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Journal articles on the topic 'Discontinuous Hamiltonians'

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Published: 1 June 2024

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1

Casimiro, Joyce A., and Jaume Llibre. "Limit Cycles of Discontinuous Piecewise Differential Hamiltonian Systems Separated by a Straight Line." Axioms 13, no. 3 (February 29, 2024): 161. http://dx.doi.org/10.3390/axioms13030161.

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In this article, we study the maximum number of limit cycles of discontinuous piecewise differential systems, formed by two Hamiltonians systems separated by a straight line. We consider three cases, when both Hamiltonians systems in each side of the discontinuity line have simultaneously degree one, two or three. We obtain that in these three cases, this maximum number is zero, one and three, respectively. Moreover, we prove that there are discontinuous piecewise differential systems realizing these maximum number of limit cycles. Note that we have solved the extension of the 16th Hilbert problem about the maximum number of limit cycles that these three classes of discontinuous piecewise differential systems separated by one straight line and formed by two Hamiltonian systems with a degree either one, two, or three, which such systems can exhibit.
2

Briani, Ariela, and Andrea Davini. "Monge solutions for discontinuous Hamiltonians." ESAIM: Control, Optimisation and Calculus of Variations 11, no. 2 (March 15, 2005): 229–51. http://dx.doi.org/10.1051/cocv:2005004.

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3

Jin, Shi, Hao Wu, and Zhongyi Huang. "A Hybrid Phase-Flow Method for Hamiltonian Systems with Discontinuous Hamiltonians." SIAM Journal on Scientific Computing 31, no. 2 (January 2009): 1303–21. http://dx.doi.org/10.1137/070709505.

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4

Dharmaraja, Sohan, Haneesh Kesari, Eric Darve, and Adrian J. Lew. "Time integrators based on approximate discontinuous Hamiltonians." International Journal for Numerical Methods in Engineering 89, no. 1 (July 25, 2011): 71–104. http://dx.doi.org/10.1002/nme.3236.

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5

Giga, Yoshikazu, Przemysław Górka, and Piotr Rybka. "A comparison principle for Hamilton-Jacobi equations with discontinuous Hamiltonians." Proceedings of the American Mathematical Society 139, no. 05 (May 1, 2011): 1777. http://dx.doi.org/10.1090/s0002-9939-2010-10630-5.

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6

Bensaoud, I., and A. Sayah. "Stability results for Hamilton-Jacobi equations with integro-differential terms and discontinuous Hamiltonians." Archiv der Mathematik 79, no. 5 (November 2002): 392–95. http://dx.doi.org/10.1007/pl00012462.

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7

Adimurthi, Aekta Aggarwal, and G. D. Veerappa Gowda. "Godunov-Type Numerical Methods for a Model of Granular Flow on Open Tables with Walls." Communications in Computational Physics 20, no. 4 (October 2016): 1071–105. http://dx.doi.org/10.4208/cicp.290615.060516a.

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AbstractWe propose and analyse finite volume Godunov type methods based on discontinuous flux for a 2×2 system of non-linear partial differential equations proposed by Hadeler and Kuttler to model the dynamics of growing sandpiles generated by a vertical source on a flat bounded rectangular table. The problem considered here is the so-called partially open table problem where sand is blocked by a wall (of infinite height) on some part of the boundary of the table. The novelty here is the corresponding modification of boundary conditions for the standing and the rolling layers and generalization of the techniques of the well-balancedness proposed in [1]. Presence of walls may lead to unbounded or discontinuous surface flow density at equilibrium resulting in solutions with singularities propagating from the extreme points of the walls. A scheme has been proposed to approximate efficiently the Hamiltonians with the coefficients which can be unbounded and discontinuous. Numerical experiments are presented to illustrate that the proposed schemes detect these singularities in the equilibrium solutions efficiently and comparisons are made with the previously studied finite difference and Semi-Lagrangian approaches by Finzi Vita et al.
8

Setayeshgar, Leila, and Hui Wang. "Large deviations for a feed-forward network." Advances in Applied Probability 43, no. 2 (June 2011): 545–71. http://dx.doi.org/10.1239/aap/1308662492.

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We consider a feed-forward network with a single-server station serving jobs with multiple levels of priority. The service discipline is preemptive in that the server always serves a job with the current highest level of priority. For this system with discontinuous dynamics, we establish the sample path large deviation principle using a weak convergence argument. In the special case where jobs have two different levels of priority, we also explicitly identify the exponential decay rate of the total population overflow probabilities by examining the geometry of the zero-level sets of the system Hamiltonians.
9

Setayeshgar, Leila, and Hui Wang. "Large deviations for a feed-forward network." Advances in Applied Probability 43, no. 02 (June 2011): 545–71. http://dx.doi.org/10.1017/s0001867800004985.

