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Journal articles on the topic 'Discontinuous Hamiltonians'

Author: Grafiati

Published: 1 June 2024

Last updated: 31 July 2025

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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 (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 pro

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2

Briani, Ariela, and Andrea Davini. "Monge solutions for discontinuous Hamiltonians." ESAIM: Control, Optimisation and Calculus of Variations 11, no. 2 (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 (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 (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 (2011): 1777. http://dx.doi.org/10.1090/s0002-9939-2010-10630-5.

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6

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 (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 generalizatio

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7

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

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8

Setayeshgar, Leila, and Hui Wang. "Large deviations for a feed-forward network." Advances in Applied Probability 43, no. 2 (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

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Setayeshgar, Leila, and Hui Wang. "Large deviations for a feed-forward network." Advances in Applied Probability 43, no. 02 (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

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10

vom Ende, Frederik. "Which bath Hamiltonians matter for thermal operations?" Journal of Mathematical Physics 63, no. 11 (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

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11

Nishimura, Akihiko, David B. Dunson, and Jianfeng Lu. "Discontinuous Hamiltonian Monte Carlo for discrete parameters and discontinuous likelihoods." Biometrika 107, no. 2 (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.

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12

Cardaliaguet, Pierre, Nicolas Forcadel, and Régis Monneau. "A class of germs arising from homogenization in traffic flow on junctions." Journal of Hyperbolic Differential Equations 21, no. 02 (2024): 189–254. http://dx.doi.org/10.1142/s0219891624500073.

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We consider traffic flows described by conservation laws. We mainly study a 2:1 junction (with two incoming roads and one outgoing road). At the mesoscopic level, the priority law at the junction is given by traffic lights, which are periodic in time; the traffic can also be slowed down by periodic in time flux-limiters. Looking at long-time behavior and on large space scale, we intuitively expect an effective junction condition to emerge, thus deriving a macroscopic model from the mesoscopic one. At the limit of the rescaling, we show rigorous homogenization of the problem and identify the ef

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13

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.

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14

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

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15

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 (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 analysi

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16

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 (2022): A3317—A3340. http://dx.doi.org/10.1137/22m147952x.

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17

Battelli, Flaviano, Michal Fečkan, and JinRong Wang. "Heteroclinic solutions in singularly perturbed discontinuous differential equations: a non-generic case." Electronic Journal of Qualitative Theory of Differential Equations, no. 27 (2024): 1–30. http://dx.doi.org/10.14232/ejqtde.2024.1.27.

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We derive Melnikov type conditions for the persistence of heteroclinic solutions in perturbed slowly varying discontinuous differential equations. Opposite to [J. Differential Equations 400(2024), 314–375] we assume that the unperturbed (frozen) equation has a parametric system of heteroclinic solutions and extend a result in [SIAM J. Math. Anal. 18(1987), 612–629] and [SIAM J. Math. Anal. 19(1988), 1254–1255] to higher dimensional non-Hamiltonian discontinuous singularly perturbed differential equations.

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18

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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19

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 (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.

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20

Zhao, Jianli, Qingjie Hu, Xinxin Xie та Chao Zhang. "Local Discontinuous Galerkin Methods for the Two-component μ-Camassa-Holm Equations". Journal of Physics: Conference Series 2890, № 1 (2024): 012018. http://dx.doi.org/10.1088/1742-6596/2890/1/012018.

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Abstract In this paper, we have developed two local discontinuous Galerkin (LDG) methods for the two-component μ-Camassa-Holm equations: a conservative scheme and a dissipative scheme. Exploiting the bi-Hamiltonian structure of the two-component μ-Camassa-Holm system, we introduce two significant Hamiltonian invariants and demonstrate that both schemes preserve discrete versions of these invariants. Additionally, we provide and prove a priori error estimates for both LDG schemes. Numerical experiments are conducted to validate the accuracy and effectiveness of the proposed methods.

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21

Llibre, Jaume, and Claudia Valls. "Limit Cycles of Planar Piecewise Differential Systems with Linear Hamiltonian Saddles." Symmetry 13, no. 7 (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 th

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22

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 (2008): 241–65. http://dx.doi.org/10.1007/s10915-008-9191-y.

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23

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 (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 sp

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24

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 (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

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25

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

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26

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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27

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 (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.

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28

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 (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.

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29

Battelli, Flaviano, and Michal Fečkan. "Periodic Solutions in Slowly Varying Discontinuous Differential Equations: The Generic Case." Mathematics 9, no. 19 (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.

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30

COCLITE, GIUSEPPE MARIA, and NILS HENRIK RISEBRO. "VISCOSITY SOLUTIONS OF HAMILTON–JACOBI EQUATIONS WITH DISCONTINUOUS COEFFICIENTS." Journal of Hyperbolic Differential Equations 04, no. 04 (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

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31

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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32

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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33

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 (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

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34

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.

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35

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.

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36

Jagla, E. A. "Discontinuous yielding transition of amorphous materials with low bulk modulus." Journal of Statistical Mechanics: Theory and Experiment 2021, no. 12 (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 t

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37

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 (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 th

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38

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 (2008): P05009. http://dx.doi.org/10.1088/1742-5468/2008/05/p05009.

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39

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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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 (2011): 943–56. http://dx.doi.org/10.1007/s10483-011-1471-8.

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41

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 (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

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42

Han, Ning, and Mingjuan Liu. "Dynamic Behavior Analysis of a Rotating Smooth and Discontinuous Oscillator with Irrational Nonlinearity." Modern Applied Science 12, no. 7 (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 heterocl

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43

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 (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 sc

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44

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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45

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 (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, it

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Liu, Di, and Yong Xu. "Random Disordered Periodical Input Induced Chaos in Discontinuous Systems." International Journal of Bifurcation and Chaos 29, no. 01 (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 fun

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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 (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 th

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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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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 (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 pie

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50

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 syste

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