Subharmonic Melnikov Function Research Articles

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.

Sub-harmonic Melnikov function for a high-dimensional non-smooth coupled system

In this paper, a double pendulum model with multi-point collision is established to study the sub-harmonic bifurcation of high-dimensional coupled non-smooth systems. Considering the coupling and non-smoothness of the system, a two-step decoupling method is proposed to detect the sub-harmonic bifurcation of a two-degree-of-freedom non-smooth coupled system. The core view is to introduce energy-time scale transformation to overcome the obstacle of the system coupled term. In the first step, a reversible transformation is introduced to decouple the system. This transformation enables the coupled form of the impact term, which presents novel obstacles to the high-dimensional non-smooth system. By introducing energy-time scale transformation in the second step, the system is expressed as a smooth decoupling form of the energy coordinate, and the trouble of impact term coupled is solved. Furthermore, the sub-harmonic Melnikov function which depends on frequency, amplitude of excitation and impact recovery coefficient is derived by using the two-step decoupling method. Hence, the sub-harmonic Melnikov function is extended to the high-dimensional non-smooth system, which reveals the influence of the impact recovery coefficient on the existence of sub-harmonic periodic orbits. The innovation of this method is that it solves the coupled problem of non-smooth terms, quantifies the impact of impact recovery coefficient on the dynamic behavior of the system, and provides a theoretical basis for the actual parameter design and control in engineering. The obtained theoretical results are verified through the numerical simulations.

Subharmonic Bifurcation for a Non-smooth Double Pendulum with Unilateral Impact

A unilateral impact double pendulum model with hinge links is constructed to detect subharmonic bifurcation for the high dimensional non-smooth system. The non-smooth and nonlinear coupled factors lead a barrier for high dimensional conventional nonlinear techniques. By introducing reversible transformation and energy time scale transformation, the system is expressed as a smooth decoupling form of energy coordinates. Thus, the concept of subharmonic Melnikov function is extended to high-dimensional nonsmooth systems, and the influence of impact recovery coefficient on the existence of subharmonic periodic orbits of double pendulum is revealed. The efficiency of the theoretical results is verified by phase portraits, time process portraits and Poincaré section.

Persistence of periodic and homoclinic orbits, first integrals and commutative vector fields in dynamical systems

We study persistence of periodic and homoclinic orbits, first integrals and commutative vector fields in dynamical systems depending on a small parameter ɛ > 0 and give several necessary conditions for their persistence. Here we treat homoclinic orbits not only to equilibria but also to periodic orbits. We also discuss some relationships of these results with the standard subharmonic and homoclinic Melnikov methods for time-periodic perturbations of single-degree-of-freedom Hamiltonian systems, and with another version of the homoclinic Melnikov method for autonomous perturbations of multi-degree-of-freedom Hamiltonian systems. In particular, we show that a first integral which converges to the Hamiltonian or another first integral as the perturbation tends to zero does not exist near the unperturbed periodic or homoclinic orbits in the perturbed systems if the subharmonic or homoclinic Melnikov functions are not identically zero on connected open sets. We illustrate our theory for four examples: the periodically forced Duffing oscillator, two identical pendula coupled with a harmonic oscillator, a periodically forced rigid body and a three-mode truncation of a buckled beam.

Subharmonic solution of a vibration system with two mass elastic connecting rods

In this article, the dynamic characteristic of linkage bobbing machine with balanced double-mass bearing periodic excitation and damping is studied. Subharmonic Melnikov function of the oscillating periodic orbits is computed through Melnikov method. And, the relationship of parameters is given when the subharmonic bifurcation occurs. The periodic motion is simulated. Through analysis, as the excitation frequency varies, the periodic motion undergoes flip bifurcations and subharmonic bifurcation occurs, which finally leads to chaos. In addition, the ultra-subharmonic solution of the system is given.

Subharmonic Bifurcation for a Nonsmooth Oscillator

A nonsmooth pendulum model with multiple impulse effect is constructed to detect the bifurcation of a periodic orbit with multiple jump discontinuous points. Subharmonic Melnikov function of this kind of nonsmooth systems is studied. Differences of subharmonic Melnikov function between the nonsmooth system with multiple jump discontinuities and the smooth system are analyzed by using the Hamiltonian function and piecewise integral method. Applying the recursive method and perturbation principle, the effects of the jump discontinuous points on the subharmonic Melnikov function are converted to integral items which can be easily calculated. Hence, the subharmonic Melnikov function for the subharmonic orbit with multiple jump discontinuous points is obtained. Finally, the existence conditions for periodic motion of the subharmonic orbit are derived and the efficiency of the conclusions is verified via numerical simulations.