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We consider a feed-forward network with a single-server station serving jobs with multiple levels of priority. The service discipline is preemptive in that the server always serves a job with the current highest level of priority. For this system with discontinuous dynamics, we establish the sample path large deviation principle using a weak convergence argument. In the special case where jobs have two different levels of priority, we also explicitly identify the exponential decay rate of the total population overflow probabilities by examining the geometry of the zero-level sets of the system Hamiltonians.
10

vom Ende, Frederik. "Which bath Hamiltonians matter for thermal operations?" Journal of Mathematical Physics 63, no. 11 (November 1, 2022): 112202. http://dx.doi.org/10.1063/5.0117534.

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In this article, we explore the set of thermal operations from a mathematical and topological point of view. First, we introduce the concept of Hamiltonians with a resonant spectrum with respect to some reference Hamiltonian, followed by proving that when defining thermal operations, it suffices to only consider bath Hamiltonians, which satisfy this resonance property. Next, we investigate the continuity of the set of thermal operations in certain parameters, such as energies of the system and temperature of the bath. We will see that the set of thermal operations changes discontinuously with respect to the Hausdorff metric at any Hamiltonian, which has the so-called degenerate Bohr spectrum, regardless of the temperature. Finally, we find a semigroup representation of (enhanced) thermal operations in two dimensions by characterizing any such operation via three real parameters, thus allowing for a visualization of this set. Using this, in the qubit case, we show commutativity of (enhanced) thermal operations and convexity of thermal operations without the closure. The latter is done by specifying the elements of this set exactly.
11

Nishimura, Akihiko, David B. Dunson, and Jianfeng Lu. "Discontinuous Hamiltonian Monte Carlo for discrete parameters and discontinuous likelihoods." Biometrika 107, no. 2 (March 7, 2020): 365–80. http://dx.doi.org/10.1093/biomet/asz083.

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Summary Hamiltonian Monte Carlo has emerged as a standard tool for posterior computation. In this article we present an extension that can efficiently explore target distributions with discontinuous densities. Our extension in particular enables efficient sampling from ordinal parameters through the embedding of probability mass functions into continuous spaces. We motivate our approach through a theory of discontinuous Hamiltonian dynamics and develop a corresponding numerical solver. The proposed solver is the first of its kind, with a remarkable ability to exactly preserve the Hamiltonian. We apply our algorithm to challenging posterior inference problems to demonstrate its wide applicability and competitive performance.
12

Allahverdiev, Bilender, and Hüseyin Tuna. "Discontinuous linear Hamiltonian systems." Filomat 36, no. 3 (2022): 813–27. http://dx.doi.org/10.2298/fil2203813a.

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We study a discontinuous linear Hamiltonian system. We obtain a result on the existence and uniqueness of solutions. Later, we introduce the corresponding maximal and minimal operators for this problem and the self-adjoint extensions of such a minimal operator are established. Finally, we obtain an eigenfunction expansion.
13

Pessoa, Claudio, and Ronisio Ribeiro. "Limit cycles of planar piecewise linear Hamiltonian differential systems with two or three zones." Electronic Journal of Qualitative Theory of Differential Equations, no. 27 (2022): 1–19. http://dx.doi.org/10.14232/ejqtde.2022.1.27.

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In this paper, we study the existence of limit cycles in continuous and discontinuous planar piecewise linear Hamiltonian differential system with two or three zones separated by straight lines and such that the linear systems that define the piecewise one have isolated singular points, i.e. centers or saddles. In this case, we show that if the planar piecewise linear Hamiltonian differential system is either continuous or discontinuous with two zones, then it has no limit cycles. Now, if the planar piecewise linear Hamiltonian differential system is discontinuous with three zones, then it has at most one limit cycle, and there are examples with one limit cycle. More precisely, without taking into account the position of the singular points in the zones, we present examples with the unique limit cycle for all possible combinations of saddles and centers.
14

CHO, SANG-SOON, HOON HUH, and KWANG-CHUN PARK. "ANALYSIS OF ELASTO-PLASTIC STRESS WAVES BY A TIME-DISCONTINUOUS VARIATIONAL INTEGRATOR OF HAMILTONIAN." International Journal of Modern Physics B 22, no. 31n32 (December 30, 2008): 6259–64. http://dx.doi.org/10.1142/s0217979208051881.

