non-Hamiltonian Perturbations Research Articles

Sensitivity of modeled tracer motion in tidal areas to numerics and to non-Hamiltonian perturbations

This study focuses on the motion of passive tracers induced by the joint action of tidal and residual currents in shallow seas with an irregular bottom topography. Interest in this problem has rapidly increased in recent years, because of the detection of large-scale pollution of marine waters by plastics. Early simplified models considered advection of tracers by a two-dimensional depth-averaged velocity field that is solenoidal, thereby resulting in a system that is Hamiltonian and nonintegrable. Here, two new aspects are considered. First, the sensitivity of solutions to three different numerical schemes is investigated. To quantify the behavior of orbits, both the largest Lyapunov exponent and the K-coefficient of the zero-one test for chaos were calculated. It turns out that a new scheme, which extends a known symplectic scheme to systems that also contain non-Hamiltonian terms, performs best. The second aspect concerns the fact that a depth-averaged velocity field is actually divergent, thereby rendering the model of tracer motion to be non-Hamiltonian. It is demonstrated that the divergent velocity components, no matter how small, cause the appearance of attractors in the system and thus they have a strong impact on the fate of tracers. Interpretation of the numerical results is given by deriving and analyzing approximate analytical solutions of the system.

Homoclinic chaos in coupled SQUIDs

An rf superconducting quantum interference device (SQUID) consists of a superconducting ring interrupted by a Josephson junction (JJ). The induced supercurrents around the ring are determined by the JJ through the celebrated Josephson relations. We study the dynamics of a pair of parametrically-driven coupled SQUIDs lying on the same plane with their axes in parallel. The drive is through the alternating critical current of the JJs. This system exhibits rich nonlinear behavior, including chaotic effects. We take advantage of the weak damping that characterizes these systems to perform a multiple-scales analysis and obtain amplitude equations, describing the slow dynamics of the system. This picture allows us to expose the existence of homoclinic orbits in the dynamics of the integrable part of the slow equations of motion. Using high-dimensional Melnikov theory, we are able to obtain explicit parameter values for which these orbits persist in the full system, consisting of both Hamiltonian and non-Hamiltonian perturbations, to form so called Shilnikov orbits, indicating a loss of integrability and the existence of chaos.

Homoclinic chaos in a pair of parametrically-driven coupled SQUIDs

An rf superconducting quantum interference device (SQUID) consists of a superconducting ring interrupted by a Josephson junction (JJ). When driven by an alternating magnetic field, the induced supercurrents around the ring are determined by the JJ through the celebrated Josephson relations. This system exhibits rich nonlinear behavior, including chaotic effects. We study the dynamics of a pair of parametrically-driven coupled SQUIDs arranged in series. We take advantage of the weak damping that characterizes these systems to perform a multiple-scales analysis and obtain amplitude equations, describing the slow dynamics of the system. This picture allows us to expose the existence of homoclinic orbits in the dynamics of the integrable part of the slow equations of motion. Using high-dimensional Melnikov theory, we are able to obtain explicit parameter values for which these orbits persist in the full system, consisting of both Hamiltonian and non-Hamiltonian perturbations, to form so-called Silnikov orbits, indicating a loss of integrability and the existence of chaos.

Periodic solutions of fully nonlinear autonomous equations of Benjamin–Ono type

We prove the existence of time-periodic, small amplitude solutions of autonomous quasi-linear or fully nonlinear completely resonant pseudo-PDEs of Benjamin–Ono type in Sobolev class. The result holds for frequencies in a Cantor set that has asymptotically full measure as the amplitude goes to zero.At the first order of amplitude, the solutions are the superposition of an arbitrarily large number of waves that travel with different velocities (multimodal solutions).The equation can be considered as a Hamiltonian, reversible system plus a non-Hamiltonian (but still reversible) perturbation that contains derivatives of the highest order.The main difficulties of the problem are: an infinite-dimensional bifurcation equation, and small divisors in the linearized operator, where also the highest order derivatives have non-constant coefficients.The main technical step of the proof is the reduction of the linearized operator to constant coefficients up to a regularizing rest, by means of changes of variables and conjugation with simple linear pseudo-differential operators, in the spirit of the method of Iooss, Plotnikov and Toland for standing water waves (ARMA 2005). Other ingredients are a suitable Nash–Moser iteration in Sobolev spaces, and Lyapunov–Schmidt decomposition.

