Melnikov Analysis Research Articles

Melnikov analysis in nonsmooth differential systems with nonlinear switching manifold

We study the family of piecewise linear differential systems in the plane with two pieces separated by a cubic curve. Our main result is that 7 is a lower bound for the Hilbert number of this family. In order to get our main result, we develop the Melnikov functions for a class of nonsmooth differential systems, which generalizes, up to order 2, some previous results in the literature. Whereas the first order Melnikov function for the nonsmooth case remains the same as for the smooth one (i.e. the first order averaged function) the second order Melnikov function for the nonsmooth case is different from the smooth one (i.e. the second order averaged function). We show that, in this case, a new term depending on the jump of discontinuity and on the geometry of the switching manifold is added to the second order averaged function.

On the analysis of nonlinear dynamic behavior of an isolation system with irrational restoring force and fractional damping

An archetypal isolation system with rational restoring force and fractional damping is proposed and investigated based on the nonlinear mechanism of geometric kinematics. The equations of motion of this nonlinear isolator subject to nonlinear damping and external excitation are derived based on the Lagrange equation. For the free vibration system, the nonlinear irrational restoring force, nonlinear stiffness behaviors, and fractional damping are investigated to show the complex transitions of multi-stability. For the forced vibration system, the analytical expressions of force transmissibility of the nonlinear isolator with single-well potential under the perturbation of viscous damping and harmonic forcing are formulated by applying the harmonic balance method. The shock response spectra of the perturbed system subject to half-sine input are evaluated by the maximum responses. The Melnikov analysis and empirical method are employed to determine the analytical criteria of chaotic thresholds for the homoclinic orbit of the perturbed system with symmetric double-well characteristics. The numerical simulations are carried out to demonstrate periodic solutions, periodic doubling bifurcation, and chaotic solutions. The maximum displacements have been obtained to show the isolation characteristics in the case of chaotic vibration.

Efficient targeted energy transfer of bistable nonlinear energy sink: application to optimal design

This paper is dedicated to the optimal design of a bistable nonlinear energy sink (NES) for the vibration control of a periodically excited linear oscillator. This system with negative linear and cubic nonlinear coupling is analytically studied with the method of multiple scales. As a result, a slow invariant manifold is obtained and is applied to predict four typical response regimes at different energy levels. Moreover, asymptotic analysis and Melnikov analysis are, respectively, used to obtain the thresholds of these typical responses. Through their efficiency comparison, it is observed that the bistable NES can be efficient and robust in a broad range of excitation amplitude. With the Hilbert transform and wavelet transform, targeted energy transfer with transient or permanent 1:1 resonance is found to be responsible for the effectiveness of such responses as strongly modulated response and 1:1 resonance. Finally, an optimal design criterion and a corresponding parameter configuration are proposed to guide the application of this type of NES.

Gain–loss-driven travelling waves in PT-symmetric nonlinear metamaterials

In this work we investigate a one-dimensional parity-time (PT)-symmetric magnetic metamaterial consisting of split-ring dimers having gain or loss. Employing a Melnikov analysis we study the existence of localized travelling waves, i.e. homoclinic or heteroclinic solutions. We find conditions under which the homoclinic or heteroclinic orbits persist. Our analytical results are found to be in good agreement with direct numerical computations. For the particular nonlinearity admitting travelling kinks, numerically we observe homoclinic snaking in the bifurcation diagram. The Melnikov analysis yields a good approximation to one of the boundaries of the snaking profile.

Melnikov analysis of chaos and heteroclinic bifurcation in Josephson system driven by an amplitude-modulated force

In this paper, the heteroclinic bifurcation and chaos in Josephson system subjected to an amplitude modulated force is discussed. By applying Melnikov method, it will be obtained that the heteroclinic bifurcation conditions of existence of chaotic motion. The effects of parameters of system on dynamical behaviors is also studied by using numerical simulation. Numerical simulation including bifurcation diagram of fixed points, phase portraits, bifurcation diagrams of system, not only show the consistent with the theoretical analysis but also exhibit the interesting bifurcation diagrams and the more new complex dynamical behaviors.

Computing Melnikov Curves for Periodically Perturbed Piecewise Smooth Oscillators

Curves dividing the parameter plane into regions according to the presence or absence of homoclinic or heteroclinic tangle corresponding to the periodically perturbed saddle of the piecewise smooth oscillator are studied using Melnikov analysis. The analysis is not simplified by choosing the discontinuity plane at a convenient location. Separatrix of the unperturbed system is parametrized exactly in a piecewise manner. Switching times, i.e. parameter values at which the separatrix crosses the discontinuity plane, are obtained. Switching times split the Melnikov integral into various subintegrals which are evaluated either exactly using term-wise integration of the infinite series of the integrand or approximately using a finite-term series approximation of the integrand, the latter being computationally an extensive task. Integral evaluations though approximate, are purely analytical expressions in terms of special functions such as digamma and hypergeometric. Melnikov plots show that the boundary between three regions in the parameter plane differ qualitatively in case of parametric and external excitations, however; adding self-excitation to the external one does not much alter the boundary qualitatively and quantitatively.