Global Analysis on a Discontinuous Dynamical System

We have studied a Filippov system [Formula: see text] with small [Formula: see text], [Formula: see text] and [Formula: see text] being periodic. Since [Formula: see text] is an abstract function, the subharmonic Melnikov function cannot be computed. In other words, for this system the Melnikov method loses effectiveness. First, we proved that the equation has a unique harmonic solution, a unique [Formula: see text]-subharmonic solution for any [Formula: see text] and they are Lyapunov asymptotically stable. Moreover, this equation has no other type of periodic solutions. Further, the attractor of this system is not chaotic. Finally, some numerical examples are given.

Subharmonic Melnikov theory for degenerate resonance systems and its application

This article explores the subharmonic orbits of a four-dimensional degenerate resonance non-autonomous system. The well-known subharmonic Melnikov method is improved in this paper. Suppose the unperturbed system is a two-degree-of-freedom decoupled Hamiltonian system and possesses a family of periodic orbits. The important issue is the persistence of the periodic orbits after periodic perturbations. In order to solve this problem, we propose a four-dimensional subharmonic Melnikov function based on periodic transformations and Poincare map. Then, a main theorem, which can be used to check the existence of the periodic orbits for the four-dimensional degenerate resonance system, is presented and proved by using the implicit function theorem. In order to verify the validity and applicability of the improved subharmonic Melnikov method, we apply it to investigate the subharmonic orbits of a honeycomb sandwich plate. The theoretical result shows that the subharmonic orbits of period 2T can exist under certain conditions. Numerical simulations are carried out to verify the analytical predictions. It is found that the subharmonic orbits for the honeycomb sandwich plate do exist based on the results of numerical simulations.

The pattern of parallel edge plasma flows due to pressure gradients, recycling, and resonant magnetic perturbations in DIII-D

High resolution plasma transport simulations with the EMC3-EIRENE code have been performed to address the parallel plasma flow structure in the boundary of a poloidal divertor configuration with non-axisymmetric perturbations at DIII-D. Simulation results show that a checkerboard pattern of flows with alternating direction is generated inside the separatrix. This pattern is aligned with the position of the main resonances (i.e., where the safety factor is equal to rational values q=m/n for a perturbation field with base mode number n): m pairs of alternating forward and backward flow channel exist for each resonance. The poloidal oscillations are aligned with the subharmonic Melnikov function, which indicates that the plasma flow is generated by parallel pressure gradients along perturbed field lines. An additional scrape-off layer-like domain is introduced by the perturbed separatrix which guides field lines from the interior to the divertor targets, resulting in an enhanced outward flow that is consistent with the experimentally observed particle pump-out effect. However, while the lobe structure of the perturbed separatrix is very well reflected in the temperature profile, the same lobes can appear to be smaller in the flow profile due to a competition between high upstream pressure and downstream particle sources driving flows in opposite directions.

Multiple Buckling and Codimension-Three Bifurcation Phenomena of a Nonlinear Oscillator

In this paper, we investigate the global bifurcations and multiple bucklings of a nonlinear oscillator with a pair of strong irrational nonlinear restoring forces, proposed recently by Han et al. [2012]. The equilibrium stabilities of multiple snap-through buckling system under static loading are analyzed. It is found that complex bifurcations are exhibited of codimension-three with two parameters at the catastrophe point. The universal unfolding for the codimension-three bifurcation is also found to be equivalent to a nonlinear viscous damped system. The bifurcation diagrams and the corresponding codimension-three behaviors are obtained by employing subharmonic Melnikov functions for the existing singular closed orbits of homoclinic, tangent homoclinic, homo-heteroclinic and cuspidal heteroclinic, respectively.

BIFURCATIONS OF PERIODIC SOLUTIONS AND CHAOS IN DUFFING-VAN DER POL EQUATION WITH ONE EXTERNAL FORCING

The Duffing-Van der Pol equation withfifth nonlinear-restoring force and one external forcing term isinvestigated in detail: the existence and bifurcations of harmonicand second-order subharmonic, and third-order subharmonic,third-order superharmonic and $m$-order subharmonic under smallperturbations are obtained by using second-order averaging methodand subharmonic Melnikov function; the threshold values of existenceof chaotic motion are obtained by using Melnikov method. Thenumerical simulation results including the influences of periodicand quasi-periodic and all parameters exhibit more new complexdynamical behaviors. We show that the reverse period-doublingbifurcation to chaos, period-doubling bifurcation to chaos,quasi-periodic orbits route to chaos, onset of chaos, and chaossuddenly disappearing, and chaos suddenly converting to periodorbits, different chaotic regions with a great abundance of periodicwindows (periods:1,2,3,4,5,7,9,10,13,15,17,19,21,25,29,31,37,41, andso on), and more wide period-one window, and varied chaoticattractors including small size and maximum Lyapunov exponentapproximate to zero but positive, and the symmetry-breaking ofperiodic orbits. In particular, the system can leave chaotic regionto periodic motion by adjusting the parameters $p, \beta, \gamma, f$and $\omega$, which can be considered as a control strategy.

