for stochastic control of the perturbations. It is based on the assumption that the coefficients of the periodic terms are the probabilities of underlying distribution. As a result, the dynamics are driven by its characteristic function. Several applications are considered. We demonstrate some specialized modules for investigating the dynamics of the proposed models, along with the synthesis of radiating antenna patterns.

Modeling of Some Classes of Extended Oscillators: Simulations, Algorithms, Generating Chaos, and Open Problems

In this article, we propose some extended oscillator models. Various experiments are performed. The models are studied using the Melnikov approach. We show some integral units for researching the behavior of these hypothetical oscillators. These will be implemented as add-on sections of a thoughtful main web-based application for researching computations. One of the main goals of the study is to share the difficulties that researchers (who are not necessarily professional mathematicians) encounter in using contemporary computer algebraic systems (CASs) for scientific research to examine in detail the dynamics of modifications of classical and newer models that are emerging in the literature (for the large values of the parameters of the models). The present article is a natural continuation of the research in the direction that has been indicated and discussed in our previous investigations. One possible application that the Melnikov function may find in the modeling of a radiating antenna diagram is also discussed. Some probability-based constructions are also presented. We hope that some of these notes will be reflected in upcoming registered rectifications of the CAS. The aim of studying the design realization (scheme, manufacture, output, etc.) of the explored differential models can be viewed as not yet being met.

Simulation on a Generalized Oscillator Model: Web-Based Application

In this paper, we propose new generalized oscillator model. Considerations in the light of the Melnikov's approach are also given. We focus on some interesting simulations based on the proposed new model. This is an integral part of a planned much more general Web-based application for scientific computing.

Taming chaos in generalized Liénard systems by the fractional-order feedback based on Melnikov analysis

The dynamical behavior of Liénard systems has always been a hot topic in nonlinear analysis. In the present study, a simple fractional-order feedback controller is put forward to tame chaos for a class of forced generalized Liénard systems. Adopting harmonic balance method, the first-order approximate equivalent integer-order system of the original fractional-order system is deduced. Then the criterion for taming chaos is established by employing the Melnikov approach. Duffing-Rayleigh chaotic oscillator is taken as an example to illustrate the validity of the proposed method. Firstly, the critical feedback intensity and differential order for taming chaos are obtained by the proposed criterion. Then, multiple numerical indicators such as phase portrait, time history plot, Lyapunov exponent and bifurcation diagram are provided to assist in analyzing theoretical results.

Complex dynamics of perturbed solitary waves in a nonlinear saturable medium: A Melnikov approach

Objective:To investigate the nonlinear dynamics of a periodically perturbed second-order ordinary differential equation obtained by using traveling wave variables in the model of pulse propagation in a nonlinear medium with saturation. Method:The Melnikov function of the investigated system along its homoclinic and heteroclinic orbits is constructed. It is established that the necessary condition for the occurrence of Melnikov chaos is always met. By analogy with the well-known Duffing equation, a damping term is added to the system to control chaos. Using the numerical calculation of the Melnikov integrals, conditions are found on the parameters of the new system for which the Melnikov chaos takes place. To verify the results obtained by the Melnikov method, attraction basins of the system are constructed. Results:The results obtained by the Melnikov method go in agreement with the structure of the constructed basin boundaries.

Dynamical properties of the periodically perturbed Triki–Biswas equation

In this paper, a perturbed traveling wave reduction of the Tricky–Biswas equation, which is used to describe the propagation of pulses in nonlinear optics, is considered. A stability analysis of the investigated ODE system without perturbation is carried out. The Melnikov function along the homoclinic orbit is constructed. It is found that in the studied system the necessary condition for occurrence of homoclinic chaos is always satisfied. A perturbation is added to the system to control the chaos obtained. Constraints on the parameters of the new system, at which homoclinic chaos is realized in it, are found. The attraction basins are plotted. It is found that their structure is fractal when the damping parameter values are less than the critical ones obtained by the Melnikov approach. The results of the numerical analysis go in agreement with those acquired theoretically.

