Energy drift in molecular dynamics simulations

In molecular dynamics, Hamiltonian systems of differential equations are numerically integrated using the Störmer–Verlet method. One feature of these simulations is that there is an unphysical drift in the energy of the system over long integration periods. We study this energy drift, by considering a representative system in which it can be easily observed and studied. We show that if the system is started in a random initial configuration, the error in energy of the numerically computed solution is well modeled as a continuous-time stochastic process: geometric Brownian motion. We discuss what in our model is likely to remain the same or to change if our approach is applied to more realistic molecular dynamics simulations.     

Characterization of the Quasi-Stationary State of an Impurity Driven by Monochromatic Light I: The Effective Theory

We consider an impurity (N-level atom) driven by monochromatic light in a host environment which is a fermionic thermal reservoir. The external light source is a time-periodic perturbation of the atomic Hamiltonian stimulating transitions between two atomic energy levels E ₁ and E _N and thus acts as an optical pump. The purpose of the present work is the analysis of the effective atomic dynamics resulting from the full microscopic time-evolution of the compound system. We prove, in particular, that the atomic dynamics of population relaxes for large times to a quasi-stationary state, that is, a stationary state up to small oscillations driven by the external light source. This state turns out to be uniquely determined by a balance condition. The latter is related to “generalized Einstein relations” of spontaneous/stimulated emission/absorption rates, which are conceptually similar to the phenomenological relations derived by Einstein in 1916. As an application we show from quantum mechanical first principles how an inversion of population of energy levels of an impurity in a crystal can appear. Our results are based on the spectral analysis of the generator of the evolution semigroup related to a non-autonomous Cauchy problem effectively describing the atomic dynamics.     

Local chaos induced by spatial oscillations of a perturbation

We study motion of an one-dimensional Hamiltonian oscillator driven by an external force which is periodic in time and in coordinate as well. It is shown that dynamics of the oscillator is strongly affected by the resonance between spatial and temporal oscillations of the perturbation imposed. In particular, this resonance can induce strong but bounded chaotic diffusion in certain areas of phase space. The model of the Duffing oscillator is used as an example for the numerical simulation.     

Exponential averaging for Hamiltonian evolution equations

We derive estimates on the magnitude of non-adiabatic interaction between a Hamiltonian partial differential equation and a high-frequency nonlinear oscillator. Assuming spatial analyticity of the initial conditions, we show that the dynamics can be transformed to the uncoupled dynamics of an infinite-dimensional Hamiltonian system and an anharmonic oscillator, up to coupling terms which are exponentially small in a certain power of the frequency of the oscillator. The result is derived from an abstract averaging theorem for infinite-dimensional analytic evolution equations in Gevrey spaces. Refining upon a similar result by Neishtadt for analytic ordinary differential equations, the temporal estimate crucially depends on the spatial regularity of the initial condition. The result shows to what extent the strong resonances between rapid forcing and highly oscillatory spatial modes can be suppressed by the choice of sufficiently smooth initial data. An application is provided by a system of nonlinear Schrödinger equations, coupled to a rapidly forcing single mode, representing small-scale oscillations. We provide an example showing that the estimates for partial differential equations we derive here are necessarily different from those in the context of ordinary differential equations.

     

On dispersive stability of Hamiltonian systems on lattices

We study the long-time dynamics of oscillations in lattices of infinitely many particles interacting via certain non-linear potentials. The aim is to proof dispersive stability of such Hamiltonian systems analogously to results known for PDEs. To do so we first recapitulate the dynamics of linear Hamiltonian systems on an infinite chain and give optimal decay rates based on the dispersion relation. Based on this we proof that if the non-linearity is weak enough, the non-linear system shows a similar behaviour like its linearization. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Energy Minimization Problem in Two-Level Dissipative Quantum Control: Meridian Case

We analyze the energy-minimizing problem for a two-level dissipative quantum system described by the Kossakowsky–Lindblad equation. According to the Pontryagin maximum principle (PMP), minimizers can be selected among normal and abnormal extremals whose dynamics are classified according to the values of the dissipation parameters. Our aim is to improve our previous analysis from [5] concerning 2D solutions in the case where the Hamiltonian dynamics are integrable.     

