Hamiltonian Dynamical Systems Research Articles

Port-Hamiltonian discontinuous Galerkin finite element methods

Abstract A port-Hamiltonian (pH) system formulation is a geometrical notion used to formulate conservation laws for various physical systems. The distributed parameter port-Hamiltonian formulation models infinite dimensional Hamiltonian dynamical systems that have a nonzero energy flow through the boundaries. In this paper, we propose a novel framework for discontinuous Galerkin (DG) discretizations of pH-systems. Linking DG methods with pH-systems gives rise to compatible structure preserving semidiscrete finite element discretizations along with flexibility in terms of geometry and function spaces of the variables involved. Moreover, the port-Hamiltonian formulation makes boundary ports explicit, which makes the choice of structure and power preserving numerical fluxes easier. We state the Discontinuous Finite Element Stokes–Dirac structure with a power preserving coupling between elements, which provides the mathematical framework for a large class of pH discontinuous Galerkin discretizations. We also provide an a priori error analysis for the port-Hamiltonian discontinuous Galerkin Finite Element Method (pH-DGFEM). The port-Hamiltonian discontinuous Galerkin finite element method is demonstrated for the scalar wave equation showing optimal rates of convergence.

Asymptotic motions converging to arbitrary dynamics for time-dependent Hamiltonians

Dynamical systems subject to perturbations that decay over time are relevant in the description of many physical models, e.g. when considering the effect of a laser pulse on a molecule, in epidemiological studies, as well as in celestial mechanics. For this reason, we consider a Hamiltonian dynamical system having an invariant torus supporting arbitrary dynamics, and we study its evolution under a perturbation decaying exponentially over time. By applying a strategy based on a refined analysis of the Banach spaces and functionals involved in the resolution of suitable non-linear invariant equations, we show the existence of orbits converging in time to the arbitrary motions associated with the unperturbed system. As a corollary, an analogous statement for time-dependent vector fields on the torus is also obtained. This result extends to the important case of arbitrary Hamiltonian dynamics a previous work of Canadell and de la Llave where only asymptotic quasi-periodic motions were considered.

Novel solitonic structure, Hamiltonian dynamics and lie symmetry algebra of biofilm

In this study, the Lie point symmetries and optimal system have been established. We discuss the biofilm model’s soliton solutions. To examine a nonlinear dynamical biofilm system, which is simply a bistable Allen–Cahn equation with quartic potential, to determine solitary wave profiles using the considered equation, a new auxiliary equation technique is used. A suitable variable transformation is used to convert the governing equation into a nonlinear ordinary differential equation. The unified approach is utilized to evaluate periodic solutions, solitary and soliton solutions as well as several newly discovered exact solitary wave solutions, it can be accomplished via Mathematica (https://www.wolfram.com/mathematica/online/). The new auxiliary equation approach is an efficient method for creating unique wave profiles based on a variety of soliton families. Also, the results are graphically visualized, by using the appropriate parametric settings. The outcomes are shown graphically in two dimensions, three dimensions, and contour form. The Hamiltonian conditions are satisfied by the planer dynamical framework of equations to ensure as the system that was generated is a conservative Hamiltonian dynamical system which includes all traveling wave structures. A sensitivity evaluation is used to explore the governing model’s thoroughly dynamical properties. It is demonstrated that the model becomes more dependent on the beginning conditions than the variables. The strategy could be used to look for exact solutions to various nonlinear evolution equations. We believe that this study will have important applications in different areas of science.

3D Generating Surfaces in Hamiltonian Systems with Three Degrees of Freedom – I

In our earlier research (see [Katsanikas & Wiggins, 2021a , 2021b , 2023a , 2023b , 2023c ]), we developed two methods for creating dividing surfaces, either based on periodic orbits or two-dimensional generating surfaces. These methods were specifically designed for Hamiltonian systems with three or more degrees of freedom. Our prior work extended these dividing surfaces to more complex structures such as tori or cylinders, all within the energy surface of the Hamiltonian system. In this paper, we introduce a new method for constructing dividing surfaces. This method differs from our previous work in that it is based on 3D surfaces or geometrical objects, rather than periodic orbits or 2D generating surfaces (see [Katsanikas & Wiggins, 2023a ]). To explain and showcase the new method and to present the structure of these 3D surfaces, the paper provides examples involving Hamiltonian systems with three degrees of freedom. These examples cover both uncoupled and coupled cases of a quadratic normal form Hamiltonian system. Our current paper is the first in a series of two papers on this subject. This research is likely to be of interest to scholars and researchers in the field of Hamiltonian systems and dynamical systems, as it introduces innovative approaches to constructing dividing surfaces and exploring their applications.

