Dissipative Perturbations Research Articles

A new solution method for the gyrokinetic Fokker‐Planck equation

A new non‐perturbative and non‐variational solution method is proposed for the gyrokinetic equation, based on a suitable approximation for the linearized Fokker‐Planck collision operator. The approach, which allows, in principle, the accurate evaluation of arbitrary moments of the distribution function for a weakly collisional magnetoplasma with toroidal equilibria of arbitrary aspect‐ratio, is susceptible of numerous significant applications, both for transport calculations, i.e., in particular, leading to the evaluation of neoclassical fluxes, as well for the investigation of linear stability analyses of dissipative perturbations.

From dark solitons in the defocusing nonlinear Schro¨dinger to holes in the complex Ginzburg-Landau equation

Perturbation approaches developed so far for the dark soliton solutions of the (fully integrable) defocusing nonlinear Schro¨dinger equation cannot describe important aspects of the dynamics resulting from dissipative perturbations of the Ginzburg-Landau type. The reason is that spatially slowly decaying changes of the background wavenumber occur, which requires the use of matching technics. We show how the perturbation selects a 1 or 2-parameter subfamily from the 3-parameter family of dark solitons of the nonlinear Schro¨dinger equation. The dynamics of the perturbed system can then be described analytically as motion within this selected subfamily yielding interesting scenarios. Interaction with shocks occurring in the complex Ginzburg-Landau equation can be included in a straightforward way.

Logarithmic correction to the probability of capture for dissipatively perturbed Hamiltonian systems.

Hamiltonian systems are analyzed with a double homoclinic orbit connecting a saddle to itself. Competing centers exist. A small dissipative perturbation causes the stable and unstable manifolds of the saddle point to break apart. The stable manifolds of the saddle point are the boundaries of the basin of attraction for the competing attractors. With small dissipation, the boundaries of the basins of attraction are known to be tightly wound and spiral-like. Small changes in the initial condition can alter the equilibrium to which the solution is attracted. Near the unperturbed homoclinic orbit, the boundary of the basin of attraction consists of a large sequence of nearly homoclinic orbits surrounded by close approaches to the saddle point. The slow passage through an unperturbed homoclinic orbit (separatrix) is determined by the change in the value of the Hamiltonian from one saddle approach to the next. The probability of capture can be asymptotically approximated using this change in the Hamiltonian. The well-known leading-order change of the Hamiltonian from one saddle approach to the next is due to the effect of the perturbation on the homoclinic orbit. A logarithmic correction to this change of the Hamiltonian is shown to be due to the effect of the perturbation on the saddle point itself. It is shown that the probability of capture can be significantly altered from the well-known leading-order probability for Hamiltonian systems with double homoclinic orbits of the twisted type, an example of which is the Hamiltonian system corresponding to primary resonance. Numerical integration of the perturbed Hamiltonian system is used to verify the accuracy of the analytic formulas for the change in the Hamiltonian from one saddle approach to the next. (c) 1995 American Institute of Physics.

Multisoliton perturbation theory for the Benjamin-Ono equation and its application to real physical systems.

A direct perturbation theory is developed to study the effects of small perturbations on the interaction process of algebraic solitons of the Benjamin-Ono (BO) equation. Using the method of multiple scales, the modulation equations for the amplitude and the phase of each soliton are derived in the lowest approximation. As practical applications of the theory, the interaction of two solitons is investigated for the two different types of perturbations that appear in real physical systems. One is a dissipative perturbation (BO--Burgers equation) and the other is a dispersive perturbation (higher-order BO equation). In both cases, the changes of the soliton parameters due to small perturbation are calculated by numerical integrations and their characteristics are elucidated in detail. Among them, the phase shift caused by the dispersive perturbation is a remarkable feature that has never been observed in the collision process of algebraic solitons.

