Melnikov Function Research Articles

On the number of limit cycles for a perturbed cubic reversible Hamiltonian system.

This paper is concerned with the limit cycle problem of a cubic reversible Hamiltonian system under perturbation of polynomials of degree n with a switching line x=0. The upper and lower bounds of the number of limit cycles are obtained using the first order Melnikov function and its expansion. The method for calculating the Melnikov function relies upon some iterative formulas, which differs from other approaches.

Bounding the number of limit cycles for perturbed piecewise linear Hamiltonian system

In this paper, we investigate the number of limit cycles for piecewise linear Hamiltonian system with a homoclinic loop under perturbations of piecewise smooth polynomials. By studying the number of zeros of the first order Melnikov function, we obtain the exact number of limit cycles bifurcating from the periodic annulus, which improves the results given by Liu and Xiong [13], Yang and Zhang [20].

A class of nonlinear ship stability analysis: Stochastic dynamics with time-delayed control in crosswind and wave conditions

The impact of ocean waves and crosswinds on ships has long been a focal point of research for many scholars. This paper proposes a stochastic ship rolling model influenced by crosswinds, ocean waves, and time delays. The safe operating region for ship navigation is presented through the phase space of the system. A quantitative discussion of the system is conducted using the stochastic Melnikov function, and the stochastic P-D bifurcation of the system is discussed using topological data analysis techniques. The research indicates that incorporating time delay feedback can effectively enhance the system’s stability.

Proof of two conjectures for perturbed piecewise linear Hamiltonian systems

In this paper, we study the number of limit cycles bifurcating from the centers of piecewise linear Hamiltonian systems having either a homoclinic loop or a heteroclinic loop under the perturbations of piecewise smooth polynomials. By investigating the Chebyshev properties of generating functions of the first order Melnikov functions, we obtain the sharp bounds of the number of limit cycles bifurcating from the periodic annuluses, which confirm the conjectures proposed by Liang, Han and Romanovski (2012) and Liang and Han (2016).

Energy harvesting for a vibration isolation system with negative stiffness under harmonic excitation

ABSTRACT In this paper, an energy harvesting system with a negative stiffness structure is established, whose main purpose is to achieve good vibration isolation, and at the same time, the system can also achieve the purpose of energy harvesting. In order to analyse the system model, the amplitude–frequency response equations for the displacement, induced voltage and electric power of the system when the system is in the main resonance state are obtained by using the multiscale method, and their amplitude–frequency response curves are plotted. In addition, we obtained the existence conditions for the system to satisfy Smale's horseshoe chaos using the Melnikov function. And the phase diagram and Poincaré mapping of the system were obtained using the fourth-order Runge–Kutta method. Numerical results show that the system has some energy harvesting performance along with vibration isolation. Variations in the parameters of the system affect the resonance frequency band and co-oscillation amplitude of the system as well as the nonlinear characteristics of the system itself.

Chaos detection and control of a fractional piecewise-smooth system with nonlinear damping

Chaotic response is a robust effect in natural systems, and it is usually unfavorable for applications owing to uncertainty. In this paper, we propose several control strategies to stabilize the chaotic rhythm of a fractional piecewise-smooth oscillator. First, the Melnikov analysis is applied to the system, and the critical condition for the occurrence of homoclinic chaos is scrupulously established. Then, by applying appropriate control mechanisms, including delayed feedback control and periodic excitations, to the system, we can eliminate the zeros in the original Melnikov function, which serve as sufficient criteria for chaos suppression. Numerical simulations further demonstrate the accuracy of the theoretical results and the validity of the control schemes. Finally, the effects of parameter variations on the efficiency of control strategies are investigated. Note that we use the complex Simpson formula to calculate the complicated Melnikov functions presented in this paper. The current work may open a new innovative path to detect and control the chaotic dynamics of fractional non-smooth models.

On the number of limit cycles in piecewise smooth generalized Abel equations with many separation lines

This paper investigates generalized Abel equations of the form dx/dθ=A(θ)xp+B(θ)xq, where p, q∈Z≥2, p≠q, and A(θ) and B(θ) are piecewise trigonometrical polynomials of degree m with n−1∈N+ separation lines 0<θ1<θ2<⋯<θn−1<2π. The main objective is to obtain the maximum number of non-zero limit cycles (i.e., non-zero isolated periodic solutions) that the equation can have, denoted by Hθ1,θ2,…,θn−1(m), and to analyze how the number and location of separation lines {θi}i=1n−1 affect Hθ1,θ2,…,θn−1(m). By using the theories of Melnikov functions and ECT-systems, we obtain lower bounds for Hθ1,θ2,…,θn−1(m). Our result extend those of Huang et al. who studied the special case of n=2, and reveal that the lower bounds decrease in the presence of pairs of symmetrical separation lines.

