Hamiltonian Systems Research Articles

Optimal control of three-dimensional unsteady partial differential equations with convection term in continuous casting

In this paper, we investigate an optimal control problem (OCP) for three-dimensional (3D) unsteady partial differential equations with convection term (UPDECT), extensively applied to the solidification process of billets. Firstly, we introduce the mathematical model of OCP and adopt a life cycle model method (LCMM) to numerically solve the 3D UPDECT. Secondly, the Lipschitz continuity of the cost function gradient is proved using the adjoint approach as a numerical method for designing optimal control systems. The cost function gradient of OCP for 3D UPDECT is analyzed based on a Hamiltonian costate system. Thirdly, we propose a modified trinomial conjugate gradient method (MTCGM) to solve the OCP for 3D UPDECT and prove the global convergence of MTCGM. Finally, the MTCGM is applied to the experimental simulation of continuous casting. According to experiment results, the optimizer based on MTCGM can significantly accelerate the convergence speed and reduce the iterative numbers, and the MTCGM provides a more stable temperature control effect.

Uncertainty quantification of phase transitions in magnetic materials lattices

This Perspective article aims to emphasize the crucial role of uncertainty quantification (UQ) in understanding magnetic phase transitions, which are pivotal in various applications, especially in the transportation and energy sectors [D. C. Jiles, Acta Mater. 51, 5907–5939 (2003) and Gutfleisch et al., Adv. Mater. 23, 821–842 (2011)]. Magnetic materials undergoing phase transitions, particularly due to high temperatures, pose challenges related to the loss of their inherent properties. However, pinpointing a definitive phase transition temperature proves challenging due to the diverse and uncertain nanostructure of materials. Deterministic approaches are limited when seeking a precise threshold. As a result, there is a need to develop probabilistic methods to improve the understanding of this physical problem. In this study, UQ is explored within the context of magnetic phase transitions. In addition, the broader applications of UQ in relation to microstructures and Hamiltonian systems are discussed to highlight its significance in materials science. Furthermore, this study discusses the potential future work on the integration of quantum computing to achieve more efficient UQ solutions of magnetic phase transitions using Ising models.

Global properties of generic real–analytic nearly–integrable Hamiltonian systems

We introduce a new class Gsn of generic real analytic potentials on Tn and study global analytic properties of natural nearly–integrable Hamiltonians 12|y|2+εf(x), with potential f∈Gsn, on the phase space M=B×Tn with B a given ball in Rn. The phase space M can be covered by three sets: a ‘non–resonant’ set, which is filled up to an exponentially small set of measure e−cK (where K is the maximal size of resonances considered) by primary maximal KAM tori; a ‘simply resonant set’ of measure εKa and a third set of measure εKb which is ‘non perturbative’, in the sense that the H–dynamics on it can be described by a natural system which is not nearly–integrable. We then focus on the simply resonant set – the dynamics of which is particularly interesting (e.g., for Arnol'd diffusion, or the existence of secondary tori) – and show that on such a set the secular (averaged) 1 degree–of–freedom Hamiltonians (labeled by the resonance index k∈Zn) can be put into a universal form (which we call ‘Generic Standard Form’), whose main analytic properties are controlled by only one parameter, which is uniform in the resonance label k.

KAM theorem for degenerate infinite-dimensional reversible systems

In this article, we establish a Kolmogorov-Arnold-Moser (KAM) theorem for degenerate infinite-dimensional reversible systems under a non-degenerate condition of Russmann type. This theorem broadens the scope of applicability of degenerate KAM theory, previously confined to Hamiltonian systems, by incorporating infinite-dimensional reversible systems. Using this theorem, we obtain the existence and linear stability of quasi-periodic solutions for a class of non-Hamiltonian but reversible beam equations with non-linearities in derivatives. For more information see https://ejde.math.txstate.edu/Volumes/2024/02/abstr.html

Periodic solutions for second-order even and noneven Hamiltonian systems

AbstractIn this paper, we consider the second-order Hamiltonian system $$ \ddot{x}+V^{\prime}(x)=0,\quad x\in \mathbb{R}^{N}. $$ x ¨ + V ′ ( x ) = 0 , x ∈ R N . We use the monotonicity assumption introduced by Bartsch and Mederski (Arch. Ration. Mech. Anal. 215:283–306, 2015). When V is even, we can release the strict convexity hypothesis, which is used by Bartsch and Mederski combined with the monotonicity assumption. When V is noneven, we weaken the strict convexity assumption and introduce another hypothesis (see (V10)). Then in both cases, we can build the homomorphism between the Nehari manifold and the unit sphere of some suitable space. Using the Nehari manifold method introduced by Szulkin (J. Funct. Anal. 257:3802–3822 2009), we prove the existence of T-periodic solutions with minimal period T.