N-pulse homoclinic orbits in perturbations of resonant hamiltonian systems

In this paper we develop an analytical method to detect orbits doubly asymptotic to slow manifolds in perturbations of integrable, two-degree-of-freedom resonant Hamiltonian systems. Our energy-phase method applies to both Hamiltonian and dissipative perturbations and reveals families of multi-pulse solutions which are not amenable to Melnikov-type methods. As an example, we study a two-mode approximation of the nonlinear, nonplanar oscillations of a parametrically forced inextensional beam. In this problem we find unusually complicated mechanisms for chaotic motions and verify their existence numerically.

Birkhoff-Hénon attractors for dissipative perturbations of area-preserving twist maps

AbstractWe prove that an area-preserving twist map having an invariant curve, can be approximated by a twist map exhibiting a Birkhoff-Hénon attractor. This is done by showing that the invariant curve can be perturbed into a saddle-node cycle with criticalities and by using a recent result reported by Diaz, Rocha and Viana.

On the relationship between thermo field dynamics and quantum statistical mechanics

The relationship between two approaches to the study of non-equilibrium field theories, namely quantum statistical mechanics and thermo field dynamics, is investigated. The formalism of superoperators acting in the Liouville space of density matrices is used to provide a detailed translation between the two approaches. It is found that thermo field dynamics is exactly equivalent to a restricted version of quantum statistical mechanics, in which the initial density matrix is constrained to be Gaussian. The dissipative perturbation theory developed within thermo field dynamics is translated into the language of statistical mechanics, and is found to be equivalent to that devised by the author using the statistical-mechanical closed-time-path method, except that the latter theory is not restricted to Gaussian initial states.

EFFECT OF PERTURBATION ON THE AREA PRESERVATION MAP

Subharmonic bifurcation patterns in area-preserving maps change their forms dramatically under dissipative perturbation. The studies of such changes are done by the use of the normal form theory and the Lyapunov-Schmidt reduction method where the eigenvalues of the linearized map move along a circle of radius (1−∊)1/2 in the complex plane. The results are: (i) one set of three-cycle branches, which is hyperbolic, becomes separated from the origin under the dissipative perturbation; (ii) a pair of four-cycle branches, which can be hyperbolic for both branches, or hyperbolic for one branch and stable for the other one depending on the condition, is separated from the origin also under the same perturbation; (iii) one set of n-cycle (n≥5) branches, which is elliptic for the area-preserving case, is retained under the perturbation for sufficiently small ∊.

Connection across a Separatrix with Dissipation

Dissipative perturbations of strongly nonlinear oscillators that correspond to slowly varying double‐well potentials are considered. The method of averaging, which describes the solution as nearly periodic, fails as the trajectory approaches the unperturbed separatrix, a homoclinic orbit of the saddle point, significantly before it is captured in either well. Nevertheless, perturbed initial conditions corresponding to the boundary of the basin of attraction for each well, which are the perturbed stable manifolds of the saddle point, are accurately determined using only the method of averaging modified by Melnikov energy ideas near the separatrix. To determine the amplitude and phase of the captured oscillations after crossing the separatrix, a transition region is constructed consisting of a large sequence of nearly solitary pulses along the separatrix. The amplitude and phases of the slowly varying nonlinear oscillations away from the separatrix, both before and after capture, are matched to this transition region. In this way, analytic connection formulas across the separatrix are obtained and are shown to depend on the perturbed initial conditions.

Higher-order change of the Hamiltonian (energy) for nearly homoclinic orbits

Conservative dynamical systems with a homoclinic orbit are analyzed under a small dissipative perturbation. The asymptotic expansions of a nearly ho- moclinic orbit are shown to fail as the surrounding saddle regions are approached. A weakly nonlinear analysis of the surrounding saddle regions is derived assuming the dissipation can be approximated by linear damping in the neighborhood of the saddle point. By matching (to sufficiently high order) the asymptotic expansion of the nearly homoclinic orbit to the solutions in the surrounding saddle regions, the change in the Hamiltonian (the energy dissipation) over a complete nearly homoclinic orbit is obtained. We show there is a new higher-order logarithmic correction to the well-known Melnikov integral for the change of the Hamiltonian from one saddle approach to the next. Numerical computations of this change in the Hamiltonian for a double-well potential are shown to compare favorably with the asymptotic result.

