Hamiltonian Systems Research Articles

Non-isochronicity on piecewise Hamiltonian differential systems with homogeneous nonlinearities

In this paper, we prove the non-isochronicity of Σ-centres for a class of planar piecewise smooth differential systems with a straight switching line, whose two sub-systems are Hamiltonian differential systems with a non-degenerated centre and only homogeneous nonlinearities. This result generalizes a classical result of non-isochronicity on the smooth Hamiltonian systems with homogeneous nonlinearities.

Krylov complexity for Jacobi coherent states

We develop computational tools necessary to extend the application of Krylov complexity beyond the simple Hamiltonian systems considered thus far in the literature. As a first step toward this broader goal, we show how the Lanczos algorithm that iteratively generates the Krylov basis can be augmented to treat coherent states associated with the Jacobi group, the semi-direct product of the 3-dimensional real Heisenberg-Weyl group H1, and the symplectic group, Sp(2, ℝ) ≃ SU(1, 1). Such coherent states are physically realized as squeezed states in, for example, quantum optics [1]. With the Krylov basis for both the SU(1, 1) and Heisenberg-Weyl groups being well understood, their semi-direct product is also partially analytically tractable. We exploit this to benchmark a scheme to numerically compute the Lanczos coefficients which, in principle, generalizes to the more general Jacobi group Hn ⋊ Sp(2n, ℝ).

Integrable systems of the ellipsoidal, paraboloidal and conical type with magnetic field

We construct integrable Hamiltonian systems with magnetic fields of the ellipsoidal, paraboloidal and conical type, i.e. systems that generalize natural Hamiltonians separating in the respective coordinate systems to include nonvanishing magnetic field. In the ellipsoidal and paraboloidal case each this classification results in three one–parameter families of systems, each involving one arbitrary function of a single variable and a parameter specifying the strength of the magnetic field of the given fully determined form. In the conical case the results are more involved, there are two one–parameter families like in the other cases and one class which is less restrictive and so far resists full classification.

Nonlinear Stability of Equilibria in Hamiltonian Systems with Multiple Resonances without Interactions

In this work, we advance in the study of the Lyapunov stability and instability of equilibrium solutions of Hamiltonian flows. More precisely, we study the nonlinear stability in the Lyapunov sense of equilibrium solutions in autonomous Hamiltonian systems with n-degrees of freedom, assuming the existence of two resonance vectors k1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{k}^1$$\\end{document} and k2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{k}^2$$\\end{document} without interaction (|k1|≤|k2|\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|\ extbf{k}^1|\\le |\ extbf{k}^2|$$\\end{document}). We provide conditions to obtain a type of formal stability, called Lie stability. In particular, we need to normalize the Hamiltonian function to any arbitrary order, and our results take into account the sign of the components of the resonance vectors. Subsequently, we guarantee some sufficient conditions to obtain exponential stability in the sense of Nekhoroshev for Lie stable systems. In addition, we give sufficient conditions for the instability in the Lyapunov sense of the full system. For this, it is necessary to normalize the Hamiltonian function to an adequate order, and assuming that the components of at least one resonance vector change of sign.

The soliton and periodic solutions for the perturbed sine–cosine–Gordon equation

In the paper, the existence of soliton and periodic solutions is studied for the perturbed sine–cosine–Gordon equation. The corresponding traveling wave system is converted to a regularly Hamilton system via using bifurcation theory of differential equations and the geometric singular perturbation theory. Then by using Melnikov's method and symbolic computation, periodic wave solutions, solitary solutions, and kink and antikink solutions are obtained. The wave speed selection principles are given, respectively. Numerical simulations are performed to illustrate our theoretical analysis.

