Hamiltonian Systems Research Articles

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.

Smooth transformations and ruling out closed orbits in planar systems

This work deals with planar dynamical systems with and without noise. In the first part, we seek to gain a refined understanding of such systems by studying their differential-geometric transformation properties under an arbitrary smooth mapping. Using elementary techniques, we obtain a unified picture of different classes of dynamical systems, some of which are classically viewed as distinct. We specifically give two examples of Hamiltonian systems with first integrals, which are simultaneously gradient systems. Potential applications of this apparent duality are discussed. The second part of this study is concerned with ruling out closed orbits in steady planar systems. We reformulate Bendixson’s criterion using the coordinate-independent Helmholtz decomposition derived in the first part, and we derive another, similar criterion. Our results allow for automated ruling out of closed orbits in certain regions of phase space and could be used in the future for efficient seeding of initial conditions in numerical algorithms to detect periodic solutions.

Energy-Preserving Parareal-RKN Algorithms for Hamiltonian Systems

Energy-Preserving Parareal-RKN Algorithms for Hamiltonian Systems

Generalized time-dependent SIS Hamiltonian models: Exact solutions and quantum deformations

The theory of Lie–Hamilton systems is used to construct generalized time-dependent SIS epidemic Hamiltonians with a variable infection rate from the ‘book’ Lie algebra. Although these are characterized by a set of non-autonomous nonlinear and coupled differential equations, their corresponding exact solution is explicitly found. Moreover, the quantum deformation of the book algebra is also considered, from which the corresponding deformed SIS Hamiltonians are obtained and interpreted as perturbations in terms of the quantum deformation parameter of previously known SIS systems. The exact solutions for these deformed systems are also obtained.

On the characteristic functions and Dirchlet-integrable solutions of singular left-definite Hamiltonian systems

In this work, a singular left-definite Hamiltonian system is considered and the characteristic-matrix theory for this Hamiltonian system is constructed. Using the results of this theory we introduce a lower bound for the number of Dirichlet-integrable solutions. Moreover we share a relation between the kernel of the solution of the nonhomogeneous boundary value problem and the characteristic-matrix.

Study on dynamic features and traveling wave solutions of perturbed nonlinear Biswas–Milovic equation combining Kudryashov's law of refractive index

This article mainly explores the Biswas–Milovic equation containing Kudryashov's refractive index law and nonlinear perturbation terms. We take the complete discrimination system for polynomial method and the trial equation method to implement qualitative and quantitative studies for the equation. We verify that the Hamiltonian of the dynamic system is conserved and apply the complete discrimination system of polynomial method to give the associated global phase diagrams, which are qualitatively analyzed to illustrate the presence of periodic solutions and solitons. In addition, we gain a classification of all single traveling wave solutions, provide optical wave patterns with specific parameters and validate the above findings.

Perturbation to conformal invariance and adiabatic invariants of generalized perturbed Hamiltonian system with additional terms

In this article, by considering the generalized perturbed Hamiltonian systems with additional terms, approximate conformal invariance and adiabatic invariants are studied. These invariants are obtained using approximate Lie point symmetries and without solving the structural equation. The necessary and sufficient condition for the relation between approximate conformal invariance and approximate Lie symmetry is obtained. The fundamental relationship between integrals and variables is presented herein, which is applicable to perturbed dynamic systems of both linear and non-linear nature. The approach is implemented on perturbed Hamiltonian systems of a generalized nature, featuring supplementary components of even dimensions. Additionally, the study derives the Hojman adiabatic invariants for generalized perturbed Hamiltonian systems with additional terms. Finally, an illustrative example is given to demonstrate the practical implementation of the findings.

Multi-control commercial aircraft trajectory optimization in a vertical plane with state-inequality constraints via singular control theory

We present a computational framework based on singular control theory to optimize commercial aircraft flights in a vertical plane. The optimization problem involves the lift coefficient (a regular control), and throttle setting (a singular control). However, given a small aerodynamic path angle typically associated with commercial aircraft flights and the significant time-scale separation in flight dynamics, it is customary to treat the aerodynamic path angle as a singular control and replace the lift coefficient as a function of states. This results in a model that is singular on both controls. This research sheds light on several key challenges that have either been unexplored, such as the possibility of switches in both controls even during a specified flight phase, or underexplored, such as state-inequality constraints and wind shear, in relation to the pure singular model. Adhering to the Pontryagin’s maximum principle, we develop a switching-point algorithm to solve the pure singular model for the direct operative cost. Extensive analysis is conducted to understand the impact of various relaxed parameters. The simulation results obtained from the pure singular model are validated through first and second order optimality conditions. Finally, we compare the solutions obtained from the pure singular model with those of the original model, ultimately concluding that the pure singular model yields exceptionally accurate results. In conclusion, the paper highlights the computational time and accuracy advantages of the proposed optimization framework. Furthermore, it underscores the noteworthy observation that the sensitivity of the associated Hamiltonian system with respect to aircraft mass is marginal.