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This paper proposes a numerical algorithm of a time-discontinuous variational integrator based on the Hamiltonian in order to obtain more accurate results in the analysis of elasto-plastic stress wave. The algorithm proposed adopts both a time-discontinuous variational integrator and space-continuous Hamiltonian so as to capture discontinuities of stress waves. The algorithm also adopts the limited kinetic energy to enhance the stability of the numerical algorithm so as to solve the discontinuities such as elastic unloading and internal reflection in plastic deformation. Finite element analysis of one dimensional elasto-plastic stress waves is carried out in order to demonstrate the accuracy of the algorithm proposed.
15

Ye, Boyang, Shi Jin, Yulong Xing, and Xinghui Zhong. "Hamiltonian-Preserving Discontinuous Galerkin Methods for the Liouville Equation With Discontinuous Potential." SIAM Journal on Scientific Computing 44, no. 5 (October 2022): A3317—A3340. http://dx.doi.org/10.1137/22m147952x.

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16

Tang, Wensheng, Yajuan Sun, and Wenjun Cai. "Discontinuous Galerkin methods for Hamiltonian ODEs and PDEs." Journal of Computational Physics 330 (February 2017): 340–64. http://dx.doi.org/10.1016/j.jcp.2016.11.023.

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17

Li, Shuangbao, Wei Zhang, and Yuxin Hao. "Melnikov-Type Method for a Class of Discontinuous Planar Systems and Applications." International Journal of Bifurcation and Chaos 24, no. 02 (February 2014): 1450022. http://dx.doi.org/10.1142/s0218127414500229.

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In this paper, we extend the well-known Melnikov method for smooth systems to a class of periodic perturbed piecewise smooth planar system. We assume that the unperturbed system is a piecewise Hamiltonian system which possesses a piecewise smooth homoclinic solution transversally crossing the switching manifold. The Melnikov-type function is explicitly derived by using the Hamiltonian function to measure the distance of the perturbed stable and unstable manifolds. Finally, we apply the obtained results to study the chaotic dynamics of a concrete piecewise smooth system.
18

Llibre, Jaume, and Claudia Valls. "Limit Cycles of Planar Piecewise Differential Systems with Linear Hamiltonian Saddles." Symmetry 13, no. 7 (June 24, 2021): 1128. http://dx.doi.org/10.3390/sym13071128.

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We provide the maximum number of limit cycles for continuous and discontinuous planar piecewise differential systems formed by linear Hamiltonian saddles and separated either by one or two parallel straight lines. We show that when these piecewise differential systems are either continuous or discontinuous and are separated by one straight line, or are continuous and are separated by two parallel straight lines, they do not have limit cycles. On the other hand, when these systems are discontinuous and separated by two parallel straight lines, we prove that the maximum number of limit cycles that they can have is one and that this maximum is reached by providing an example of such a system with one limit cycle. When the line of discontinuity of the piecewise differential system is formed by one straight line, the symmetry of the problem allows to take this straight line without loss of generality as the line x=0. Similarly, when the line of discontinuity of the piecewise differential system is formed by two parallel straight lines due to the symmetry of the problem, we can assume without loss of generality that these two straight lines are x=±1.
19

Xu, Yan, Jaap J. W. van der Vegt, and Onno Bokhove. "Discontinuous Hamiltonian Finite Element Method for Linear Hyperbolic Systems." Journal of Scientific Computing 35, no. 2-3 (March 1, 2008): 241–65. http://dx.doi.org/10.1007/s10915-008-9191-y.

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20

Bertsch, Michiel, Flavia Smarrazzo, Andrea Terracina, and Alberto Tesei. "Discontinuous viscosity solutions of first-order Hamilton–Jacobi equations." Journal of Hyperbolic Differential Equations 18, no. 04 (December 2021): 857–98. http://dx.doi.org/10.1142/s0219891621500259.

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We study the Cauchy problem for the simplest first-order Hamilton–Jacobi equation in one space dimension, with a bounded and Lipschitz continuous Hamiltonian which only depends on the spatial derivative. Uniqueness of discontinuous viscosity solutions is proven, if the initial data function has a finite number of jump discontinuities. Main ingredients of the proof are the barrier effect of spatial discontinuities of a solution (which is linked to the boundedness of the Hamiltonian), and a comparison theorem for semicontinuous viscosity subsolution and supersolution. These are defined in the spirit of the paper [H. Ishii, Perron’s method for Hamilton–Jacobi equations, Duke Math. J. 55 (1987) 368–384], yet using essential limits to introduce semicontinuous envelopes. The definition is shown to be compatible with Perron’s method for existence and is crucial in the uniqueness proof. We also describe some properties of the time evolution of spatial jump discontinuities of the solution, and obtain several results about singular Neumann problems which arise in connection with the above referred barrier effect.
21

Tian, R. L., Z. J. Zhao, X. W. Yang, and Y. F. Zhou. "Subharmonic Bifurcation for a Nonsmooth Oscillator." International Journal of Bifurcation and Chaos 27, no. 10 (September 2017): 1750163. http://dx.doi.org/10.1142/s0218127417501632.