Homoclinic orbits and chaos in a pair of parametrically driven coupled nonlinear resonators

We study the dynamics of a pair of parametrically driven coupled nonlinear mechanical resonators of the kind that is typically encountered in applications involving microelectromechanical systems (MEMS) and nanoelectromechanical systems (NEMS). We take advantage of the weak damping that characterizes these systems to perform a multiple-scales analysis and obtain amplitude equations, describing the slow dynamics of the system. This picture allows us to expose the existence of homoclinic orbits in the dynamics of the integrable part of the slow equations of motion. Using a version of the high-dimensional Melnikov approach, developed by G. Kovačič and S. Wiggins [Physica D 57, 185 (1992)], we are able to obtain explicit parameter values for which these orbits persist in the full system, consisting of both Hamiltonian and non-Hamiltonian perturbations, to form so-called Šilnikov orbits, indicating a loss of integrability and the existence of chaos. Our analytical calculations of Šilnikov orbits are confirmed numerically.

Perturbation of purely imaginary eigenvalues of Hamiltonian matrices under structured perturbations

The perturbation theory for purely imaginary eigenvaluesof Hamiltonian matrices under Hamiltonian and non-Hamiltonian perturbations is discussed. It is shown that there is a substantial difference in the behavior under these perturbations. The perturbation of real eigenvalues of real skew-Hamiltonian matrices under structured perturbations is discussed as well and these results are used to analyze the properties of the URV method for computing the eigenvalues of Hamiltonian matrices.

PERSISTENCE OF RESONANT INVARIANT TORI WITH NON-HAMILTONIAN PERTURBATION

A persistence theorem for resonant invariant tori with non-Hamiltonian perturbation is proved. The method is a combination of the theory of normally hyperbolic invariant manifolds and an appropriate continuation method. The results obtained are extensions of Chicone's for the three dimensional non-Hamiltonian systems.

Optical solitons: Quasi-stationarity versus Lie transform

This paper is a comparitive study of the two approaches namely the quasi-stationarity and the Lie transform that are used for studying the classical optical solitons along with its perturbations governed by the non-linear Schrodinger's equation with Kerr law of non-linearity. The relative advantages and disadvantages of the two methods are enumerated. The study is conducted with both, Hamiltonian as well as non-Hamiltonian type perturbations.

On Averaging in Two-Frequency Systems with Small Hamiltonian and Much Smaller Non-Hamiltonian Perturbations

On Averaging in Two-Frequency Systems with Small Hamiltonian and Much Smaller Non-Hamiltonian Perturbations

Obstructions to the continuation of analytic integrals of Hamiltonian systems under non-Hamiltonian perturbations

In this Letter we consider n degrees-of-freedom integrable Hamiltonian systems subjected to a non-Hamiltonian perturbation controlled by a small parameter ε. An obstruction to the analytic continuation of the integrals of motion of the unperturbed system with respect to ε is developed for sufficiently small perturbations. The theory is applied to a perturbed system of Morse oscillators.

CHAOTIC ATTITUDE MOTION OF SATELLITES UNDER SMALL PERTURBATION TORQUES

In this paper, the chaotic attitude motion of dissipative satellites under small perturbation torques is investigated by using the Deprit's canonical variables in the standard Hamiltonian form. Melnikov's integral is used to predict the transversal intersections of the stable and unstable manifolds for the satellites in perturbation. The theoretical criterion of chaotic attitude motion of the perturbed satellites will be derived from the Melnikov integral. Two models of satellites are studied. The first model is a quasi-rigid, energy- dissipating satellite subject to the time-periodic, non-Hamiltonian perturbation torques. The second model is a gyrostat satellite under small perturbation torques. It will be shown that, in terms of Deprit's variables, the equations of the attitude motion of the satellites can be easily transformed into the Hamiltonian form which is suitable for the application of Melnikov's method to some cases of complex, small perturbation torques.

Homoclinic Chaos in 2-DOF Robot Manipulators Driven by PD Controllers

The existence of Smale horseshoes in the dynamics of twodegrees of freedom (2-DOF) robot manipulators with viscous damping anddriven by classical proportional-derivative (PD) controllers is proved.The situation where an actuator failure occurs is also analyzed. Thecontrollers correspond to the classical PD and the PD with gravitycompensation. They are considered as non-Hamiltonian perturbations of anundriven 2-DOF robot; then a technique developed by Holmes and Marsden,which uses a combination of a reduction scheme and Melnikov's methodwith an energy balance argument, is applied.