Machine Learning Paves the Way for High Entropy Compounds Exploration: Challenges, Progress, and Outlook.

Machine learning (ML) has emerged as a powerful tool in the research field of high entropy compounds (HECs), which have gained worldwide attention due to their vast compositional space and abundant regulatability. However, the complex structure space of HEC poses challenges to traditional experimental and computational approaches, necessitating the adoption of machine learning. Microscopically, machine learning can model the Hamiltonian of the HEC system, enabling atomic-level property investigations, while macroscopically, it can analyze macroscopic material characteristics such as hardness, melting point, and ductility. Various machine learning algorithms, both traditional methods and deep neural networks, can be employed in HEC research. Comprehensive and accurate data collection, feature engineering, and model training and selection through cross-validation are crucial for establishing excellent ML models. ML also holds promise in analyzing phase structures and stability, constructing potentials in simulations, and facilitating the design of functional materials. Although some domains, such as magnetic and device materials, still require further exploration, machine learning's potential in HEC research is substantial. Consequently, machine learning has become an indispensable tool in understanding and exploiting the capabilities of HEC, serving as the foundation for the new paradigm of Artificial-intelligence-assisted material exploration.

A scheme for realizing nonreciprocal interlayer coupling in bilayer topological systems

Nonreciprocal interlayer coupling is difficult to practically implement in bilayer non-Hermitian topological photonic systems. In this work, we identify a similarity transformation between the Hamiltonians of systems with nonreciprocal interlayer coupling and on-site gain/loss. The similarity transformation is widely applicable, and we show its application in one- and two-dimensional bilayer topological systems as examples. The bilayer non-Hermitian system with nonreciprocal interlayer coupling, whose topological number can be defined using the gauge-smoothed Wilson loop, is topologically equivalent to the bilayer system with on-site gain/loss. We also show that the topological number of bilayer non-Hermitian C6v-typed domain-induced topological interface states can be defined in the same way as in the case of the bilayer non-Hermitian Su–Schrieffer–Heeger model. Our results show the relations between two microscopic provenances of the non-Hermiticity and provide a universal and convenient scheme for constructing and studying nonreciprocal interlayer coupling in bilayer non-Hermitian topological systems. This scheme is useful for observation of non-Hermitian skin effect in three-dimensional systems.Graphical

Orbital stability of periodic wave solution for Eckhaus-Kundu equation

In this paper, we mainly study the orbital stability of periodic traveling wave solution for the Eckhaus-Kundu equation with quintic nonlinearity, which is not a standard Hamilton system. Considering the studied equation is not a standard Hamilton system, the method presented by M. Grillakis and others for proving orbital stability cannot be applied directly, and this equation has two higher order nonlinear terms. So, by constructing three conserved quantities, using detailed spectral analysis and appropriate techniques, we overcome the complexity of the studied equation developed in calculation and proof, then, a conclusion on the orbital stability of the dn periodic wave solution for the Eckhaus-Kundu equation is obtained. As an extension of the proof for the above results, we also prove the orbital stability of the solitary wave for the studied Eckhaus-Kundu equation.

Quasiperiodic Poincaré persistence at high degeneracy

For Hamiltonian systems with high-order degenerate perturbation, we study the persistence of resonant invariant tori, where the resonant tori might be elliptic, hyperbolic or mixed types. As a consequence, we prove a quasiperiodic Poincaré theorem at high degeneracy. This answers a long standing conjecture on the persistence of resonant invariant tori in general situations.

Quasi-periodic solutions of a discrete integrable equation with a finite-dimensional integrable symplectic structure

Through the paper, we research the quasi-periodic solutions to a semi-discrete hierarchy which has Hamiltonian structure and integrable symplectic map. Firstly, a semi-discrete hierarchy is derived by use of the discrete zero-curvature representation and then its integrability is proved under the Liouville condition. Using the binary nonlinearization approach, the integrable symplectic map and finite-dimensional Hamiltonian system of the hierarchy are obtained. Moreover, the trigonal curve Kq−1 is denoted through the characteristic polynomials for the Lax pair, as well as the related Baker–Akhiezer function and meromorphic function are introduced, from which the asymptotic properties and divisors of the two functions mentioned above are analyzed. Finally, we introduce the three kinds of Abel differentials and straighten out of corresponding continuous and discrete flows, the quasi-periodic solutions of the equations are received via the Riemann theta function.

On the Interaction of Massive Photons and Mechanical Oscillators in Cavity Optomechanics: Basic Model and Quantization

AbstractThis study investigates the theoretical aspects of the interaction between photons with mass and a mechanical oscillator as drawn within the framework of cavity optomechanics. The study employs Proca theory as the mathematical framework to initially describe the dynamics of massive photons in a Fabry‐Perot cavity with a movable mass, both in classical and quantum scenarios. It quantifies the modifications induced by the nonzero photon mass, considering first‐ and second‐order effects, and derives expressions for the amplification of radiation pressure resulting from the presence of nonzero photon mass. Additionally, it derives the Hamiltonian of the quantum optomechanical system, incorporating the effects of photon mass at first and second order. It anticipates that experimental realization of massive optomechanics can be achieved by utilizing Proca material, which is a spatio‐temporally dispersive material that exhibits behavior equivalent to Proca theory in a vacuum, thus enabling the study of the interaction between massive photons and mechanical systems in cavity‐based optomechanical setups (referred to as massive cavity optomechanics). The study presented here caters to a diverse audience with an interest in the analysis and measurement of interactions among massive objects at the quantum scale.