The pitch vulnerability of a typical wave energy converter geometry based on Melnikov and Markov approaches

ABSTRACT Many wave energy converter (WEC) designs emerging over the recent years are floating cylindrical geometries with position-keeping mooring system. However, the dynamic stability of these novel floating structures under random sea states has not been investigated adequately. The Melnikov function model and the Markov process model are two efficient approaches providing quantitative predictions of capsizing. In this study, application of the two methods to a moored floating cylinder representing a generic WEC, has been explored to predict the risks of pitchpoling under random excitation. Using the Melnikov approach, the rate of phase space flux was evaluated to quantify the dynamic stability. This approach is compared with the Markov approach, which evaluates the mean first escape (capsize) rate to quantify the dynamic stability. The two methods are investigated through studies using various parameters. The similarity of their results and the differences of their approximations will be explored.

General Melnikov Approach to Implicit ODE’s

Existence of solutions connecting a singularity of a perturbed implicit differential equations is studied. It is assumed that the unperturbed differential equation has a solution of the same kind. By a suitable, nonlinear, change of coordinates these kind of solutions are associated to homoclinic solutions to a fixed point of an ordinary differential equation with a one-dimensional centre manifold. Then we obtain a Melnikov condition for the persistence of homoclinic orbits which is simpler than the one obtained in Battelli and Feckan (J Differ Equ 256:1157–1190, 2014). This difference is due to the fact that this method does not distinguish solutions according to the rate of convergence to the fixed point and then another assumption on the perturbation term is needed.

Regular and chaotic behavior of collective atomic motion in two-component Bose-Einstein condensates

We theoretically study binary Bose-Einstein condensates trapped in a single-well harmonic potential to probe the dynamics of collective atomic motion. The idea is to choose tunable scattering lengths through Feshbach resonances such that the ground-state wave function for two types of the condensates are spatially immiscible where one of the condensates, located at the center of the potential trap, can be effectively treated as a potential barrier between bilateral condensates of the second type of atoms. In the case of small wave function overlap between bilateral condensates, one can parametrize their spatial part of the wave functions in the two-mode approximation together with the time-dependent population imbalance $z$ and the phase difference $\phi$ between two wave functions. The condensate in the middle can be approximated by a Gaussian wave function with the displacement of the condensate center $\xi$. As driven by the time-dependent displacement of the central condensate, we find the Josephson oscillations of the collective atomic motion between bilateral condensates as well as their anharmonic generalization of macroscopic self-trapping effects. In addition, with the increase in the wave function overlap of bilateral condensates by properly choosing tunable atomic scattering lengths, the chaotic oscillations are found if the system departs from the state of a fixed point. The Melnikov approach with a homoclinic solution of the derived $z,\,\phi$, and $\xi$ equations can successfully justify the existence of chaos. All results are consistent with the numerical solutions of the full time-dependent Gross-Pitaevskii equations.

Homoclinic bifurcations and chaotic dynamics of non-planar waves in axially moving beam subjected to thermal load

The homoclinic bifurcations and nonplanar chaotic waves in axially moving beam (AMB) under thermal excitation are investigated. By the multiple scale technique, the equivalent nonlinear system is derived to explore qualitatively the dynamical characteristics of AMB system for the case of primary resonance. Using Melnikov approach as well as geometric analysis, the criterion for homoclinic chaos and complex nonplanar motions for AMB system is discussed. The theoretical predictions are tested by the numerical approach. For the design and application of the AMB, some inspiration and guidance are provided by the results from theory and simulation.