Homoclinic bifurcations and chaotic dynamics for a piecewise linear system under a periodic excitation and a viscous damping

In this paper, homoclinic bifurcations and chaotic dynamics of a piecewise linear system subjected to a periodic excitation and a viscous damping are investigated by the Melnikov analysis for nonsmooth systems in detail. The piecewise linear system can be seen as a simple linear feedback control system with dead zone and saturation constrains. The unperturbed system is a piecewise linear Hamiltonian system, which contains two parameters and exhibits quintuple well characteristic. The discontinuous unperturbed system, which is obtained by reducing the two parameters to zero, has saddle-like singularity and homoclinic-like orbit. Analytical expressions for the unperturbed homoclinic and heteroclinic orbits are derived by using Hamiltonian function for the piecewise linear system. The Melnikov analysis for nonsmooth planar systems is first described briefly, and the theorem for homoclinic bifurcations for the nonsmooth planar systems is also obtained and then employed to detect the homoclinic and heteroclinic tangency under the perturbation of a viscous damping and a periodic excitation. Finally, the chaotic attractors and the bifurcation diagrams are presented to show the bifurcations and chaotic dynamics of the piecewise linear system.

Bifurcation Results for Traveling Waves in Nonlinear Magnetic Metamaterials

In this work, we study a model of a one-dimensional magnetic metamaterial formed by a discrete array of nonlinear resonators. We focus on periodic and localized traveling waves of the model, in the presence of loss and an external drive. Employing a Melnikov analysis we study the existence and persistence of such traveling waves, and study their linear stability. We show that, under certain conditions, the presence of dissipation and/or driving may stabilize or destabilize the solutions. Our analytical results are found to be in good agreement with direct numerical computations.

Identification and control of chaos in nonlinear gear dynamic systems using Melnikov analysis

In this paper, the Melnikov analysis is extended to develop a practical model of gear system to control and eliminate the chaotic behavior. To this end, a nonlinear dynamic model of a spur gear pair with backlash, time-varying stiffness and static transmission error is established. Based on the Melnikov analysis the global homoclinic bifurcation and transition to chaos in this model are predicted. Then non-feedback control method is used to eliminate the chaos by applying an additional control excitation. The regions of the parameter space for the control excitation are obtained analytically. The accuracy of the theoretical predictions and also the performance of the proposed control system are verified by the comparison with the numerical simulations. The simulation results show effectiveness of the proposed control system and present some useful information to analyze and control the gear dynamical systems.

Melnikov analysis and chaos control of nonlinear dispersive KdV equation under external periodic perturbation

The behavior of non-smooth solitary waves switching to chaos is studied. Firstly, we present some singular homoclinic orbits of an unperturbed system. These singular homoclinic orbits correspond to non-smooth solutions. Secondly, we find that the peculiar solitary waves are more likely to be chaos by using the Melnikov theory. Finally, chaos thresholds under different amplitudes and frequencies of a periodic perturbation are given. One interesting finding is that there exists a peculiar perturbation frequency, which has significant effect on the system. The system is not well-controlled under this frequency. However, the system can be well controlled, when the frequency of the perturbation surpasses the peculiar perturbation frequency with fixed parameters of the unperturbed system.

Homoclinic, subharmonic, and superharmonic bifurcations for a pendulum with periodically varying length

Dynamic behavior of a weightless rod with a point mass sliding along the rod axis according to periodic law is studied. This is the simplest model of child's swing. Melnikov's analysis is carried out to find bifurcations of homoclinic, subharmonic oscillatory, and subharmonic rotational orbits. For the analysis of superharmonic rotational orbits the averaging method is used and stability of obtained approximate solution is checked. The analytical results are compared with numerical simulation results.

Amplitude modulation control of escape from a potential well

We demonstrate the effectiveness of periodic amplitude modulations in controlling (suppressing and enhancing) escape from a potential well through the universal model of a damped Helmholtz oscillator subjected to an external periodic excitation (the escape-inducing excitation) whose amplitude is periodically modulated (the escape-controlling excitation). Analytical and numerical results show that this multiplicative control works reliably for different subharmonic resonances between the two periodic excitations involved, and that its effectiveness is comparable to those of different methods of additive control. Additionally, we demonstrate the robustness of the multiplicative control against the presence of low-intensity Gaussian noise. • Multiplicative control of escape from a potential well has been demonstrated. • Theoretical predictions are obtained from a Melnikov analysis. • It has been shown the robustness of the multiplicative control against noise.

PHASE CONTROL IN THE MASS-SPRING MODEL WITH NONSMOOTH STIFFNESS AND EXTERNAL EXCITATION

The control of chaotic dynamics in a nonlinear mass-spring model with nonsmooth stiffness is analyzed here. This is carried out by applying the phase control technique, which uses a periodic perturbation of a suitable phase ϕ. For this purpose, we take as prototype model a system consisting of a double-well potential with an additional spring component, which acts into the system only for large enough displacements. The crucial role of the phase is evidenced by using numerical simulations and also by using analytical methods, such as the Melnikov analysis. We expect that our results might be fruitful with implications in some mechanical problems such as suspension of vehicles, among others, where similar models are extensively used.