Heteroclinic loop bifurcations by perturbing a class of Z2-equivariant quadratic switching Hamiltonian systems with nilpotent singular points

This paper investigates the problem of heteroclinic loop bifurcation by perturbing a class of Z2-equivariant quadratic switching systems with nilpotent singular points. Firstly, it provides sufficient and necessary conditions for the occurrence of a heteroclinic loop. Next, the system is perturbed by piecewise polynomial systems of degree n≥1. The paper then considers the lower bound for the maximum number of limit cycles that bifurcate from the generalized heteroclinic loop, finding at least n+[n2] limit cycles.

Power-law decay of the fraction of the mixed eigenstates in kicked top model with mixed-type classical phase space.

The properties of mixed eigenstates in a generic quantum system with a classical counterpart that has mixed-type phase space, although important to understand several fundamental questions that arise in both theoretical and experimental studies, are still not clear. Here, following a recent work [Č. Lozej, D. Lukman, and M. Robnik, Phys. Rev. E 106, 054203 (2022)2470-004510.1103/PhysRevE.106.054203], we perform an analysis of the features of mixed eigenstates in a time-dependent Hamiltonian system, the celebrated kicked top model. As a paradigmatic model for studying quantum chaos, the kicked top model is known to exhibit both classical and quantum chaos. The types of eigenstates are identified by means of the phase-space overlap index, which is defined as the overlap of the Husimi function with regular and chaotic regions in classical phase space. We show that the mixed eigenstates appear due to various tunneling precesses between different phase-space structures, while the regular and chaotic eigenstates are, respectively, associated with invariant tori and chaotic components in phase space. We examine how the probability distribution of the phase-space overlap index evolves with increasing system size for different kicking strengths. In particular, we find that the relative fraction of mixed states exhibits a power-law decay as the system size increases, indicating that only purely regular and chaotic eigenstates are left in the strict semiclassical limit. We thus provide further verification of the principle of uniform semiclassical condensation of Husimi functions and confirm the correctness of the Berry-Robnik picture.

Direct Poisson neural networks: learning non-symplectic mechanical systems

In this paper, we present neural networks learning mechanical systems that are both symplectic (for instance particle mechanics) and non-symplectic (for instance rotating rigid body). Mechanical systems have Hamiltonian evolution, which consists of two building blocks: a Poisson bracket and an energy functional. We feed a set of snapshots of a Hamiltonian system to our neural network models which then find both the two building blocks. In particular, the models distinguish between symplectic systems (with non-degenerate Poisson brackets) and non-symplectic systems (degenerate brackets). In contrast with earlier works, our approach does not assume any further a priori information about the dynamics except its Hamiltonianity, and it returns Poisson brackets that satisfy Jacobi identity. Finally, the models indicate whether a system of equations is Hamiltonian or not.

Untangling dissipative and Hamiltonian effects in bulk and boundary-driven systems.

Using the theory of large deviations, macroscopic fluctuation theory provides a framework to understand the behavior of nonequilibrium dynamics and steady states in diffusive systems. We extend this framework to a minimal model of a nonequilibrium nondiffusive system, specifically an open linear network on a finite graph. We explicitly calculate the dissipative bulk and boundary forces that drive the system towards the steady state, and the nondissipative bulk and boundary forces that drive the system in orbits around the steady state. Using the fact that these forces are orthogonal in a certain sense, we provide a decomposition of the large-deviation cost into dissipative and nondissipative terms. We establish that the purely nondissipative force turns the dynamics into a Hamiltonian system. These theoretical findings are illustrated by numerical examples.

High-order symmetric and energy-preserving collocation integrators for the second-order Hamiltonian system

High-order symmetric and energy-preserving collocation integrators for the second-order Hamiltonian system

Gravitoelectric dynamical tides at second post-Newtonian order

We present a gravitoelectric quadrupolar dynamical tidal-interaction Hamiltonian for a compact binary system, that is valid to second order in the post-Newtonian expansion. Our derivation uses the diagrammatic effective field theory approach, and involves Feynman integrals up to two loops, evaluated with the dimensional regularization scheme. We also derive the effective Hamiltonian for adiabatic tides, obtained by taking the appropriate limit of the dynamical effective Hamiltonian, and we check its validity by verifying the complete Poincaré algebra. In the adiabatic limit, we also calculate two gauge-invariant observables, namely, the binding energy for a circular orbit and the scattering angle in a hyperbolic scattering. Our results are important for developing accurate gravitational waveform models for neutron-star binaries for present and future gravitational-wave observatories.