Multiple Timescales, Mixed Mode Oscillations and Canards in Models of Intracellular Calcium Dynamics

A range of representative models of intracellular calcium dynamics are surveyed, with the aim of determining which model characteristics are qualitatively unchanged by changes to details of the model components. Techniques from geometric singular perturbation theory are used to investigate the role of separation of timescales in determining model dynamics, with particular emphasis on identifying parameter regimes in which mixed mode oscillations are present as a result of the separation of timescales. We find that the number of distinct timescales and the number of variables evolving on each timescale varies between models and depends on both the model assumptions and on the parameter regime of interest within the model, but in all cases, the presence of canards and associated mixed mode oscillations provides a mechanism by which the models can robustly exhibit complex oscillations, with the frequency of oscillation depending sensitively on parameter values. We find that analysis of the number and nature of the distinct timescales in a model allows us to make useful predictions about the dynamics associated with the model, and that this may give us more information about the model dynamics than a classification according to the modelling assumptions made about different cellular mechanisms in deriving the models.     

Collective behavior of interacting locally synchronized oscillations in neuronal networks

Local circuits in the cortex and hippocampus are endowed with resonant, oscillatory firing properties which underlie oscillations in various frequency ranges (e.g. gamma range) frequently observed in the local field potentials, and in electroencephalography. Synchronized oscillations are thought to play important roles in information binding in the brain. This paper addresses the collective behavior of interacting locally synchronized oscillations in realistic neural networks. A network of five neurons is proposed in order to produce locally synchronized oscillations. The neuron models are Hindmarsh–Rose type with electrical and/or chemical couplings. We construct large-scale models using networks of such units which capture the essential features of the dynamics of cells and their connectivity patterns. The profile of the spike synchronization is then investigated considering different model parameters such as strength and ratio of excitatory/inhibitory connections. We also show that transmission time-delay might enhance the spike synchrony. The influence of spike-timing-dependence-plasticity is also studies on the spike synchronization.     

Adiabatic Invariance and Applications: From Molecular Dynamics to Numerical Weather Prediction

A wide class of Hamiltonian systems exhibit a mixture of slow motion with superimposed fast oscillations. Under the assumption of scale separation, these systems can be investigated using the principle of adiabatic invariance. In this paper, we start with a review of some of the main theoretical and numerical findings. We then briefly summarize a few important implications for molecular dynamics (MD) before we provide a more extensive discussion of numerical weather prediction (NWP). In particular, the conservative Hamiltonian particle-mesh (HPM) method is extended to Euler's equation and the fundamental concepts of geostrophic and hydrostatic balance are illustrated on the level of fluid blobs. We also demonstrate numerically that symplectic time-stepping methods are able to maintain hydrostatic balance to high accuracy.     

Dynamical Systems Related to Fractional Hamiltonians on the Two-Dimensional Sphere

We consider a class of fractional Hamiltonian systems generalizing the classical problem of motion in a central field. Our analysis is based on transforming an integrable Hamiltonian system with two degrees of freedom on the plane into a dynamical system that is defined on the sphere and inherits the integrals of motion of the original system. We show that in the four-dimensional space of structural parameters, there exists a one-dimensional manifold (containing the case of the planar Kepler problem) along which the closedness of the orbits of all finite motions and the third Kepler law are preserved. Similarly, there exists a one-dimensional manifold (containing the case of the two-dimensional isotropic harmonic oscillator) along which the closedness of the orbits and the isochronism of oscillations are preserved. Any deformation of orbits on these manifolds does not violate the hidden symmetry typical of the two-dimensional isotropic oscillator and the planar Kepler problem. We also consider two-dimensional manifolds on which all systems are characterized by the same rotation number for the orbits of all finite motions.Deceased     

Hamiltonian Systems near Relative Equilibria

We give explicit differential equations for the dynamics of Hamiltonian systems near relative equilibria. These split the dynamics into motion along the group orbit and motion inside a slice transversal to the group orbit. The form of the differential equations that is inherited from the symplectic structure and symmetry properties of the Hamiltonian system is analysed and the effects of time reversing symmetries are included. The results will be applicable to the stability and bifurcation theories of relative equilibria of Hamiltonian systems.     