Bifurcation theory of limit cycles by higher order Melnikov functions and applications

In this paper, we study Poincaré, Hopf and homoclinic bifurcations of limit cycles for planar near-Hamiltonian systems. Our main results establish Hopf and homoclinic bifurcation theories by higher order Melnikov functions, obtaining conditions on upper bounds and lower bounds of the maximum number of limit cycles. As an application, we concern a cubic near-Hamiltonian system, and study Hopf and homoclinic bifurcations in detail, finding more limit cycles than [26].

Nonintegrability of time-periodic perturbations of single-degree-of-freedom Hamiltonian systems near homo- and heteroclinic orbits

We consider time-periodic perturbations of single-degree-of-freedom Hamiltonian systems and study their real-meromorphic nonintegrability in the Bogoyavlenskij sense using a generalized version due to Ayoul and Zung of the Morales–Ramis theory. The perturbation terms are assumed to have finite Fourier series in time, and the perturbed systems are rewritten as higher-dimensional autonomous systems having the small parameter as a state variable. We show that if the Melnikov functions are not constant, then the autonomous systems are not real-meromorphically integrable near homo- and heteroclinic orbits. Our result is not just an extension of previous results for homoclinic orbits to heteroclinic orbits and provides a stronger conclusion than them for the case of homoclinic orbits. We illustrate the theory for two periodically forced Duffing oscillators and a periodically forced two-dimensional system.

Homoclinic Bifurcations and Chaotic Dynamics in a Bistable Vibro-Impact SD Oscillator Subject to Gaussian White Noise

This paper studies the effect of Gaussian white noise on homoclinic bifurcations and chaotic dynamics of a bistable, vibro-impact Smooth-and-Discontinuous (SD) oscillator. First, the SD oscillator is reproduced and generalized by installing a slider on a fixed rod, so the slider is connected by a pair of linear springs initially pre-compressed in the vertical direction to achieve bistable vibration characteristics, and two screw nuts are installed on the rod as two adjustable bilateral rigid constraints to generate the vibro-impact. A discontinuous dynamical equation with a map defined on switching boundaries to represent velocity loss during each collision is derived to describe the vibration pattern of the bistable, vibro-impact SD oscillator through studying the persistence of the unique, unperturbed, nonsmooth, homoclinic structure. Second, the general framework of random Melnikov process for a class of bistable, vibro-impact systems contaminated with Gaussian white noise is derived and employed through the corresponding Melnikov function to obtain the necessary parameter thresholds for homoclinic tangency and possible chaos of the bistable, vibro-impact SD oscillator. Third, the effectiveness of a semi-analytical prediction by the Melnikov function is verified using the largest Lyapunov exponent, bifurcation series, and 0–1 test. Finally, the sensitivity to the initial values of chaos is verified by the fractal attractor basins, and the influence of the Gaussian white noise on periodic and chaotic structures is studied through Poincaré mapping to show the rich dynamical geometric structures.

Estimates for the number of limit cycles of the planar polynomial differential systems with homogeneous nonlinearities

This paper devotes to the study of planar polynomial differential systems with homogeneous nonlinearities of degree n>1. We are concerned with the maximum number of limit cycles surrounding the origin of such systems, denoted by Ho(n). By means of the second order analysis using the theories of Melnikov functions, we provide new estimates for Ho(n) restricted to the cases where the origin is a focus, a node, a saddle or a nilpotent singularity. In particular, Ho(n)≥n for each n in the case of focus. To the best of our knowledge, this improves the previous works in the literature.

Existence of Kink and Antikink Wave Solutions of Singularly Perturbed Modified Gardner Equation

In this paper, the singularly perturbed modified Gardner equation is considered. Firstly, for the unperturbed equation, under certain parameter conditions, we obtain the exact expressions of kink wave solution and antikink wave solution by using the bifurcation method of dynamical systems. Then, the persistence of the kink and antikink wave solutions of the perturbed modified Gardner equation is studied by exploiting the geometric singular perturbation theory and the Melnikov function method. When the perturbation parameter is sufficiently small, we obtain the sufficient conditions to guarantee the existence of kink and antikink wave solutions.

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.

Subharmonic Melnikov functions and nonintegrability for autonomous and non-autonomous perturbations of single-degree-of-freedom Hamiltonian systems near periodic orbits

We study autonomous and non-autonomous perturbations of single-degree-of-freedom Hamiltonian systems and give sufficient conditions for their real-analytic non-integrability near periodic orbits of the unperturbed systems such that the first integrals and commutative vector fields depend analytically on the small parameter by using the subharmonic Melnikov functions. Moreover, we show that autonomous dissipative perturbations prevent real-analytic integrability of these systems. Our results reveal that the perturbed systems can be real-analytically non-integrable even if there is no homoclinic/heteroclinic orbit in the unperturbed systems. We illustrate our theory with a periodically forced duffing equation and a damped Morse oscillator.