On parametric generalizations of the Kardar-Parisi-Zhang equation and their integrability

There are analyzed two physically reasonable generalizations of the Kardar-Parisi-Zhang equation describing the spin glasses growth models and possessing important from physical point of view properties. The first one proved to be a completely integrable Hamiltonian dynamical system with an infinite hierarchy of commuting to each other conservation laws, and the second one proved to be linearized modulo some nonlinear constraints, imposed on its solutions.

The vicinity of Earth–Moon [formula omitted] and [formula omitted] in the Hill restricted 4-body problem

The Hill restricted 4-body problem (HR4BP) is a periodic time-dependent Hamiltonian dynamical system that is both a generalization of the circular restricted 3-body problem (CR3BP) and Hill’s problem. In this work, the HR4BP is used to model the motion of an infinitesimal particle under the mutual gravitation of the Sun, Earth, and Moon. In the HR4BP, the equilibria L1 and L2 of the CR3BP are replaced by π-periodic orbits, the so-called EM L1,2 dynamical equivalents. We compute center manifold reductions about EM L1,2 to obtain a local overview of the dynamics in their vicinity. In this way, we show the existence of families of 2- and 3-dimensional quasi-periodic invariant tori around EM L1,2. Additionally, we exploit the first integral constructed in the center manifold reduction to compute hyperbolic invariant manifolds. We draw comparisons to the bicircular and quasi-bicircular restricted 4-body problems.

Machine learning of independent conservation laws through neural deflation.

We introduce a methodology for seeking conservation laws within a Hamiltonian dynamical system, which we term "neural deflation." Inspired by deflation methods for steady states of dynamical systems, we propose to iteratively train a number of neural networks to minimize a regularized loss function accounting for the necessity of conserved quantities to be in involution and enforcing functional independence thereof consistently in the infinite-sample limit. The method is applied to a series of integrable and nonintegrable lattice differential-difference equations. In the former, the predicted number of conservation laws extensively grows with the number of degrees of freedom, while for the latter, it generically stops at a threshold related to the number of conserved quantities in the system. This data-driven tool could prove valuable in assessing a model's conserved quantities and its potential integrability.

Motion of a spinning particle under the conservative piece of the self-force is Hamiltonian to first order in mass and spin

We consider the motion of a point particle with spin in a stationary spacetime. We define, following Witzany et al. [Classical Quantum Gravity 36, 075003 (2019)] and later Ramond [arXiv:2210.03866], a 12-dimensional Hamiltonian dynamical system whose orbits coincide with the solutions of the Mathisson-Papapetrou-Dixon equations of motion with the Tulczyjew-Dixon spin supplementary condition, to linear order in spin. We then perturb this system by adding the conservative pieces of the leading order gravitational self-force and self-torque sourced by the particle's mass and spin. We show that this perturbed system is Hamiltonian and derive expressions for the Hamiltonian function and symplectic form. This result extends a previous result for spinless point particles [F. M. Blanco and E. E. Flanagan, Phys. Rev. Lett. 130, 051201 (2023)].

Classical and quantum particles from nongeneric conformal orbits

The nongeneric six- and eightdimensional orbits of SO(4,2) are described in explicitly covariant way. The relevant Hamiltonian dynamical systems are constructed and canonically quantized. It is shown that the resulting unitary representations of conformal group fit into the classification described by Mack (1977) [31].

Contact geometric approach to Glauber dynamics near a cusp and its limitation

We study a nonequilibrium mean field Ising model in the low temperature phase regime, where metastable equilibrium states develop a cuspidal (spinodal) singularity. We focus on celebrated Glauber dynamics, and design a contact Hamiltonian flow which captures some of its rough features in this regime. We prove, however, that there is an inevitable discrepancy between the scaling laws for the relaxation time in the Glauber and the contact Hamiltonian dynamical systems.

NySALT: Nyström-type inference-based schemes adaptive to large time-stepping

Large time-stepping is important for efficient long-time simulations of deterministic and stochastic Hamiltonian dynamical systems. Conventional structure-preserving integrators, while being successful for generic systems, have limited tolerance to time step size due to stability and accuracy constraints. We propose to use data to innovate classical integrators so that they can be adaptive to large time-stepping and are tailored to each specific system. In particular, we introduce NySALT, Nyström-type inference-based schemes adaptive to large time-stepping. The NySALT has optimal parameters for each time step learnt from data by minimizing the one-step prediction error. Thus, it is tailored for each time step size and the specific system to achieve optimal performance and tolerate large time-stepping in an adaptive fashion. We prove and numerically verify the convergence of the estimators as data size increases. Furthermore, analysis and numerical tests on the deterministic and stochastic Fermi-Pasta-Ulam (FPU) models show that NySALT enlarges the maximal admissible step size of linear stability, and quadruples the time step size of the Störmer–Verlet and the BAOAB when maintaining similar levels of accuracy.