On the orbital stability of pendulum-like oscillations and rotations of a symmetric rigid body with a fixed point

We deal with the problem of orbital stability of planar periodic motions of a dynamically symmetric heavy rigid body with a fixed point. We suppose that the center of mass of the body lies in the equatorial plane of the ellipsoid of inertia. Unperturbed periodic motions are planar pendulum-like oscillations or rotations of the body around a principal axis keeping a fixed horizontal position. Local coordinates are introduced in a neighborhood of the unperturbed periodic motion and equations of the perturbed motion are obtained in Hamiltonian form. Regions of orbital instability are established by means of linear analysis. Outside the above-mentioned regions, nonlinear analysis is performed taking into account terms up to degree 4 in the expansion of the Hamiltonian in a neighborhood of unperturbed motion. The nonlinear problem of orbital stability is reduced to analysis of stability of a fixed point of the symplectic map generated by the equations of the perturbed motion. The coefficients of the symplectic map are determined numerically. Rigorous results on the orbital stability or instability of unperturbed motion are obtained by analyzing these coefficients. The orbital stability is investigated analytically in two limiting cases: small amplitude oscillations and rotations with large angular velocities when a small parameter can be introduced.     

A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results

We present a general mechanism to establish the existence of diffusing orbits in a large class of nearly integrable Hamiltonian systems. Our approach is based on following the “outer dynamics” along homoclinic orbits to a normally hyperbolic invariant manifold. The information on the outer dynamics is encoded by a geometrically defined “scattering map.” We show that for every finite sequence of successive iterations of the scattering map, there exists a true orbit that follows that sequence, provided that the inner dynamics is recurrent. We apply this result to prove the existence of diffusing orbits that cross large gaps in a priori unstable models of arbitrary degrees of freedom, when the unperturbed Hamiltonian is not necessarily convex and the induced inner dynamics is not necessarily a twist map, and the perturbation satisfies explicit conditions that are generic. We also mention several other applications where this mechanism is easy to verify (analytically or numerically), such as the planar elliptic restricted three-body problem and the spatial circular restricted three-body problem. Our method differs, in several crucial aspects, from earlier works. Unlike the well-known “two-dynamics” approach, the method we present here relies on the outer dynamics alone. There are virtually no assumptions on the inner dynamics, such as on existence of its invariant objects (e.g., primary and secondary tori, lower-dimensional hyperbolic tori, and their stable/unstable manifolds, Aubry-Mather sets), which are not used at all. © 2019 Wiley Periodicals, Inc.     

Capture into resonance and escape from it in a forced nonlinear pendulum

We study the dynamics of a nonlinear pendulum under a periodic force with small amplitude and slowly decreasing frequency. It is well known that when the frequency of the external force passes through the value of the frequency of the unperturbed pendulum’s oscillations, the pendulum can be captured into resonance. The captured pendulum oscillates in such a way that the resonance is preserved, and the amplitude of the oscillations accordingly grows. We consider this problem in the frames of a standard Hamiltonian approach to resonant phenomena in slow-fast Hamiltonian systems developed earlier, and evaluate the probability of capture into resonance. If the system passes through resonance at small enough initial amplitudes of the pendulum, the capture occurs with necessity (so-called autoresonance). In general, the probability of capture varies between one and zero, depending on the initial amplitude. We demonstrate that a pendulum captured at small values of its amplitude escapes from resonance in the domain of oscillations close to the separatrix of the pendulum, and evaluate the amplitude of the oscillations at the escape.     

Trajectory Segmentation and Symbolic Representation of Dynamics of Delayed Recurrent Inhibitory Neural Loops

We develop a general symbolic dynamics framework to examine the dynamics of an analogue of the integrate-and-fire neuron model of recurrent inhibitory loops with delayed feedback, which incorporates the firing procedure and absolute refractoriness. We first show that the interaction of the delay, the inhibitory feedback and the absolute refractoriness can generate three basic types of oscillations, and these oscillations can be pinned together to form interesting coexisting periodic patterns in the case of short feedback duration. We then develop a natural symbolic dynamics formulation for the segmentation of a typical trajectory in terms of the basic oscillatory patterns, and use this to derive general principles that determine whether a periodic pattern can and should occur.     