A collision operator for describing dissipation in noncanonical phase space

The phase space of a noncanonical Hamiltonian system is partially inaccessible due to dynamical constraints (Casimir invariants) arising from the kernel of the Poisson tensor. When an ensemble of noncanonical Hamiltonian systems is allowed to interact, dissipative processes eventually break the phase space constraints, resulting in a thermodynamic equilibrium described by a Maxwell–Boltzmann distribution. However, the time scale required to reach Maxwell–Boltzmann statistics is often much longer than the time scale over which a given system achieves a state of thermal equilibrium. Examples include diffusion in rigid mechanical systems, as well as collisionless relaxation in magnetized plasmas and stellar systems, where the interval between binary Coulomb or gravitational collisions can be longer than the time scale over which stable structures are self-organized. Here, we focus on self-organizing phenomena over spacetime scales such that particle interactions respect the noncanonical Hamiltonian structure, but yet act to create a state of thermodynamic equilibrium. We derive a collision operator for general noncanonical Hamiltonian systems, applicable to fast, localized interactions. This collision operator depends on the interaction exchanged by colliding particles and on the Poisson tensor encoding the noncanonical phase space structure, is consistent with entropy growth and conservation of particle number and energy, preserves the interior Casimir invariants, reduces to the Landau collision operator in the limit of grazing binary Coulomb collisions in canonical phase space, and exhibits a metriplectic structure. We further show how thermodynamic equilibria depart from Maxwell–Boltzmann statistics due to the noncanonical phase space structure, and how self-organization and collisionless relaxation in magnetized plasmas and stellar systems can be described through the derived collision operator.

Quantum Tunneling and Complex Dynamics in the Suris's Integrable Map.

Quantum tunneling in a two-dimensional integrable map is studied. The orbits of the map are all confined to the curves specified by the one-dimensional Hamiltonian. It is found that the behavior of tunneling splitting for the integrable map and the associated Hamiltonian system is qualitatively the same, with only a slight difference in magnitude. However, the tunneling tails of the wave functions, obtained by superposing the eigenfunctions that form the doublet, exhibit significant differences. To explore the origin of the difference, we observe the classical dynamics in the complex plane and find that the existence of branch points appearing in the potential function of the integrable map could play the role of yielding non-trivial behavior in the tunneling tail. The result highlights the subtlety of quantum tunneling, which cannot be captured in nature only by the dynamics in the real plane.

Gradient preserving Operator Inference: Data-driven reduced-order models for equations with gradient structure

Hamiltonian Operator Inference has been introduced in Sharma et al. (2022) to learn structure-preserving reduced-order models (ROMs) for Hamiltonian systems. This approach constructs a low-dimensional model using only data and knowledge of the Hamiltonian function. Such ROMs can keep the intrinsic structure of the system, allowing them to capture the physics described by the governing equations. In this work, we extend this approach to more general systems that are either conservative or dissipative in energy, and which possess a gradient structure. We derive the optimization problems for inferring structure-preserving ROMs that preserve the gradient structure. We further derive an a priori error estimate for the reduced-order approximation. To test the algorithms, we consider semi-discretized partial differential equations with gradient structure, such as the parameterized wave and Korteweg–de-Vries equations, and equations of three-dimensional linear elasticity in the conservative case and the one- and two-dimensional Allen-Cahn equations in the dissipative case. The numerical results illustrate the accuracy, structure-preservation properties, and predictive capabilities of the gradient-preserving Operator Inference ROMs.

A new canonical affine bracket formulation of Hamiltonian classical field theories of first order

It has been a long standing question how to extend, in the finite-dimensional setting, the canonical Poisson bracket formulation from classical mechanics to classical field theories, in a completely general, intrinsic, and canonical way. In this paper, we provide an answer to this question by presenting a new completely canonical bracket formulation of Hamiltonian Classical Field Theories of first order on an arbitrary configuration bundle. It is obtained via the construction of the appropriate field-theoretic analogues of the Hamiltonian vector field and of the space of observables, via the introduction of a suitable canonical Lie algebra structure on the space of currents (the observables in field theories). This Lie algebra structure is shown to have a representation on the affine space of Hamiltonian sections, which yields an affine analogue to the Jacobi identity for our bracket. The construction is analogous to the canonical Poisson formulation of Hamiltonian systems although the nature of our formulation is linear-affine and not bilinear as the standard Poisson bracket. This is consistent with the fact that the space of currents and Hamiltonian sections are respectively, linear and affine. Our setting is illustrated with some examples including Continuum Mechanics and Yang–Mills theory.