On the geometry of Hamiltonian vector fields

It is known that vector fields play an important role in many areas of mathematics and technology, in particular, in the theory of dynamical systems. Hamiltonian dynamical systems are generated by Hamiltonian vector fields. The paper studies the geometry of a singular foliation of a four-dimensional Euclidean space generated by the orbits of two Hamiltonian vector fields. It is proved that the regular leaf of this foliation is a surface of positive normal curvature and nonzero normal torsion.

Metric Gravity in the Hamiltonian Form—Canonical Transformations—Dirac’s Modifications of the Hamilton Method and Integral Invariants of the Metric Gravity

Two different Hamiltonian formulations of the metric gravity are discussed and applied to describe a free gravitational field in the d dimensional Riemann space-time. Theory of canonical transformations, which relates equivalent Hamiltonian formulations of the metric gravity, is investigated in detail. In particular, we have formulated the conditions of canonicity for transformation between the two sets of dynamical variables used in our Hamiltonian formulations of the metric gravity. Such conditions include the ordinary condition of canonicity known in classical Hamilton mechanics, i.e., the exact coincidence of the Poisson (or Laplace) brackets which are determined for both the new and old dynamical Hamiltonian variables. However, in addition to this, any true canonical transformations defined in the metric gravity, which is a constrained dynamical system, must also guarantee the exact conservation of the total Hamiltonians Ht (in both formulations) and preservation of the algebra of first-class constraints. We show that Dirac’s modifications of the classical Hamilton method contain a number of crucial advantages, which provide an obvious superiority of this method in order to develop various non-contradictory Hamiltonian theories of many physical fields, when a number of gauge conditions are also important. Theory of integral invariants and its applications to the Hamiltonian metric gravity are also discussed. For Hamiltonian dynamical systems with first-class constraints this theory leads to a number of peculiarities some of which have been investigated.

The fractional wave propagation, dynamical investigation, and sensitive visualization of the continuum isotropic bi-quadratic Heisenberg spin chain process

This paper deals with the Lakshmanan-Porsezian-Daniel equation which delineates the continuum isotropic bi-quadratic Heisenberg spin chain phenomenon. A new auxiliary equation method is exerted on the considered equation to find solitary wave profiles. It is a simple and powerful approach for developing innovative wave profiles based on diverse soliton families such as trigonometric functions, rational, hyperbolic trigonometric function and exponential function etc. As a result, the solitonic wave patterns attain such as dark, bright, dark-bright, singular, rational, periodic-singular, exponential, and periodic solitons etc. The deep dynamical aspects of the governing model study by performing the chaos and sensitivity analysis. The planer dynamical system of equation develop and satisfy the Hamiltonian criteria to assure that, the developed system is Hamiltonian dynamical system and contains all traveling wave structures and the system is conservative. The graphical explanation of energy levels presents the significant insights and the existence of closed-form solutions to the model. The periodic, quasi-periodic, and quasi-periodic-chaotic profiles are present to see the deep dynamics of the continuum isotropic bi-quadratic Heisenberg spin chain system. The graphically visualization for sensitivity analysis of the governing equation portraits by taking some initial values to verify its dependence. It is shown that, the model is more sensitive regarding to initial conditions rather then parameters. The graphical two dimensional, three dimensional, and contour visualization of the obtained results are presented to express the pulse propagation behavior by assuming the appropriate values of the involved parameters. The impact of fractional parameter is displayed in the graphical sense. The fractional order controls the soliton behaviour which means that, the prediction and precautions can be constructed about the physical phenomenon of the continuum isotropic bi-quadratic Heisenberg spin chain. As a results, the fractional order exhibits the states of distortion in continuum bi-quadratic magnetic system with non-zero vector on which the form evaluates to zero. The graphical two dimensional, three dimensional, and contour visualization of the obtained results are presented to express the pulse propagation behavior by assuming the appropriate values of the involved parameters.