Isomorphism and Hamilton representation of some nonholonomic systems

We consider some questions connected with the Hamiltonian form of the two problems of nonholonomic mechanics: the Chaplygin ball problem and the Veselova problem. For these problems we find representations in the form of the generalized Chaplygin systems that can be integrated by the reducing multiplier method. We give a concrete algebraic form of the Poisson brackets which, together with an appropriate change of time, enable us to write down the equations of motion of the problems under study. Some generalization of these problems are considered and new ways of implementation of nonholonomic constraints are proposed. We list a series of nonholonomic systems possessing an invariant measure and sufficiently many first integrals for which the question about the Hamiltonian form remains open even after change of time. We prove a theorem on isomorphism of the dynamics of the Chaplygin ball and the motion of a body in a fluid in the Clebsch case.     

Hamiltonian dynamics on convex symplectic manifolds

We study the dynamics of Hamiltonian diffeomorphisms on convex symplectic manifolds. To this end we first establish an explicit isomorphism between the Floer homology and the Morse homology of such a manifold, and then use this isomorphism to construct a biinvariant metric on the group of compactly supported Hamiltonian diffeomorphisms analogous to the metrics constructed by Viterbo, Schwarz and Oh. These tools are then applied to prove and reprove results in Hamiltonian dynamics. Our applications comprise a uniform lower estimate for the slow entropy of a compactly supported Hamiltonian diffeomorphism, the existence of infinitely many non-trivial periodic points of a compactly supported Hamiltonian diffeomorphism of a subcritical Stein manifold, new cases of the Weinstein conjecture, and, most noteworthy, new existence results for closed trajectories of a charge in a magnetic field on almost all small energy levels. We shall also obtain some new Lagrangian intersection results. Partially supported by the Swiss National Foundation. Supported by the Swiss National Foundation and the von Roll Research Foundation.     

Non-linear oscillations of a hamiltonian system with one degree of freedom and fourth-order resonance

Non-linear oscillations of a 2π-periodic Hamiltonian system with one degree of freedom are considered . It is assumed that the origin of coordinates is an equilibrium position, the linearized system is assumed to be stable, its characteristic exponents ±iv are pure imaginary, and the value of 4v is close to an integer. When the methods of classical perturbation theory are used, the investigation reduces to an analysis of a model system which can be described by the typical Hamiltonian of problems on the motion of Hamiltonian systems with one degree of freedom in the case of fourth-order resonance. The system is analysed in detail. The results for the model system are applied to the total system using Poincaré's theory of periodic motion and the KAM-theory. The existence, number and stability of 8π-periodic motions of the initial system are investigated. Trajectories of motion which start in a fairly small neighbourhood of the origin of coordinates are bounded. An estimate of the size of that neighbourhood is given. The examples considered are of a point mass above a curve in the shape of an ellipse which collides with the curve, and plane non-linear oscillations of a satellite in an elliptical orbit in the case of fourth-order resonance.     

Time averaging for the strongly confined nonlinear Schrödinger equation, using almost-periodicity

We study the limiting behavior of a nonlinear Schrödinger equation describing a 3-dimensional gas that is strongly confined along the vertical, z direction. The confinement induces fast oscillations in time, that need to be averaged out. Since the Hamiltonian in the z direction is merely assumed confining, without any further specification, the associated spectrum is discrete but arbitrary, and the fast oscillations induced by the nonlinear equation entail countably many frequencies that are arbitrarily distributed. For that reason, averaging cannot rely on small denominator estimates or like.To overcome these difficulties, we prove that the fast oscillations are almost-periodic in time, with values in a Sobolev-like space that we completely identify. We then exploit the existence of long-time averages for almost-periodic functions to perform the necessary averaging procedure in our nonlinear problem.