THE BOGDANOV MAP: BIFURCATIONS, MODE LOCKING, AND CHAOS IN A DISSIPATIVE SYSTEM

We investigate the bifurcations and basins of attraction in the Bogdanov map, a planar quadratic map which is conjugate to the Hénon area-preserving map in its conservative limit. It undergoes a Hopf bifurcation as dissipation is added, and exhibits the panoply of mode locking, Arnold tongues, and chaos as an invariant circle grows out, finally to be destroyed in the homo-clinic tangency of the manifolds of a remote saddle point. The Bogdanov map is the Euler map of a two-dimensional system of ordinary differential equations first considered by Bogdanov and Arnold in their study of the versal unfolding of the double-zero-eigenvalue singularity, and equivalently of a vector field invariant under rotation of the plane by an angle 2π. It is a useful system in which to observe the effect of dissipative perturbations on Hamiltonian structure. In addition, we argue that the Bogdanov map provides a good approximation to the dynamics of the Poincare maps of periodically forced oscillators.

DYNAMICS OF ROTATIONALLY SYMMETRIC SINE-GORDON SOL1TONS WITH APPLLCATIONS FOR A LARGE AREA JOSEPHSON JUNCTION

The dynamics of rotationally symmetric sine Gordon solitons of large radius under poten-tial and dissipative perturbations is considered, with the help of a collective-coordinate desc-ription of the soliton and the Lagrangian variation method. The particle-like character of the soliton is emphasized The equation of motlon and in particular, the general momentum of the soliton are obtained in a canonical manner. It is possible to discuss the dynamics of the soliton even without applying the explicit form of the perturbating potential All dyna-mical regimes in the phase space are explored. Such phenomena as the soliron return effect, soliton escaping and saddle point are addressed In the presence of dissipation, the corrected equation of soliton motion is obtained from the generalized variational equation. Finally, the application of the theoretical treatment is considered for the fluxon dynamics in a circularly symmetric Josephson junction. The analytical results are examined by direct numerical simu-lations, fairly good agreements being achieved. it turns out that the presented ueaMnem provides a reliable description of the dynamics of the sol tons considered.

Soliton switching in nonlinear couplers

We present a theoretical overview of soliton switching phenomena in two-mode nonlinear couplers. By complementing numerical studies with perturbative or exact solitary wave solutions, one finds that nonlinear Schrodinger or sine-Gordon solitons tend to maintain their identity in the coupled systems. Moreover, the coupling itself may originate novel vector solitary waves, such as gap solitons in periodic media. The switching dynamics in the presence of dissipative perturbations such as linear gain or intrapulse Raman scattering is also discussed.

Non-Hamiltonian Perturbations of Integrable Systems and Resonance Trapping

This paper studies general, non-Hamiltonian perturbations of integrable systems with two degrees of freedom and derives conditions for temporary and permanent resonance trapping. The analysis involves a noncanonical transformation of variables near the resonant manifold and averaging with respect to the fast phase to investigate oscillatory behavior on the intermediate timescale. The resulting reduced system is Hamiltonian to leading order and permits, after averaging on the intermediate, or libration, timescale, a canonical transformation to action-angle variables in the oscillation zone. The final system so obtained reveals the possible existence of two- and three-dimensional invariant tori in the vicinity of the resonant manifold. An explicit divergence condition for general perturbations to be dissipative on the slow timescale follows from the analysis. An application of this approach to the problem of resonant trapping and escape is outlined for the restricted problem of three bodies subject to dissipative perturbations with a radial symmetry.