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A nonsmooth pendulum model with multiple impulse effect is constructed to detect the bifurcation of a periodic orbit with multiple jump discontinuous points. Subharmonic Melnikov function of this kind of nonsmooth systems is studied. Differences of subharmonic Melnikov function between the nonsmooth system with multiple jump discontinuities and the smooth system are analyzed by using the Hamiltonian function and piecewise integral method. Applying the recursive method and perturbation principle, the effects of the jump discontinuous points on the subharmonic Melnikov function are converted to integral items which can be easily calculated. Hence, the subharmonic Melnikov function for the subharmonic orbit with multiple jump discontinuous points is obtained. Finally, the existence conditions for periodic motion of the subharmonic orbit are derived and the efficiency of the conclusions is verified via numerical simulations.
22

Hájíček, P., and J. Kijowski. "Lagrangian and Hamiltonian formalism for discontinuous fluid and gravitational field." Physical Review D 57, no. 2 (January 15, 1998): 914–35. http://dx.doi.org/10.1103/physrevd.57.914.

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23

Jin, Shi, and Xin Wen. "Hamiltonian-Preserving Schemes for the Liouville Equation with Discontinuous Potentials." Communications in Mathematical Sciences 3, no. 3 (2005): 285–315. http://dx.doi.org/10.4310/cms.2005.v3.n3.a2.

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24

García, Belén, Jaume Llibre, Jesús S. Pérez del Río, and Set Pérez-González. "Limit Cycles of Polynomially Integrable Piecewise Differential Systems." Axioms 12, no. 4 (March 31, 2023): 342. http://dx.doi.org/10.3390/axioms12040342.

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In this paper, we study how many algebraic limit cycles have the discontinuous piecewise linear differential systems separated by a straight line, with polynomial first integrals on both sides. We assume that at least one of the systems is Hamiltonian. Under this assumption, piecewise differential systems have no more than one limit cycle. This study characterizes linear differential systems with polynomial first integrals.
25

Castillo, P. E., and S. A. Gómez. "Conservación de invariantes de la ecuación de Schrödinger no lineal por el método LDG." Revista Mexicana de Física E 64, no. 1 (April 10, 2018): 52. http://dx.doi.org/10.31349/revmexfise.64.52.

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Conservation of the energy and the Hamiltonian of a general non linear Schr¨odinger equation is analyzed for the finite element method “Local Discontinuous Galerkin” spatial discretization. Conservation of the discrete analogue of these quantities is also proved for the fully discrete problem using the modified Crank-Nicolson method as time marching scheme. The theoretical results are validated on a series of problemsfor different nonlinear potentials.
26

Battelli, Flaviano, and Michal Fečkan. "Periodic Solutions in Slowly Varying Discontinuous Differential Equations: The Generic Case." Mathematics 9, no. 19 (October 2, 2021): 2449. http://dx.doi.org/10.3390/math9192449.

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We study persistence of periodic solutions of perturbed slowly varying discontinuous differential equations assuming that the unperturbed (frozen) equation has a non singular periodic solution. The results of this paper are motivated by a result of Holmes and Wiggins where the authors considered a two dimensional Hamiltonian family of smooth systems depending on a scalar variable which is the solution of a singularly perturbed equation.
27

COCLITE, GIUSEPPE MARIA, and NILS HENRIK RISEBRO. "VISCOSITY SOLUTIONS OF HAMILTON–JACOBI EQUATIONS WITH DISCONTINUOUS COEFFICIENTS." Journal of Hyperbolic Differential Equations 04, no. 04 (December 2007): 771–95. http://dx.doi.org/10.1142/s0219891607001355.

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We consider Hamilton–Jacobi equations, where the Hamiltonian depends discontinuously on both the spatial and temporal location. Our main result is the existence of viscosity solution to the Cauchy problem, and that the front tracking algorithm yields an L∞ contractive semigroup. We define a viscosity solution by treating the discontinuities in the coefficients analogously to "internal boundaries". The existence of viscosity solutions is established constructively via a front tracking approximation, whose limits are viscosity solutions, where by "viscosity solution" we mean a viscosity solution that posses some additional regularity at the discontinuities in the coefficients. We then show a comparison result that is valid for these viscosity solutions.
28

Tao, Molei, and Shi Jin. "Accurate and efficient simulations of Hamiltonian mechanical systems with discontinuous potentials." Journal of Computational Physics 450 (February 2022): 110846. http://dx.doi.org/10.1016/j.jcp.2021.110846.