2D Generating Surfaces and Dividing Surfaces in Hamiltonian Systems with Three Degrees of Freedom

In our previous work, we developed two methods for generalizing the construction of a periodic orbit dividing surface for a Hamiltonian system with three or more degrees of freedom. Starting with a periodic orbit, we extend it to form a torus or cylinder, which then becomes a higher-dimensional object within the energy surface (see [Katsanikas & Wiggins, 2021a, 2021b, 2023a, 2023b]). In this paper, we present two methods to construct dividing surfaces not from periodic orbits but by using 2D surfaces (2D geometrical objects) in a Hamiltonian system with three degrees of freedom. To illustrate the algorithm for this construction, we provide benchmark examples of three-degree-of-freedom Hamiltonian systems. Specifically, we employ the uncoupled and coupled cases of the quadratic normal form of a Hamiltonian system with three degrees of freedom.

Periodic and Antiperiodic Solutions for a Second‐Order Hamiltonian System With Nonlinearity Depending on Derivative

Some existence results of periodic solution are obtained for a class of second‐order Hamiltonian systems with nonlinearity depending on derivative. We prove that there exists T0 > 0 such that, for any T < T0, the provided Hamiltonian system has a nontrivial T‐periodic and T/2‐antiperiodic solution via linking theorem and iteration method.

Jet space extensions of infinite-dimensional Hamiltonian systems

We analyze infinite-dimensional Hamiltonian systems corresponding to partial Differential equations on one-dimensional spatial domains formulated with formally skew-adjoint Hamiltonian operators and non-quadratic Hamiltonian density. In various applications, the Hamiltonian density can depend on spatial derivatives of the state such that these systems can not straightforwardly be formulated as boundary port-Hamiltonian system using a Stokes-Dirac structure. In this work, we show that any Hamiltonian system of the above class can be reformulated as a Hamiltonian system on the jet space, in which the Hamiltonian density only depends on the extended state variable itself and not on its derivatives. Consequently, well-known geometric formulations with Stokes-Dirac structures are applicable. Additionally, we provide a similar result for dissipative systems. We illustrate the developed theory by means of the the Boussinesq equation, the dynamics of an elastic rod and the Allen-Cahn equation.

Exploiting Equivariance in the Design of Tracking Controllers for Euler-Poincare Systems on Matrix Lie Groups

The trajectory tracking problem is a fundamental control task in the study of mechanical systems. Hamiltonian systems are posed on the cotangent bundle of configuration space of a mechanical system, however, symmetries for the full cotangent bundle are not commonly used in geometric control theory. In this paper, we propose a group structure on the cotangent bundle of a Lie group and leverage this to define momentum and configuration errors for trajectory tracking, drawing on recent work on equivariant observer design. We show that this error definition leads to error dynamics that are themselves “Euler-Poincare like” and use these to derive simple, almost global trajectory tracking control for fully-actuated Euler-Poincare systems on a Lie group state space.

Hydrodynamic Substitution and Dynamics of Non–Hamiltonian Systems

The possibility of transforming the system of ordinary differential equations to the system of hydrodynamic Euler-type equations with application of hydrodynamic substitution for the Liouville transport equation for phase density equivalent to the original system of equations. Conditions for integrating of Euler-type equations are obtained by modifying HamiltonJacobi method for non-Hamiltonian dynamical systems.

Solutions of Analogs of Time-Dependent Schrödinger Equations Corresponding to a Pair of $$H^{2+2+1}$$ Hamiltonian Systems in the Hierarchy of Degenerations of an Isomonodromic Garnier System

Solutions of Analogs of Time-Dependent Schrödinger Equations Corresponding to a Pair of $$H^{2+2+1}$$ Hamiltonian Systems in the Hierarchy of Degenerations of an Isomonodromic Garnier System

Nonlinear stability of elliptic equilibria in Hamiltonian systems with resonances of order four with interactions

Nonlinear stability of elliptic equilibria in Hamiltonian systems with resonances of order four with interactions

Melnikov functions and limit cycle bifurcations for a class of piecewise Hamiltonian systems

<abstract><p>This study evaluated the number of limit cycles for a class of piecewise Hamiltonian systems with two zones separated by two semi-straight lines. First, we obtained explicit expressions of higher Melnikov functions. Then we applied these expressions to find the upper bounds of the number of limit cycles bifurcated from a period annulus of a piecewise polynomial Hamiltonian system.</p></abstract>