Dissipative dynamics of orbits homoclinic to a resonance band

This paper presents an example of geometric singular perturbation theory for a near-integrable dissipative system with two degrees of freedom. In this example, small dissipative perturbations yield orbits homoclinic to an equilibrium in a resonance band. The solution method for this example illustrates a new combination of the Melnikov method and geometric singular perturbation theory.

New Developments in Dynamics: Hyperbolicity and Chaotic Dynamics

Two important theories in Dynamical Systems were constructed in the sixties: the hyperbolic theory for general systems and the KAM (after Kolmogorov, Arnold and Moser) theory for conservative systems as the ones that appear in Celestial Mechanics. Most of our discussions here concern dissipative (or locally dissipative) systems, although most questions are now being posed for area preserving maps (symplectic maps in higher dimensions). Moreover, one can argue that understanding dynamically small dissipative perturbations of conservative systems is of much importance: indeed it has been recently shown that a KAM curve (tori in higher dimension) can be destroyed and in fact engulfed in the basin of attraction of a Hénon-like strange attractor as defined below.

Standard Form and a Method of Averaging for Strongly Nonlinear Oscillatory Dispersive Traveling Waves

A method of averaging that is equivalent to but different from the method of multiple scales is introduced to analyze perturbations to strongly nonlinear partial differential equations with oscillatory solutions. Standard form is derived for the amplitude and phase (angle) corresponding to a slowly varying traveling wave satisfying the usual nonlinear dispersion relation. The standard form is linear in spatial and temporal derivatives of the wavenumber and frequency, a nontrivial generalization of standard form for ordinary differential equations. A near identity transformation is determined that eliminates secular terms. To leading order after averaging, conservation of waves and the wave action equation are obtained. Higher-order terms that determine the phase shift are more complicated, but are shown to simplify for purely dissipative perturbations.

Time decay of solutions of semilinear strongly damped generalized wave equations

AbstractWe study the asymptotic behaviour in time of the solutions of dissipative perturbations of wave‐type equations in ℝN, utt + But + Au + G(u) = 0, with commuting positive operators A, B and a power like non‐linearity G(u). First we give some (pseudo) conformal invariants of the linear operator in the equation. This allows us to derive optimal decay rates for the solutions of the linearized problems. We then prove some decay estimates for the non‐linear problems using the tools of scattering theory and the aforementioned conformal invariants.

On higher-order corrections in soliton perturbation theory

A scheme is proposed for calculating higher-order corrections to a soliton in the presence of perturbations. This approach, which is based on the matrix Riemann problem, has made it possible to avoid the use of integral equations. As an example, second-order corrections are calculated to the soliton of the nonlinear Schrodinger equation with a dissipative perturbation.

Exponential Decay of Energy of Evolution Equations with Locally Distributed Damping

Consider an evolution equation with energy dissipation, \[ \frac{\partial }{{\partial t}}y( {x,t} ) = Ay( {x,t} ) + By( {x,t} ) \] on a bounded x-domain $\Omega $, where $By$ signifies a dissipative perturbation to an otherwise energy-conserving system. This dissipation may be due to medium impurities, viscous effects, or artificially imposed dampers and stabilizers. It is distributed over only part of the domain $\Omega $. The question of when the dissipation is effective enough to cause uniform exponential decay of energy is examined. Because of the locally distributed nature of energy dissipation, the problem lacks coercivity and is not directly solvable by energy identities. Thus, to get conditions sufficient for uniform exponential decay, a different approach needs to be taken. Provided here is a set of tight sufficient conditions in terms of the influence of the dissipative operatorBon the separated eigenmodes or clustered eigenmodes ofA. The main theorems are general enough to treat the wave and beam equations in one space dimension and the Schrödinger equation on a two-dimensional rectangle or disk. In particular, for the Schrödinger equation, it shown how such phenomena as the “whispering gallery”and “bouncing ball” eigenmodes dictate the supports of effective damping functions. Thus, the theory presented is also capable of providing valuable information about the placement and design of actuators and sensors in modern distributed parameter control theory.