The Melnikov Method and Subharmonic Orbits in a Piecewise-Smooth System

We consider a two-dimensional piecewise-smooth system defined in two domains separated by a switching manifold $\Sigma$. We assume that there exists a piecewise-defined continuous Hamiltonian that is a first integral of the system. We also suppose that the system possesses an invisible fold-fold at the origin and two heteroclinic orbits connecting two hyperbolic critical points on either side of $\Sigma$. Finally, we assume that the region enclosed by these heteroclinic connections is fully covered by periodic orbits surrounding the origin, whose periods monotonically increase as they approach the heteroclinic connection. For a nonautonomous ($T$-periodic) Hamiltonian perturbation of amplitude $\varepsilon$, we rigorously prove, for every $n$ and $m$ relatively prime and $\varepsilon>0$ small enough, that there exists an $nT$-periodic orbit impacting $2m$ times with the switching manifold at every period if a modified subharmonic Melnikov function possesses a simple zero. We also prove that if the orbits ...

The chaotic vibration of a metal plate with large deflection under a transverse magnetic field

By using Karman’s plate theory of large deflection, the nonlinear equation of motion of a thin metal plate with the coaction of a transverse uniform magnetic field and a transverse load is established. These equations consider the magnetic Lorentz force induced by the eddy current. Based on the Bubnov-Galerkin method, the nonlinear partial differential equation is transformed into a third-order nonlinear ordinary differential equation. By using the sub-harmonic orbit Melnikov function method, the criterion of the Smale-horseshoe chaos is also acquired. Furthermore, the chaotic motion is numerically simulated with Matlab. The bifurcation diagram, the phase curve, the Poincaré map and the evolution curve are calculated. The digital characteristics of the chaotic motions are provided based on the analysis. The analysis results show that the magnetic induction intensity and the external load may affect the vibration of the system.

Dynamic interactions of traveling waves propagating in a linear chain with an local essentially nonlinear attachment

We study strongly nonlinear dynamical interactions between traveling waves propagating in a linear spring-mass chain with a strongly nonlinear, lightweight local attachment. We analyze the dynamics of this system by constructing a reduced model in the form of a strongly nonlinear integro-differential equation with inhomogeneous terms representing local and non-local interactions between the chain and the nonlinear attachment. Then we construct homoclinic and subharmonic Melnikov functions and prove the existence of chaotic motions and subharmonic periodic orbits in the combined chain-attachment system. In the limit of weak coupling between the particles of the chain we study the bifurcations that generate stable-unstable pairs of subharmonic motions. Generalizations of the methodology to a more general class of linear systems with local essentially nonlinear attachments are discussed. This work provides further evidence that the break of symmetry of an otherwise linear chain by a strongly nonlinear (even lightweight) attachment can give rise to complex (even chaotic) dynamics. The underlying dynamical mechanism of this complexity is nonlinear resonant energy transfer from the traveling waves to the nonlinear attachment. The presented results contribute towards the study of the dynamic and resonance interactions of waves propagating in extended media with strongly nonlinear local attachments.

Melnikov theory to all orders and Puiseux series for subharmonic solutions

We study the problem of subharmonic bifurcations for analytic systems in the plane with perturbations depending periodically on time, in the case in which we only assume that the subharmonic Melnikov function has at least one zero. If the order of zero is odd, then there is always at least one subharmonic solution, whereas if the order is even, in general, other conditions have to be assumed to guarantee the existence of subharmonic solutions. Even when such solutions exist, in general, they are not analytic in the perturbation parameter. We show that they are analytic in a fractional power of the perturbation parameter. To obtain a fully constructive algorithm which allows us not only to prove existence but also to obtain bounds on the radius of analyticity and to approximate the solutions within any fixed accuracy, we need further assumptions. The method we use to construct the solution—when this is possible—is based on a combination of the Newton–Puiseux algorithm and the tree formalism. This leads to a graphical representation of the solution in terms of diagrams. Finally, if the subharmonic Melnikov function is identically zero, we show that it is possible to introduce higher order generalizations, for which the same kind of analysis can be carried out.