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29

Thoma, Tobias, and Paul Kotyczka. "Structure preserving discontinuous Galerkin approximation of one-dimensional port-Hamiltonian systems." IFAC-PapersOnLine 56, no. 2 (2023): 6783–88. http://dx.doi.org/10.1016/j.ifacol.2023.10.386.

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30

Damene, Loubna, and Rebiha Benterki. "Limit cycles of discontinuous piecewise linear differential systems formed by centers or Hamiltonian without equilibria separated by irreducible cubics." Moroccan Journal of Pure and Applied Analysis 7, no. 2 (January 29, 2021): 248–76. http://dx.doi.org/10.2478/mjpaa-2021-0017.

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Abstract The main goal of this paper is to provide the maximum number of crossing limit cycles of two different families of discontinuous piecewise linear differential systems. More precisely we prove that the systems formed by two regions, where, in one region we define a linear center and in the second region we define a Hamiltonian system without equilibria can exhibit three crossing limit cycles having two or four intersection points with the cubic of separation. After we prove that the systems formed by three regions, where, in two noadjacent regions we define a Hamiltonian system without equilibria, and in the third region we define a center, can exhibit six crossing limit cycles having four and two simultaneously intersection points with the cubic of separation.
31

Jagla, E. A. "Discontinuous yielding transition of amorphous materials with low bulk modulus." Journal of Statistical Mechanics: Theory and Experiment 2021, no. 12 (December 1, 2021): 123201. http://dx.doi.org/10.1088/1742-5468/ac3d36.

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Abstract The yielding transition of amorphous materials is studied with a two-dimensional Hamiltonian model that allows both shear and volume deformations. The model is investigated as a function of the relative value of the bulk modulus B with respect to the shear modulus μ. When the ratio B/μ is small enough, the yielding transition becomes discontinuous, yet reversible. If the system is driven at constant strain rate in the coexistence region, a spatially localized shear band is observed while the rest of the system remains blocked. The crucial role of volume fluctuations in the origin of this behavior is clarified in a mean field version of the model.
32

Wang, An Mei. "Research on Information Applied Technology in the Study of Two-Layer Quantum Dots System." Applied Mechanics and Materials 473 (December 2013): 133–36. http://dx.doi.org/10.4028/www.scientific.net/amm.473.133.

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We study a two-electron system in a double-layer quantum dot under a magnetic field by means of the exact diagonalization of the Hamiltonian matrix. We find that discontinuous ground-state energy transitions are induced by an external magnetic field in the case of strong coupling. However, in the case of weak coupling, the angular momentum of the true ground state does not change in accordance with the change of the magnetic field B and remains = 0.
33

Wang, An Mei, Peng Wang, and Li Bo Fan. "Four Interacting Electrons in a Vertically Coupled Quantum Dots." Advanced Materials Research 468-471 (February 2012): 1810–13. http://dx.doi.org/10.4028/www.scientific.net/amr.468-471.1810.

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We studied the ground-state-transition of a vertically coupled four-layer single electron QDs system under a magnetic field by the exact diagonalization of the Hamiltonian matrix. For S=0, the energy spectra of the Dots are calculated as a function of applied magnetic field. We found discontinuous ground-state-transition induced by an external magnetic field in the case of strong coupling. However, in the case of weak coupling, such a transition does not occur.
34

Lu, Tiao, Wei Cai, Jianguo Xin, and Yinglong Guo. "Linear Scaling Discontinuous Galerkin Density Matrix Minimization Method with Local Orbital Enriched Finite Element Basis: 1-D Lattice Model System." Communications in Computational Physics 14, no. 2 (August 2013): 276–300. http://dx.doi.org/10.4208/cicp.290212.240812a.

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AbstractIn the first of a series of papers, we will study a discontinuous Galerkin (DG) framework for many electron quantum systems. The salient feature of this framework is the flexibility of using hybrid physics-based local orbitals and accuracy-guaranteed piecewise polynomial basis in representing the Hamiltonian of the many body system. Such a flexibility is made possible by using the discontinuous Galerkin method to approximate the Hamiltonian matrix elements with proper constructions of numerical DG fluxes at the finite element interfaces. In this paper, we will apply the DG method to the density matrix minimization formulation, a popular approach in the density functional theory of many body Schrödinger equations. The density matrix minimization is to find the minima of the total energy, expressed as a functional of the density matrix ρ(r,r′), approximated by the proposed enriched basis, together with two constraints of idempotency and electric neutrality. The idempotency will be handled with the McWeeny’s purification while the neutrality is enforced by imposing the number of electrons with a penalty method. A conjugate gradient method (a Polak-Ribiere variant) is used to solve the minimization problem. Finally, the linear-scaling algorithm and the advantage of using the local orbital enriched finite element basis in the DG approximations are verified by studying examples of one dimensional lattice model systems.
35