Quaternionic Dirac oscillator

We construct an elementary quaternionic slow–fast Hamiltonian dynamical system with one formal control parameter and two slow degrees of freedom as half-integer spin in resonance 1:1:2 with two slow oscillators. Invariant under spin reversal and having a codimension-5 crossing of its fast Kramers-degenerate semi-quantum eigenvalues, our system is the dynamical equivalent of the spin-quadrupole model by Avron et al (1989 Commun. Math. Phys. 124 595–627), exhibiting non-Abelian geometric phases. The equivalence is uncovered through the equality of the spectral flow between quantum superbands and Chern numbers c 2 computed by Avron et al.

Quantum mechanical approach to bifurcation point detection in Hamiltonian dynamical systems

Abstract The energy level statistics of a bounded quantum system whose classical dynamical system exhibits bifurcations is investigated using the two-point correlation function (TPCF), which at the bifurcation points exhibits periodic spike oscillations owing to the accumulation of levels called the shell effect. The spike oscillations of the TPCF are analyzed by the reduced chi-squared value, which exhibits abrupt increases at bifurcation points, thereby yielding a novel detection approach. Using this method, we attempt to numerically detect the bifurcation points of a lemon-shaped billiard.

Discontinuous Galerkin Methods with Time-Operators in Their Numerical Traces for Time-Dependent Electromagnetics

Abstract We present a new class of discontinuous Galerkin methods for the space discretization of the time-dependent Maxwell equations whose main feature is the use of time derivatives and/or time integrals in the stabilization part of their numerical traces. These numerical traces are chosen in such a way that the resulting semidiscrete schemes exactly conserve a discrete version of the energy. We introduce four model ways of achieving this and show that, when using the mid-point rule to march in time, the fully discrete schemes also conserve the discrete energy. Moreover, we propose a new three-step technique to devise fully discrete schemes of arbitrary order of accuracy which conserve the energy in time. The first step consists in transforming the semidiscrete scheme into a Hamiltonian dynamical system. The second step consists in applying a symplectic time-marching method to this dynamical system in order to guarantee that the resulting fully discrete method conserves the discrete energy in time. The third and last step consists in reversing the above-mentioned transformation to rewrite the fully discrete scheme in terms of the original variables.

About Linearization of Infinite-Dimensional Hamiltonian Systems

This article is concerned with analytic Hamiltonian dynamical systems in infinite dimension in a neighborhood of an elliptic fixed point. Given a quadratic Hamiltonian, we consider the set of its analytic higher order perturbations. We first define the subset of elements which are formally symplectically conjugacted to a (formal) Birkhoff normal form. We prove that if the quadratic Hamiltonian satisfies a Diophantine-like condition and if such a perturbation is formally symplectically conjugated to the quadratic Hamiltonian, then it is also analytically symplectically conjugated to it. Of course what is an analytic symplectic change of variables depends strongly on the choice of the phase space. Here we work on periodic functions with Gevrey regularity.

Global dynamics and evolution for the Szekeres system with nonzero cosmological constant term

The Szekeres system with cosmological constant term describes the evolution of the kinematic quantities for Einstein field equations in \({\mathbb {R}}^4\). In this study, we investigate the behavior of trajectories in the presence of cosmological constant. It has been shown that the Szekeres system is a Hamiltonian dynamical system. It admits at least two conservation laws, h and \(I_{0}\) which indicate the integrability of the Hamiltonian system. We solve the Hamilton–Jacobi equation, and we reduce the Szekeres system from \({\mathbb {R}}^4\) to an equivalent system defined in \({\mathbb {R}}^2\). Global dynamics are studied where we find that there exists an attractor in the finite regime only for positive valued cosmological constant and \(I_0<-2 .08\). Otherwise, trajectories reach infinity. For \(I_ {0}>0\) the origin of trajectories in \({\mathbb {R}}^2\) is also at infinity. Finally, we investigate the evolution of physical properties by using dimensionless variables different from that of Hubble-normalization conducing to a dynamical system in \({\mathbb {R}}^5\). We see that the attractor at the finite regime in \({\mathbb {R}}^5\) is related with the de Sitter universe for a positive cosmological constant.

Superfluid vortex dynamics on an ellipsoid and other surfaces of revolution

We present a study of superfluid vortex dynamics on general axisymmetric compact surfaces with no holes. Specifically, we develop a general method to transform conformally from the axisymmetric surface to a plane, where we use familiar methods based on the hydrodynamic stream function. Our approach shows that vortices constitute a Hamiltonian dynamical system, with their angular positions on the surface as canonical variables. The dynamical motion conserves both the total energy and the angular momentum, and it features a marked anticorrelation between the vortex speed and the local curvature of the surface.