On the integrability of hybrid Hamiltonian systems

A hybrid system is a system whose dynamics are controlled by a mixture of both continuous and discrete transitions. The integrability of Hamiltonian systems is often identified with complete integrability or Liouville integrability, that is, the existence of as many independent integrals of motion in involution as the dimension of the phase space. Under certain regularity conditions, Liouville–Arnold theorem states that the invariant geometric structure associated with Liouville integrability is a fbration by Lagrangian tori, on which motions are linear. In this paper, we study an extension of the Liouville–Arnold theorem for hybrid systems whose continuous dynamics is given by a Hamiltonian vector field and we state conditions on the impact map and the switching surface under which a hybrid Hamiltonian system is, in a certain sense, completely integrable.

Implicit Nonholonomic Mechanics with Collisions

In this paper, variational techniques are used to analyze the dynamics of nonholonomic mechanical systems with impacts. Implicit nonholonomic smooth Lagrangian and Hamiltonian systems are extended to a nonsmooth context appropriate for collisions. In particular, we provide a variational formulation for implicit nonholonomic mechanical systems with collisions, for those collisions that preserve energy and momentum at the impact. Lastly, the theoretical results are illustrated by examining the example of a rolling disk hitting a wall.

Multiple coherent amplitude modes and exciton-phonon coupling in quasi-one-dimensional excitonic insulator Ta2NiSe5.

An excitonic insulator (EI) is an intriguing correlated electronic phase of condensed excitons. Ta2NiSe5 is a model material for investigating condensed excitonic states. Herein, femtosecond pump-probe spectroscopy is used to study the coherent phonon dynamics and associated exciton-phonon coupling in single-crystal Ta2NiSe5. The reflectivity time series consists of exponential decay due to hot carriers and damped oscillations due to the Ag phonon vibration. Given the in-plane anisotropic thermal conductivity of Ta2NiSe5, coherent phonon oscillations are stronger with perpendicular polarization to its quasi-one-dimensional chains. The 1-, 2-, and 4-THz vibration modes show coherent amplitude responses in the EI phase of Ta2NiSe5 with increasing temperature, totally different from those of normal coherent phonons (the 3- and 3.7-THz modes). The amplitude modes at higher frequencies decouple with the EI order parameter at lower temperatures, as supported by theoretical analysis with a model Hamiltonian of the exciton-phonon coupling system. Our work provides valuable insights into the character of the EI order parameter and its coupling to multiple coherent amplitude modes.

Exploration of solitons and analytical solutions by sub-ODE and variational integrators to Klein-Gordon model

In this paper, we use the sub-ODE method to analyze soliton solutions for the renowned nonlinear Klein-Gordon model (NLKGM). This method provides a variety of soliton solutions, including three positive solitons, three Jacobian elliptic function solutions, bright solitons, dark solitons, periodic solitons, rational solitons and hyperbolic function solutions. Applications for these solitons can be found in optical communication, fiber optic sensors, plasma physics, Bose-Einstein condensation and other areas. We also study some numerical solutions by using forward, backward, and central difference techniques. Moreover, we discuss variational integrators (VIs) using the projection technique for NLKGM. We develop a numerical solution for NLKGM using the discrete Euler lagrange equation, the Lagrangian and the Euler lagrange equation. At the end, in various dimensions, covering 3D, 2D, and contour, we will also plot several graphs for the obtained NLKGM solutions. A contour plot is a type of graphic representation that displays a three-dimensional surface on a two-dimensional plane by using contour lines. Each contour line in the plotted function represents one of the function's constant values, mapping the function's value across the plane. This model has been studied across multiple soliton solutions using various methods in the open literature, but this model for VIs and finite deference scheme (FDS) is the first time it has been studied. Within the various numerical techniques accessible for solving Hamiltonian systems, variational integrators distinguish themselves because of their symplectic quality. Here are some of the symplectic properties: symplectic orthogonality, energy conservation, area preservation, and structure preservation.

Limit cycle bifurcations by perturbing a kind of quadratic Hamiltonian system having a homoclinic loop

Limit cycle bifurcations by perturbing a kind of quadratic Hamiltonian system having a homoclinic loop

New Results for Fractional Hamiltonian Systems

New Results for Fractional Hamiltonian Systems

On a class of Hamiltonian systems in the plane involving general nonlinearities with critical exponential growth

On a class of Hamiltonian systems in the plane involving general nonlinearities with critical exponential growth