Híjar, Humberto, Jacqueline Quintana-H, and Godehard Sutmann. "Probability distributions of Hamiltonian changes in linear magnetic systems under discontinuous perturbations." Journal of Statistical Mechanics: Theory and Experiment 2008, no. 05 (May 22, 2008): P05009. http://dx.doi.org/10.1088/1742-5468/2008/05/p05009.

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36

Nurijanyan, S., J. J. W. van der Vegt, and O. Bokhove. "Hamiltonian discontinuous Galerkin FEM for linear, rotating incompressible Euler equations: Inertial waves." Journal of Computational Physics 241 (May 2013): 502–25. http://dx.doi.org/10.1016/j.jcp.2013.01.017.

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37

Li, Can-hua, and Chuan-miao Chen. "Ultraconvergence for averaging discontinuous finite elements and its applications in Hamiltonian system." Applied Mathematics and Mechanics 32, no. 7 (July 2011): 943–56. http://dx.doi.org/10.1007/s10483-011-1471-8.

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38

Baymout, Louiza, Rebiha Benterki, and Jaume Llibre. "The Solution of the Extended 16th Hilbert Problem for Some Classes of Piecewise Differential Systems." Mathematics 12, no. 3 (January 31, 2024): 464. http://dx.doi.org/10.3390/math12030464.

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The limit cycles have a main role in understanding the dynamics of planar differential systems, but their study is generally challenging. In the last few years, there has been a growing interest in researching the limit cycles of certain classes of piecewise differential systems due to their wide uses in modeling many natural phenomena. In this paper, we provide the upper bounds for the maximum number of crossing limit cycles of certain classes of discontinuous piecewise differential systems (simply PDS) separated by a straight line and consequently formed by two differential systems. A linear plus cubic polynomial forms six families of Hamiltonian nilpotent centers. First, we study the crossing limit cycles of the PDS formed by a linear center and one arbitrary of the six Hamiltonian nilpotent centers. These six classes of PDS have at most one crossing limit cycle, and there are systems in each class with precisely one limit cycle. Second, we study the crossing limit cycles of the PDS formed by two of the six Hamiltonian nilpotent centers. There are systems in each of these 21 classes of PDS that have exactly four crossing limit cycles.
39

Han, Ning, and Mingjuan Liu. "Dynamic Behavior Analysis of a Rotating Smooth and Discontinuous Oscillator with Irrational Nonlinearity." Modern Applied Science 12, no. 7 (June 22, 2018): 37. http://dx.doi.org/10.5539/mas.v12n7p37.

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This paper focuses on a novel rotating mechanical model which provides a cylindrical example of transition from smooth to discontinuous dynamics. The remarkable feature of the proposed system is a cylindrical dynamical system with strongly irrational nonlinearity exhibiting both smooth and discontinuous characteristics due to the geometry configuration. By using nonlinear dynamical technique, the unperturbed dynamics of the proposed system are studied including the irrational restoring force, stability of equilibria, Hamiltonian function and phase portraits. Note that a pair of double heteroclinic-like orbits connecting two non-standard saddle points are proposed in discontinuous case. For the perturbed system, we introduce a cylindrical approximate system for which the analytical solutions can be obtained successfully to reflect the nature of the original system without barrier of the irrationalities. Melnikov method is employed to detect the chaotic thresholds for the double heteroclinic orbits under the perturbation of viscous damping and external harmonic forcing in smooth regime. Finally, numerical simulations show the efficiency of the proposed method and demonstrate the predicated periodic solution and chaotic attractors. It is found that a good degree of correlation is demonstrated in the bifurcation diagram, the phase portraits of periodic solution, the chaotic attractor’ structures and the Lyapunov characteristics between the original system and approximate system.
40

He, Xijun, Dinghui Yang, Xiao Ma, and Yanjie Zhou. "Symplectic interior penalty discontinuous Galerkin method for solving the seismic scalar wave equation." GEOPHYSICS 84, no. 3 (May 1, 2019): T133—T145. http://dx.doi.org/10.1190/geo2018-0492.1.