Nonintegrability of time-periodic perturbations of single-degree-of-freedom Hamiltonian systems near homo- and heteroclinic orbits

We consider time-periodic perturbations of single-degree-of-freedom Hamiltonian systems and study their real-meromorphic nonintegrability in the Bogoyavlenskij sense using a generalized version due to Ayoul and Zung of the Morales–Ramis theory. The perturbation terms are assumed to have finite Fourier series in time, and the perturbed systems are rewritten as higher-dimensional autonomous systems having the small parameter as a state variable. We show that if the Melnikov functions are not constant, then the autonomous systems are not real-meromorphically integrable near homo- and heteroclinic orbits. Our result is not just an extension of previous results for homoclinic orbits to heteroclinic orbits and provides a stronger conclusion than them for the case of homoclinic orbits. We illustrate the theory for two periodically forced Duffing oscillators and a periodically forced two-dimensional system.

Periodic solutions in a 2D-symmetric Hamiltonian system through reduction and averaging method

We study a type of perturbed polynomial Hamiltonian system in 1:1 resonance. The perturbation consists of a homogeneous quartic potential invariant by rotations of π / 2 radians. The existence of periodic solutions is established using reduction and averaging theories. The different types of periodic solutions, linear stability, and bifurcation curves are characterized in terms of the parameters. Finally, some choreography of bifurcations are obtained, showing in detail the evolution of the phase flow.

Limit cycles of discontinuous piecewise differential Hamiltonian systems separated by a circle, or a parabola, or a hyperbola

Piecewise differential systems in the plane have been extensively studied in the last two decades, due to the vast application of these systems to describe natural phenomena. The knowledge of the existence or not of periodic solutions, in particular, of the limit cycles, is very important for understanding the dynamics of the differential systems. In this article we studied the maximum number of limit cycles of discontinuous piecewise differential systems, formed by two Hamiltonians systems of degree one with three distinct lines of discontinuity either a circle, or a parabola, or a hyperbola.

Fractal structures in the chaotic advection of passive scalars in leaky planar hydrodynamical flows.

The advection of passive scalars in time-independent two-dimensional incompressible fluid flows is an integrable Hamiltonian system. It becomes non-integrable if the corresponding stream function depends explicitly on time, allowing the possibility of chaotic advection of particles. We consider for a specific model (double gyre flow), a given number of exits through which advected particles can leak, without disturbing the flow itself. We investigate fractal escape basins in this problem and characterize fractality by computing the uncertainty exponent and basin entropy. Furthermore, we observe the presence of basin boundaries with points exhibiting the Wada property, i.e., boundary points that separate three or more escape basins.

The Evolution of the Phase Space Structure Along Pitchfork and Period-Doubling Bifurcations in a 3D-Galactic Bar Potential

We investigate how the phase space structure of a Three-Dimensional (3D) autonomous Hamiltonian system evolves across a series of successive Two-Dimensional (2D) and 3D pitchfork and period-doubling bifurcations, as the transition of the parent families of Periodic Orbits (POs) from stability to simple instability leads to the creation of new stable POs. Our research illustrates the consecutive alterations in the phase space structure near POs as the stability of the main family of POs changes. This process gives rise to new families of POs within the system, either maintaining the same or exhibiting higher multiplicity compared to their parent families. Tracking such a phase space transformation is challenging in a 3D system. By utilizing the color and rotation technique to visualize the Four-Dimensional (4D) Poincaré surfaces of section of the system, i.e. projecting them onto a 3D subspace and employing color to represent the fourth dimension, we can identify distinct structural patterns. Perturbations of parent and bifurcating stable POs result in the creation of tori characterized by a smooth color variation on their surface. Furthermore, perturbations of simple unstable parent POs beyond the bifurcation point, which lead to the birth of new stable families of POs, result in the formation of figure-8 structures of smooth color variations. These figure-8 formations surround well-shaped tori around the bifurcated stable POs, losing their well-defined forms for energies further away from the bifurcation point. We also observe that even slight perturbations of highly unstable POs create a cloud of mixed color points, which rapidly move away from the location of the PO. Our study introduces, for the first time, a systematic visualization of 4D surfaces of section within the vicinity of higher multiplicity POs. It elucidates how, in these cases, the coexistence of regular and chaotic orbits contributes to shaping the phase space landscape.