Parametric roll vulnerability of ships using Markov and Melnikov approaches

The designs of modern container ships, roll-on–roll-off vessels and cruise vessels have evolved over the years, and in recent times, some of them have been observed to experience dynamic instabilities during operation in the open ocean. These catastrophic events demonstrate that satisfying prescriptive stability rules set forth by International Maritime Organization (IMO), national authorities (e.g., Coast Guard) and other classification societies are not sufficient to ensure dynamic stability of ships at sea. In light of these events, IMO is organizing efforts to make way toward a second generation of intact stability criteria that are better equipped to deal with these dynamic instabilities. This paper discusses the development of such a tool for parametric rolling in a realistic random seaway, which is one of the critical phenomena identified by IMO. In this study, a previously developed analytical model for roll restoring moment, which was found to be effective in modeling the problem of parametric roll, is analyzed using the Melnikov approach. The stability of the system is quantified in terms of rate of phase space flux of the system. This approach is further compared with another technique known as the Markov approach that is based on stochastic averaging and quantifies stability in terms of mean first passage time. The sensitivity of both of these metrics to environmental parameters is investigated. Finally, the nature of random response is analyzed using Lyapunov exponents to determine whether the vessel exhibits any chaotic dynamics.

S. Balasuriya: “Barriers and Transport in Unsteady Flows: A Melnikov Approach”

S. Balasuriya: “Barriers and Transport in Unsteady Flows: A Melnikov Approach”

Beyond the Melnikov method II: Multidimensional setting

We present a Melnikov type approach for establishing transversal intersections of stable/unstable manifolds of perturbed normally hyperbolic invariant manifolds. We do not need to know the explicit formulas for the homoclinic orbits prior to the perturbation. We also do not need to compute any integrals along such homoclinics. All needed bounds are established using rigorous computer assisted numerics. Lastly, and most importantly, the method establishes intersections for an explicit range of parameters, and not only for perturbations that are ‘small enough’, as is the case in the classical Melnikov approach.

Chaos analysis of ship rolling motion in stochastic beam seas

In this paper, the chaotic motion of ship roll in stochastic beam seas, which is regarded as a bounded noise is investigated in detail. The stochastic Melnikov approach is applied to the model and the criterion for the chaos in the mean-square sense is derived. The chaotic thresholds of noise parameters obtained by means of the stochastic Melnikov process are verified by the numerical safe basin. Besides of the factors of noise disturbances, the effects of the other parameters in the system on safe basin are also discussed systematically. The varieties of coefficients of the restoring moment can also induce the erosion of safe basin and lead to the occurrence of the chaos. On the other hand, the increase of damping coefficients can enhance the safe domain of ship sailing.

Pavel Ivanovich Melnikov as Writer and Ethnologist

This article is devoted to one the aspects intellectual heritage P. Melnikov. Pavel Ivanovich Melnikov (1819-1883) was an ethnographer-fiction writer who based the descriptions on big actual and documentary material, giving it art and literary interpretation. His main ethnographic work is the Mordva sketches published in 1867 in the Russian Bulletin magazine, but not finished. The analysis his publications shows that actually the ethnography mordva had no self-sufficing value for P. I. Melnikov. Melnikov shared the idea Slavic cultural and historical type, general for Slavophiles-pochvenniks, which during historical development assimilate Finno-Ugric tribes. The article is based on application a comparative method, and comparison to N. Danilevsky's treatise Russia and Europe allowed to draw a conclusion that Melnikov and Danilevsky's approach concerning situation of foreigners in the Russian Empire coincided in a position concerning paganism. Traditional beliefs the people the Volga region were considered by them as reflection and memory ancient Slavic paganism. The historical past Slavs and finno-ugrs was considered by Melnikov-Pechersky as the general, and modern condition traditional family institutes and pagan religious representations seemed to them the direct evidence old Slavic common life. From the point view modern historical science Melnikov-Pechersky came to a right conclusion about the early beginning a Salvic incorporation the Finno-Ugric population the Volga region. Their descriptions life and religion mordva keep value the important primary source. This question occupied Danilevsky in the last turn. This study can be interest to ethnographers and researchers the Russian social thought the 19th century.