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To improve the computational accuracy and efficiency of long-time wavefield simulations, we have developed a so-called symplectic interior penalty discontinuous Galerkin (IPDG) method for 2D acoustic equation. For the symplectic IPDG method, the scalar wave equation is first transformed into a Hamiltonian system. Then, the high-order IPDG formulations are introduced for spatial discretization because of their high accuracy and ease of dealing with computational domains with complex boundaries. The time integration is performed using an explicit third-order symplectic partitioned Runge-Kutta scheme so that it preserves the Hamiltonian structure of the wave equation in long-term simulations. Consequently, the symplectic IPDG method combines the advantages of discontinuous Galerkin method and the symplectic time integration. We investigate the properties of the method in detail for high-order spatial basis functions, including the stability criteria, numerical dispersion and dissipation relationships, and numerical errors. The analyses indicate that the symplectic SIPG method is nondissipative and retains low numerical dispersion. We also find that different symplectic IPDG methods have different convergence behaviors. It is indicated that using coarse meshes with a high-order method produces smaller errors and retains high accuracy. We have applied our method to simulate the scalar wavefields for different models, including layered models, a rough topography model, and the Marmousi model. The numerical results show that the symplectic IPDG method can suppress numerical dispersion effectively and provide accurate information on the wavefields. We also conduct a long-term experiment that verifies the capability of symplectic IPDG method for long-time simulations.
41

Liu, Hailiang, and Nianyu Yi. "A Hamiltonian preserving discontinuous Galerkin method for the generalized Korteweg–de Vries equation." Journal of Computational Physics 321 (September 2016): 776–96. http://dx.doi.org/10.1016/j.jcp.2016.06.010.

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42

Liu, Di, and Yong Xu. "Random Disordered Periodical Input Induced Chaos in Discontinuous Systems." International Journal of Bifurcation and Chaos 29, no. 01 (January 2019): 1950002. http://dx.doi.org/10.1142/s0218127419500020.

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In this paper, we extend the random Melnikov method from stochastic systems with a continuous vector field to discontinuous systems driven by a random disordered periodic input under the assumption that the unperturbed system is a piecewise Hamiltonian system. By measuring the distance of the perturbed stable and unstable manifolds, the nonsmooth random Melnikov process can be derived in detail, and then the mean square criterion for the onset of chaos is established in the statistical sense. It is shown that the threshold for the onset of chaos depends on the stochastic force and a scalar function of hypersurface. Finally, an example is given to analyze the chaotic dynamics using this extended approach, and discuss the effects of noise intensity on the dynamical behaviors of the system. The results indicate that the increase of the noise intensity will result in a chaotic motion of the discontinuous stochastic system and the changes of possible chaotic degree in the phase space. At the same time, the effects of noise intensity on chaos are further investigated through the system response including time history and phase portraits, Poincaré maps and [Formula: see text]-[Formula: see text] test.
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PÉREZ, A. T., R. CHACÓN, and A. CASTELLANOS. "BEHAVIOR OF DYNAMICAL SYSTEMS SUBJECTED TO CONTINUOUS AND DISCONTINUOUS FORCING: APPLICATION TO LAMINAR CHAOTIC MIXING." International Journal of Bifurcation and Chaos 06, no. 12b (December 1996): 2627–34. http://dx.doi.org/10.1142/s0218127496001697.

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This paper studies the effect of continuous and discontinuous time dependent forcings onto dynamical systems. We compare these different forcings in the context of laminar chaotic mixing. It is shown that the response of a Hamiltonian two-dimensional system to a time periodic sinusoidal forcing differs qualitatively and quantitatively from the response to a square wave function of the same frequency. Consequently, the mixing efficiency of both types of forcings are different. Also a periodic function of the same shape as that of the velocity of the unperturbed system is tested as a forcing, its mixing efficiency being intermediate.
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Khanin, Konstantin, and Andrei Sobolevski. "Particle dynamics inside shocks in Hamilton–Jacobi equations." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 368, no. 1916 (April 13, 2010): 1579–93. http://dx.doi.org/10.1098/rsta.2009.0283.

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The characteristic curves of a Hamilton–Jacobi equation can be seen as action-minimizing trajectories of fluid particles. For non-smooth ‘viscosity’ solutions, which give rise to discontinuous velocity fields, this description is usually pursued only up to the moment when trajectories hit a shock and cease to minimize the Lagrangian action. In this paper we show that, for any convex Hamiltonian, there exists a uniquely defined canonical global non-smooth coalescing flow that extends particle trajectories and determines the dynamics inside shocks. We also provide a variational description of the corresponding effective velocity field inside shocks, and discuss the relation to the ‘dissipative anomaly’ in the limit of vanishing viscosity.
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van Oers, Alexander M., Leo R. M. Maas, and Onno Bokhove. "Hamiltonian discontinuous Galerkin FEM for linear, stratified (in)compressible Euler equations: internal gravity waves." Journal of Computational Physics 330 (February 2017): 770–93. http://dx.doi.org/10.1016/j.jcp.2016.10.032.