The number of limit cycles of a kind of piecewise quadratic systems with switching curve y = xm

We first study the expressions for the second-order Melnikov functions for planar piecewise smooth near-Hamiltonian systems with multiple zones separated by multiple switching curves through the origin. Next, up to the second-order Melnikov functions, we consider the number of limit cycles Z(m,2) of the system x˙=y,y˙=−x with two zones separated by the curve y=xm(2≤m∈N) under the piecewise perturbations of polynomials in x and y with degree 2. It is proved that Z(2,2)=9, 13≤Z(4,2)≤19, and Z(2k,2)≥14 for k≥3 and that Z(2k+1,2)=15 for k≥2. The results are new and improve some results in the literature.

Melnikov analysis of a perturbed integrable non-Hamiltonian system

This paper provides a completed Melnikov analysis of any order for a class of perturbed polynomial differential systems, where the unperturbed system is a cubic center with a straight line of singular points. The emphasis on any order Melnikov function is crucial because of two major reasons: one is that higher order Melnikov functions are not in general expressible explicitly, the other one is the essential difficulty of computing elliptic integrals due to the complexity of many iterations. The exact upper bounds of the number of limit cycles bifurcated from the period annulus under different polynomial perturbations are obtained by analyzing the algebraic structures from explicit expressions of any order Melnikov functions.

A class of discontinuous systems exhibit perturbed period doubling bifurcation

&lt;p&gt;This research considers discontinuous dynamical systems, which have related vector fields that shift over a discontinuity surface. These systems appear in a variety of applications, including ecology, medicine, neuroscience, and nonsmooth mechanics. The purpose of this paper is to develop a perturbation technique that measures the effect of a nonsmooth perturbation on the period doubling bifurcation of an unperturbed system. The unperturbed system is assumed to be close to a period doubling orbit, such that when the bifurcation parameter varies, the response changes from a period one to a period two limit cycle. The generalized determination of the Poincaré map associated with perturbed systems subjected to nonsmooth transitions is derived. The main techniques used in the proof of the results are normal forms and Melnikov functions, which are defined in two zones. Various examples are presented to show that non-smoothness is responsible for period doubling. To illustrate the interesting period doubling phenomenon that emerges from an existing flat periodic orbit via the non-smooth perturbation, a simple and novel discontinuous system is provided. An additional example is provided to show the emergence of a perturbed period doubling orbit near an unperturbed one.&lt;/p&gt;

Melnikov functions and limit cycle bifurcations for a class of piecewise Hamiltonian systems

&lt;abstract&gt;&lt;p&gt;This study evaluated the number of limit cycles for a class of piecewise Hamiltonian systems with two zones separated by two semi-straight lines. First, we obtained explicit expressions of higher Melnikov functions. Then we applied these expressions to find the upper bounds of the number of limit cycles bifurcated from a period annulus of a piecewise polynomial Hamiltonian system.&lt;/p&gt;&lt;/abstract&gt;

Double homoclinic bifurcations by perturbing a class of cubic Z2-equivariant polynomial systems with nilpotent singular points

This paper focuses on the study of the limit cycle bifurcation of a general cubic Z2-equivariant Hamiltonian differential system with two nilpotent singular points. Initially, the necessary and sufficient conditions for the occurrence of two nilpotent singularities are provided in the unperturbed system, along with a complete description of all possible phase portraits on the plane. Subsequently, using a combination of the Melnikov function method and various analytical techniques, we derive the approximate expansion of the first order Melnikov function at the interval endpoint, as well as formula for its coefficients. These results can be utilized to analyze double homoclinic loop bifurcation of the perturbed system, finding the lower bound of limit cycles.

Computation of Isolated Periodic Solutions for Forced Response Blade-Tip/Casing Contact Problems

Abstract This article introduces a numerical procedure dedicated to the identification of isolated branches of solutions for nonlinear mechanical systems. Here, it is applied to a fan blade subject to rubbing interactions and harmonic forcing. Both contact, which is initiated by means of the harmonic forcing, and dry friction are accounted for. The presented procedure relies on the computation of the system's nonlinear normal modes (NNM) and their analysis through the application of an energy principle derived from the Melnikov function. The dynamic Lagrangian frequency-time strategy associated with the harmonic balance method (DLFT-HBM) is used to predict the blade's dynamics response as well as to compute the autonomous nonlinear normal modes. The open industrial fan blade NASA rotor 67 is employed in order to avoid confidentiality issues and to promote the reproducibility of the presented results. Previous publications have underlined the complexity of NASA rotor 67's dynamics response as it undergoes structural contacts, thus making it an ideal benchmark blade when searching for isolated solutions. The application of the presented procedure considering a varying amplitude of the harmonic forcing allows to predict isolated branches of solutions featuring nonlinear resonances. With the use of the Melnikov energy principle, nonlinear modal interactions are shown to be responsible for the separation of branches of solutions from the main response curve. In the end, the application of the presented procedure on an industrial blade model with contact interactions demonstrates that it is both industry-ready and applicable to highly nonlinear mechanical systems.