Horseshoes in a PD-controlled 2-DOF robot manipulator

The existence of Smale horseshoes into the dynamics of a two degree of freedom (2-DOF) robot manipulator driven by a PD controller is proved. The PD controller is considered as a non-Hamiltonian perturbation of an undriven double pendulum, then a technique developed by Holmes and Marsden (1982). which uses a combination of a reduction scheme and the method of Melnikov with an energy balance argument, is applied.

Phase space transport in noisy Hamiltonian systems.

This paper analyzes the effect of low amplitude friction and noise in accelerating phase space transport in time-independent Hamiltonian systems that exhibit global stochasticity. Numerical experiments reveal that even very weak non-Hamiltonian perturbations can dramatically increase the rate at which an ensemble of orbits penetrates obstructions like cantori or Arnold webs, thus accelerating the approach toward an invariant measure, i.e., a microcanonical population of the accessible phase space region. An investigation of first passage times through cantori leads to three conclusions, namely: (i) that, at least for white noise, the detailed form of the perturbation is unimportant, (ii) that the presence or absence of friction is largely irrelevant, and (iii) that, overall, the amplitude of the response to weak noise scales logarithmically in the amplitude of the noise.

Off-equilibrium dynamics of a four-dimensional spin glass with asymmetric couplings

We study the off-equilibrium dynamics of the Edwards-Anderson spin glass in four dimensions under the influence of a non-Hamiltonian perturbation (i.e. with asymmetric couplings). We find that for small asymmetry the model behaves like the Hamiltonian model, while for large asymmetry the behaviour of the model can be well described by an interrupted ageing scenario. The autocorrelation function scales as , with a function of the asymmetry. For very long waiting times the previous regime crosses over to a time translational invariant regime with stretched exponential relaxation. The model does not show signs of reaching a time translational invariant regime for weak asymmetry, but in the ageing regime the exponent is always different from 1, showing a non-trivial ageing scenario also valid in the Hamiltonian model.

On the stability of the mean-field spin glass broken phase under non-Hamiltonian perturbations

We study the dynamics of the SK model modified by a small non-hamiltonian perturbation. We study aging, and we find that on the time scales investigated by our numerical simulations it survives a small perturbation (and is destroyed by a large one). If we assume we are observing a transient behavior the scaling of correlation times versus the asymmetry strength is not compatible with the one expected for the spherical model. We discuss the slow power law decay of observable quantities to equilibrium, and we show that for small perturbations power like decay is preserved. We also discuss the asymptotically large time region on small lattices.

Homoclinic orbits to parabolic points

We give a proof of the Poincare-Melnikov method in the case of non-Hamiltonian perturbations of one and a half degrees of freedom Hamiltonians, having orbits homoclinic to degenerate periodic orbits of parabolic type.

The stability of linear periodic Hamiltonian systems under non-Hamiltonian perturbations

A definition of strong stability and strong instability is proposed for a linear periodic Hamiltonian system of differential equations under a given non-Hamiltonian perturbation. Such a system is subject to the action of periodic perturbations: an arbitrary Hamiltonian perturbation and a given non-Hamiltonian one. Sufficient conditions for strong stability and strong instability are established. Using the linear periodic Lagrange equations of the second kind, the effect of gyroscopic forces and specified dissipative and non-conservative perturbing forces on strong stability and strong instability is investigated on the assumption that the critical relations of combined resonances are satisfied.

Capture and escape from resonance in the dynamics of the rigid body in viscous medium

The problem of capture and escape from resonance generally arises when certain nonlinear integrable Hamiltonian systems are subjected to non-Hamiltonian perturbation. The essential features of this problem are independent of the nature of the system under consideration. As an example we examine in detail the abovementioned resonance phenomena in rotational motions of a rigid body in a viscous medium. Our analysis is based on the averaging method. Conditions for the existence of the motions captured into resonance are derived. Scattering of the phase trajectories is estimated in the case of passage through resonance without capture.

An action principle for the Vlasov equation and associated Lie perturbation equations. Part 2. The Vlasov–Maxwell system

An action principle for the Vlasov–Maxwell system in Eulerian field variables is presented. Thus the (extended) particle distribution function appears as one of the fields to be freely varied in the action. The Hamiltonian structures of the Vlasov–Maxwell equations and of the reduced systems associated with small-ampliltude perturbation calculations are easily obtained. Previous results for the linearized Vlasov–Maxwell system are generalized. We find the Hermitian structure also when the background is time-dependent, and furthermore we may now also include the case of non-Hamiltonian perturbations within the Hamiltonian-Hermitian context. The action principle for the Vlasov–Maxwell system appears to be suitable for the derivation of reduced dynamical equations by expanding the action in various small parameters.