On the nonlinear on–off dynamics of a butterfly valve actuated by an induced electromotive force

In this paper, we study the nonlinear dynamics of a butterfly valve actuated by the induced electromotive force (emf) of a permanent magnet, with a focus on the on–off dynamics of the valve and its nonlinear response under ambient perturbation. The complex interplay between the electromagnetic, hydrodynamic and mechanical forces leads to a fundamentally multiphysical, nonlinear dynamical model for the problem. First, we analyze the stability of the on–off conditions in terms of three critical dynamical parameters – the actuating DC voltage, inlet velocity and the opening angle. Next, the response of the system to perturbations around the equilibrium points is studied in terms of the frequency response using the method of multiple scales. Finally, evidence of fractality is established using Melnikov analysis and a plot of the basins of attraction. The results reported in the article, in addition to being of fundamental theoretical interest, are expected to impact practical design considerations of electromechanical butterfly valves.

Exponentially small splitting of separatrices beyond Melnikov analysis: Rigorous results

We study the problem of exponentially small splitting of separatrices of one degree of freedom classical Hamiltonian systems with a non-autonomous perturbation fast and periodic in time. We provide a result valid for general systems which are algebraic or trigonometric polynomials in the state variables. It consists on obtaining a rigorous proof of the asymptotic formula for the measure of the splitting. We obtain that the splitting has the asymptotic behavior Kεβe−a/ε, identifying the constants K, β, a in terms of the system features.We consider several cases. In some cases, assuming the perturbation is small enough, the values of K, β coincide with the classical Melnikov approach. We identify the limit size of the perturbation for which this theory holds true. However for the limit cases, which appear naturally both in averaging and bifurcation theories, we encounter that, generically, K and β are not well predicted by Melnikov theory.

An extended high-dimensional Melnikov analysis for global and chaotic dynamics of a non-autonomous rectangular buckled thin plate

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.

Impulse-induced optimum control of escape from a metastable state by periodic secondary excitations

We characterize the role of the impulse transmitted (time integral over a half-period) by resonant secondary excitations at controlling (suppressing and enhancing) escape from a potential well, which is induced by periodic primary excitations. By using the universal model of a dissipative Helmholtz oscillator, we demonstrate numerically that optimum control of escape occurs when the impulse transmitted by the chaos-controlling excitations is maximum while keeping their amplitude and period fixed. These findings are in complete agreement with analytical predictions from two independent methods: Melnikov analysis and energy-based analysis. Additional numerical results corresponding to other alternative escape-controlling excitations demonstrate the generality of the essential role of the excitation's impulse.

When complexity leads to simplicity: Ocean surface mixing simplified by vertical convection

The effect of weak vertical motion on the dynamics of materials that are limited to move on the ocean surface is an unresolved problem with important environmental and ecological implications (e.g., oil spills and larvae dispersion). We investigate this effect by introducing into the classical horizontal time-periodic double-gyre model vertical motion associated with diurnal convection. The classical model produces chaotic advection on the surface. In contrast, the weak vertical motion simplifies this chaotic surface mixing pattern for a wide range of parameters. Melnikov analysis is employed to demonstrate that these conclusions are general and may be applicable to realistic cases. This counter intuitive result that the very weak nocturnal convection simplifies ocean surface mixing has significant outcomes.

NONLINEAR RESPONSE OF THE MASS-SPRING MODEL WITH NONSMOOTH STIFFNESS

In this paper, we study the nonlinear response of the nonlinear mass-spring model with nonsmooth stiffness. For this purpose, we take as prototype model, a system that consists of the double-well smooth potential with an additional spring component acting on the system only for large enough displacement. We focus our study on the analysis of the homoclinic orbits for such nonlinear potential for which we observe the appearance of chaotic motion in the presence of damping effects and an external harmonic force, analyzing the crucial role of the linear spring in the dynamics of our system. The results are shown by using both the Melnikov analysis and numerical simulations. We expect our work to have implications on problems concerning the suspension of vehicles, among others.

Bifurcation and Chaos Control in Thin Circular Functionally Graded Plates

In the present study, harmonic and chaotic behaviors of a thin circular functionally graded (FG) plate are studied. Using nonlinear governing equations of FG plate subjected to one-term excitation force and utilizing special shape function, Duffing equation of the system is derived. Employing Melnikov analysis, the conditions of existence of harmonic and chaotic motions for derived Duffing equation are expressed. By use of the Poincare maps, the identified dynamical behaviors are illustrated based on the numerical solutions of the governing equations. The bifurcation diagrams are presented in the case that the excitation amplitude increases, while other parameters are fixed. It is indicated that, with the increase of the excitation amplitude, the motion changes from period-1 to period-2, and then to period-4, and finally is chaotic. It is confirmed that the graded plate with more ceramic constituent has higher resistance to chaotic phenomenon in comparison with the graded plate with more metal constituent. Finally, a simple control method is employed to control the chaotic vibration of the plate.