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46

Cao, Qingjie, Marian Wiercigroch, Ekaterina E. Pavlovskaia, J. Michael T. Thompson, and Celso Grebogi. "Piecewise linear approach to an archetypal oscillator for smooth and discontinuous dynamics." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 366, no. 1865 (August 13, 2007): 635–52. http://dx.doi.org/10.1098/rsta.2007.2115.

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In a recent paper we examined a model of an arch bridge with viscous damping subjected to a sinusoidally varying central load. We showed how this yields a useful archetypal oscillator which can be used to study the transition from smooth to discontinuous dynamics as a parameter, α , tends to zero. Decreasing this smoothness parameter (a non-dimensional measure of the span of the arch) changes the smooth load–deflection curve associated with snap-buckling into a discontinuous sawtooth. The smooth snap-buckling curve is not amenable to closed-form theoretical analysis, so we here introduce a piecewise linearization that correctly fits the sawtooth in the limit at α =0. Using a Hamiltonian formulation of this linearization, we derive an analytical expression for the unperturbed homoclinic orbit, and make a Melnikov analysis to detect the homoclinic tangling under the perturbation of damping and driving. Finally, a semi-analytical method is used to examine the full nonlinear dynamics of the perturbed piecewise linear system. A chaotic attractor located at α =0.2 compares extremely well with that exhibited by the original arch model: the topological structures are the same, and Lyapunov exponents (and dimensions) are in good agreement.
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Benterki, Rebiha, Jeidy Jimenez, and Jaume Llibre. "Limit cycles of planar discontinuous piecewise linear Hamiltonian systems without equilibria separated by reducible cubics." Electronic Journal of Qualitative Theory of Differential Equations, no. 69 (2021): 1–38. http://dx.doi.org/10.14232/ejqtde.2021.1.69.

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Due to their applications to many physical phenomena during these last decades the interest for studying the discontinuous piecewise differential systems has increased strongly. The limit cycles play a main role in the study of any planar differential system, but to determine the maximum number of limits cycles that a class of planar differential systems can have is one of the main problems in the qualitative theory of the planar differential systems. Thus in general to provide a sharp upper bound for the number of crossing limit cycles that a given class of piecewise linear differential system can have is a very difficult problem. In this paper we characterize the existence and the number of limit cycles for the piecewise linear differential systems formed by linear Hamiltonian systems without equilibria and separated by a reducible cubic curve, formed either by an ellipse and a straight line, or by a parabola and a straight line parallel to the tangent at the vertex of the parabola. Hence we have solved the extended 16th Hilbert problem to this class of piecewise differential systems.
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Benterki, Rebiha, Jeidy Jimenez, and Jaume Llibre. "Limit cycles of planar discontinuous piecewise linear Hamiltonian systems without equilibria separated by reducible cubics." Electronic Journal of Qualitative Theory of Differential Equations, no. 69 (2021): 1–38. http://dx.doi.org/10.14232/ejqtde.2021.1.69.

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Abstract:
Due to their applications to many physical phenomena during these last decades the interest for studying the discontinuous piecewise differential systems has increased strongly. The limit cycles play a main role in the study of any planar differential system, but to determine the maximum number of limits cycles that a class of planar differential systems can have is one of the main problems in the qualitative theory of the planar differential systems. Thus in general to provide a sharp upper bound for the number of crossing limit cycles that a given class of piecewise linear differential system can have is a very difficult problem. In this paper we characterize the existence and the number of limit cycles for the piecewise linear differential systems formed by linear Hamiltonian systems without equilibria and separated by a reducible cubic curve, formed either by an ellipse and a straight line, or by a parabola and a straight line parallel to the tangent at the vertex of the parabola. Hence we have solved the extended 16th Hilbert problem to this class of piecewise differential systems.
49

Cai, Wenjun, Yajuan Sun, Yushun Wang, and Huai Zhang. "Local discontinuous Galerkin methods based on the multisymplectic formulation for two kinds of Hamiltonian PDEs." International Journal of Computer Mathematics 95, no. 1 (June 11, 2017): 114–43. http://dx.doi.org/10.1080/00207160.2017.1335866.

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50

Breitenbach, Tim, and Alfio Borzì. "A Sequential Quadratic Hamiltonian Method for Solving Parabolic Optimal Control Problems with Discontinuous Cost Functionals." Journal of Dynamical and Control Systems 25, no. 3 (September 19, 2018): 403–35. http://dx.doi.org/10.1007/s10883-018-9419-6.

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