BIFURCATION CURVES OF SUBHARMONIC SOLUTIONS AND MELNIKOV THEORY UNDER DEGENERACIES

We study perturbations of a class of analytic two-dimensional autonomous systems with perturbations depending periodically on time; for instance one can imagine a periodically driven or forced system with one degree of freedom. In the first part of the paper, we revisit a problem considered by Chow and Hale on the existence of subharmonic solutions. In the analytic setting, under more general (weaker) conditions on the perturbation, we prove their results on the bifurcation curves dividing the region of non-existence from the region of existence of subharmonic solutions. In particular our results apply also when one has degeneracy to the first order — i.e. when the subharmonic Melnikov function is identically constant. Moreover we can deal as well with the case in which degeneracy persists to arbitrarily high orders, in the sense that suitable generalizations to higher orders of the subharmonic Melnikov function are also identically constant. The bifurcation curves consist of four branches joining continuously at the origin, where each of them can have a singularity (although generically they do not). The branches can form a cusp at the origin: we say in this case that the curves are degenerate as the corresponding tangent lines coincide. The method we use is completely different from that of Chow and Hale, and it is essentially based on the proof of convergence of the perturbation theory. It also allows us to treat the Melnikov theory in degenerate cases in which the subharmonic Melnikov function is either identically vanishing or has a zero which is not simple. This is investigated at length in the second part of the paper. When the subharmonic Melnikov function has a non-simple zero, we consider explicitly the case where there exist subharmonic solutions, which, although not analytic, still admit a convergent fractional series in the perturbation parameter.

Bifurcations of periodic solutions and chaos in Josephson system

The Josephson equation is investigated in detail: the existence and bifurcationsfor harmonic and subharmonic solutions under small perturbations areobtained by using second-order averaging method and subharmonic Melnikov function,and the criterion of existence for chaos is proved by Melnikov analysis; thebifurcation curves about n-subharmonic and heteroclinic orbits and the driving frequency$\omega$ effects to the forms of chaotic behaviors are given by numerical simulations.

Subharmonic bifurcations in a pendulum parametrically excited by a non-harmonic perturbation

A pendulum whose support is subjected to a periodic non-harmonic oscillation in the vertical direction is considered. The subharmonic Melnikov functions for the oscillating and for the rotating motions are explicitly constructed. It is shown that both functions converge towards the homoclinic Melnikov function and furthermore all the results for the harmonic perturbation are recovered.

Lyapunov-Schmidt Reduction and Melnikov Integrals for Bifurcation of Periodic Solutions in Coupled Oscillators

We consider a system of autonomous ordinary differential equations depending on a small parameter such that the unperturbed system has an invariant manifold of periodic solutions. The problem addressed here is the determination of sufficient geometric conditions for some of the periodic solutions on this invariant manifold to survive after perturbation. The main idea is to use a Lyapunov-Schmidt reduction for an appropriate displacement function in order to obtain the bifurcation function for the problem in a form which can be recognized as a generalization of the subharmonic Melnikov function. Thus, the multidimensional bifurcation problem can be cast in a form where the geometry of the problem is clearly incorporated. An important application can be made in case the uncoupled system of differential equations is a system of oscillators in resonance. In this case the invariant manifold of periodic solutions is just the product of the uncoupled oscillations. When each of the oscillators has one degree of freedom, the bifurcation function is computed by quadrature along the unperturbed oscillations. Additional applications include the computation of entrainment domains for a sinusoidally forced van der Pol oscillator and the computation of mutual synchronization domains for a system of inductively coupled van der Pol oscillators.

Phase space structure and chaotic scattering in near-integrable systems

We investigate the bifurcation phenomena and the change in phase space structure connected with the transition from regular to chaotic scattering in classical systems with unbounded dynamics. The regular systems discussed in this paper are integrable ones in the sense of Liouville, possessing a degenerated unstable periodic orbit at infinity. By means of a McGehee transformation the degeneracy can be removed and the usual Melnikov method is applied to predict homoclinic crossings of stable and unstable manifolds for the perturbed system. The chosen examples are the perturbed radial Kepler problem and two kinetically coupled Morse oscillators with different potential parameters which model the stretching dynamics in ABC molecules. The calculated subharmonic and homoclinic Melnikov functions can be used to prove the existence of chaotic scattering and of elliptic and hyperbolic periodic orbits, to calculate the width of the main stochastic layer and of the resonances, and to predict the range of initial conditions where singularities in the scattering function are found. In the second example the value of the perturbation parameter at which channel transitions set in is calculated. The theoretical results are supplemented by numerical experiments.