Beyond the Melnikov method: A computer assisted approach

We present a Melnikov type approach for establishing transversal intersections of stable/unstable manifolds of perturbed normally hyperbolic invariant manifolds (NHIMs). The method is based on a new geometric proof of the normally hyperbolic invariant manifold theorem, which establishes the existence of a NHIM, together with its associated invariant manifolds and bounds on their first and second derivatives. We do not need to know the explicit formulas for the homoclinic orbits prior to the perturbation. We also do not need to compute any integrals along such homoclinics. All needed bounds are established using rigorous computer assisted numerics. Lastly, and most importantly, the method establishes intersections for an explicit range of parameters, and not only for perturbations that are ‘small enough’, as is the case in the classical Melnikov approach.

Nonlinear vibration of a post-buckled giant-magnetostrictive thin film laminated beam

One buckled simply supported giant magnetostrictive material (GMM) thin film laminated beam model subject to axial magnetic excitations is proposed. Themagnetostrictive force issimplified as a harmonic excitation. Applying the Hamilton principle and Galerkin approach the model is expressed as a one dimensional vibrating model with parametrical excitation. The multi-scales method is used to obtain the approximation of the solution when the vibration is around the buckled position and the amplitude of the oscillation is small. With the help of the undermined fundamental frequency method and normal form theory the Melnikov approach is improved to calculate more precise threshold value of the amplitude of excitation leading to chaos. The numerical simulation shows the performance of the new Melnikov approach is much better than the traditional way.

Vibration of a shape memory alloy beam and detecting the threshold value of chaos induced by noise

The aim of this study is to describe the hysteretic nonlinear characteristic of the strain–stress relationship of a shape memory alloy; a Van-der-Pol hysteretic cycle is applied to simulate the hysteretic loops. Then, the model of a simply supported shape memory alloy beam with nonlinear damping was proposed. The stability of the free vibration is studied. The criterion determining the stochastic chaos is obtained by the random Melnikov approach. To verify the theoretical analysis, numerical experiments are carried out. The results totally agreed with the analytical work. Clear fractal boundaries of the system's safe basin are observed.

Fourth body gravitation effect on the resonance orbit characteristics of the restricted three-body problem

In this paper, the gravitational effect of a fourth body on the resonance orbit defined in the restricted three-body problem (RTBP) is considered. In this regard, Resonance Hamiltonian of the RTBP and the Hamiltonian associated with the fourth gravitational body that perturbs the resonance orbit are computed. The Melnikov approach is utilized as a mean for the detection of chaos in resonance orbit under the influence of the fourth gravitation body. In addition, the numerical simulation of RTBP and bicircular four-body model, time–frequency analysis (TFA), and fast Lyapunov indicator (FLI) are performed to verify the results of the Melnikov approach. The results indicate that for the (2:1) resonance orbit, the Melnikov integral computed over outer loop of separatrix does not cross the zero line, and consequently chaos is unexpected. On the other hand, the Melnikov integral computed over the inner sepratrix loop crosses the zero line indicating a potential for chaos. Similarly, it is shown that inclusion of the fourth body gravitation leads the (3:1) as well as the (4:1) resonance orbits to chaos. Additionally, simulation results indicate that for some initial conditions on the separatrix, the fourth body effect bounds the amplitude of the resonance orbits while diffusing its corresponding trajectory in the bounded phase space. TFA and the FLI verify similar results.

Modeling of a Shape Memory Alloy Beam and its Stochastic Chaos

The van-der-pol hysteretic cycle was applied to describe the hysteretic nonlinear characteristic of the strain-stress relation of a shape memory alloy (SMA). A new model with nonlinear damping of a simply supported SMA beam was proposed. The Criterions determining the stochastic chaos is obtained by the random Melnikov approach. The numerical results show the effectiveness of the theoretical analysis. Clear fractal boundaries of the system's safe basin is observed.