Periodic solutions of Hamiltonian systems coupling twist with an isochronous center

Periodic solutions of Hamiltonian systems coupling twist with an isochronous center

Higher Time-Derivative Theories from Space–Time Interchanged Integrable Field Theories

We compare a relativistic and a nonrelativistic version of Ostrogradsky’s method for higher-time derivative theories extended to scalar field theories and consider as an alternative a multi-field variant. We apply the schemes to space–time rotated modified Korteweg–de Vries systems and, exploiting their integrability, to Hamiltonian systems built from space–time rotated inverse Legendre transformed higher-order charges of these systems. We derive the equal-time Poisson bracket structures of these theories, establish the integrability of the latter theories by means of the Painlevé test and construct exact analytical period benign solutions in terms of Jacobi elliptic functions to the classical equations of motion. The classical energies of these partially complex solutions are real when they respect a certain modified CPT-symmetry and complex when this symmetry is broken. The higher-order Cauchy and initial-boundary value problem are addressed analytically and numerically. Finally, we provide the explicit quantization of the simplest mKdV system, exhibiting the usual conundrum of having the choice between having to deal with either a theory that includes non-normalizable states or spectra that are unbounded from below. In our non-Hermitian system, the choice is dictated by the correct sign in the decay width.

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.

Color image encryption algorithm based on novel dynamic DNA encoding and chaotic system* *This work was supported by the Open Fund of Advanced Cryptography and System Security Key Laboratory of Sichuan Province (Grant No. SKLACSS–202208), the Science and Technology Research Program of Chongqing Municipal Education Commission (Grant no. KJZD-M202000501), Sponsored by Natural Science Foundation of Chongqing (CSTB2023NSCQ-LZX0139) and National Natural

To enhance the security of image data, prevent unauthorized access, tampering, and leakage, maintain personal privacy, protect intellectual property rights, and ensure the integrity of images during transmission and storage. This study introduces an innovative color image encryption scheme based on dynamic DNA encoding operations and chaotic systems. By simulating a quantum random walk, a random key is generated to enhance the security of the confidential system. In addition, we integrated the enhanced Josephus problem into DNA coding rules to create dynamic DNA coding rules. At the same time, we proposed a dynamic double-loop DNA XOR operation, which fully utilizes the random sequence generated by the generalized Hamiltonian chaos system to precisely control the loop direction, starting point, and number of operations. This enhances the complexity of the encryption algorithm. After sufficient experimental verification and in-depth research and analysis, our innovative design not only enhances the difficulty of cracking while ensuring image quality but also provides reliable protection for the security of image data.

Hayden-Preskill recovery in Hamiltonian systems

Hayden-Preskill recovery in Hamiltonian systems

Existence and symmetry breaking results for positive solutions of elliptic Hamiltonian systems

Abstract In this paper, we are interested in positive solutions of $$ \begin{align*}\left\{ \begin{array}{@{}ll} -\Delta u = a(x)v^{p-1}, \quad &\text{ in } \Omega,\\ -\Delta v = b(x)u^{q-1}, \quad &\text{ in } \Omega,\\ u,v>0, \quad &\text{ in } \Omega,\\ u=v=0, \quad &\text{ on } \partial\Omega, \end{array} \right. \end{align*} $$ where $\Omega $ is a bounded annular domain (not necessarily an annulus) in ${\mathbb {R}}^N (N \ge 3)$ and $ a(x), b(x)$ are positive continuous functions. We show the existence of a positive solution for a range of supercritical values of p and q when the problem enjoys certain mild symmetry and monotonicity conditions. We shall also address the symmetry breaking phenomena where the system is fully symmetric. Indeed, as a consequence of our results, we shall show that problem (1) has $\Bigl \lfloor \frac {N}{2} \Bigr \rfloor $ (the floor of $\frac {N}{2}$ ) positive non-radial solutions when $ a(x)=b(x)=1$ and $\Omega $ is an annulus with certain assumptions on the radii. In general, for the radial case where the domain is an annulus, we prove the existence of a non-radial solution provided $$ \begin{align*}(p-1)(q-1)> \Big(1+\frac{2N}{\lambda_H}\Big)^2\left(\frac{q}{p}\right),\end{align*} $$ where $\lambda _H$ is the best constant for the Hardy inequality on $\Omega .$ We remark that the best constant $\lambda _H$ for the Hardy inequality is just the characteristic of the domain, and is independent of the choices of p and $q.$ For this reason, the aforementioned inequality plays a major role to prove the existence and multiplicity of non-radial solutions when the problem is fully symmetric. Our proofs use a variational formulation on appropriate convex subsets for which the lack of compactness is recovered for the supercritical problem.