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Journal articles on the topic 'Liouville systems'

Author: Grafiati

Published: 14 March 2023

Last updated: 31 July 2025

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1

Chetverikov, V. N. "Liouville systems and symmetries." Differential Equations 48, no. 12 (2012): 1639–51. http://dx.doi.org/10.1134/s0012266112120099.

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2

Narmanov, A. Ya, and Sh R. Ergashova. "Geometry of some completely integrable Hamiltonian systems." UZBEK MATHEMATICAL JOURNAL 69, no. 1 (2025): 110–19. https://doi.org/10.29229/uzmj.2025-1-10.

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The paper studies the geometry of a Liouville foliation generated by a completely integrable Hamiltonian system. It is shown that regular leaves are two dimensional submanifolds with zero Gaussian curvature and zero Gaussian torsion. It is studied a geometry of the distribution which generates orthogonal foliation to the Liouville foliation.

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3

Belozerov, Gleb Vladimirovich, and Anatoly Timofeevich Fomenko. "Orbital invariants of billiards and linearly integrable geodesic flows." Sbornik: Mathematics 215, no. 5 (2024): 573–611. http://dx.doi.org/10.4213/sm10034e.

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Orbital invariants of integrable topological billiards with two degrees of freedom are discovered and calculated in the case of constant energy of the system. These invariants (rotation vectors) are calculated in terms of rotation functions on one-parameter families of Liouville 2-tori. An analogue of Liouville's theorem is proved for a piecewise smooth billiard in a neighbourhood of a regular level. Action-angle variables are introduced. A general formula for rotation functions is obtained. There was a conjecture due to Fomenko that the rotation functions of topological billiards are monotone

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4

Borisova, Galina. "Sturm - Liouville systems and nonselfadjoint operators, presented as couplings of dissipative and antidissipative operators with real absolutely continuous spectra." Annual of Konstantin Preslavsky University of Shumen, Faculty of mathematics and informatics XXIII C (2022): 11–21. http://dx.doi.org/10.46687/wxfc2019.

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This paper is a continuation of the considerations of the paper [1] and it presents the connection between Sturm-Liouville systems and Livšic operator colligations theory. An usefull representation of solutions of Sturm - Liouville systems is obtained using the resolvent of operators from a large class of nonselfadjoint nondissipative operators, presented as couplings of dissipative and antidissipative operators with real spectra. A connection between Sturm-Liouville systems and the inner state of the corresponding open system of operators from the considered class is presented.

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5

Vedyushkina, Viktoriya Viktorovna, and Sergey Evgen'evich Pustovoitov. "Classification of Liouville foliations of integrable topological billiards in magnetic fields." Sbornik: Mathematics 214, no. 2 (2023): 166–96. http://dx.doi.org/10.4213/sm9770e.

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The topology of the Liouville foliations of integrable magnetic topological billiards, systems in which a ball moves on piecewise smooth two-dimensional surfaces in a constant magnetic field, is considered. The Fomenko-Zieschang invariants of Liouville equivalence are calculated for the Hamiltonian systems arising, and the topology of invariant 3-manifolds, isointegral and isoenergy ones, is investigated. The Liouville equivalence of such billiards to some known Hamiltonian systems is discovered, for instance, to the geodesic flows on 2-surfaces and to systems of rigid body dynamics. In partic

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6

Rynne, Bryan P. "The asymptotic distribution of the eigenvalues of right definite multiparameter Sturm-Liouville systems." Proceedings of the Edinburgh Mathematical Society 36, no. 1 (1993): 35–47. http://dx.doi.org/10.1017/s0013091500005873.

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This paper studies the asymptotic distribution of the multiparameter eigenvalues of a right definite multiparameter Sturm–Liouville eigenvalue problem. A uniform asymptotic analysis of the oscillation number of solutions of a single Sturm–Liouville type equation with potential depending on a general parameter is given; these results are then applied to the system of multiparameter Sturm–Liouville equations to give the asymptotic eigenvalue distribution for the system as a function of a “multi-index” oscillation number.

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7

Wang, Guofang. "Moser-Trudinger inequalities and Liouville systems." Comptes Rendus de l'Académie des Sciences - Series I - Mathematics 328, no. 10 (1999): 895–900. http://dx.doi.org/10.1016/s0764-4442(99)80293-6.

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8

Lin, Chang-Shou. "Liouville Systems of Mean Field Equations." Milan Journal of Mathematics 79, no. 1 (2011): 81–94. http://dx.doi.org/10.1007/s00032-011-0149-4.

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9

Zhuo, Ran, and FengQuan Li. "Liouville type theorems for Schrödinger systems." Science China Mathematics 58, no. 1 (2014): 179–96. http://dx.doi.org/10.1007/s11425-014-4925-9.

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10

Demskoi, D. K. "One Class of Liouville-Type Systems." Theoretical and Mathematical Physics 141, no. 2 (2004): 1509–27. http://dx.doi.org/10.1023/b:tamp.0000046560.84634.8c.

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11

Battaglia, Luca, Francesca Gladiali, and Massimo Grossi. "Nonradial entire solutions for Liouville systems." Journal of Differential Equations 263, no. 8 (2017): 5151–74. http://dx.doi.org/10.1016/j.jde.2017.06.009.

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12

Chipot, M., I. Shafrir, and G. Wolansky. "On the Solutions of Liouville Systems." Journal of Differential Equations 140, no. 1 (1997): 59–105. http://dx.doi.org/10.1006/jdeq.1997.3316.

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13

Chipot, M., I. Shafrir, and G. Wolansky. "On the Solutions of Liouville Systems." Journal of Differential Equations 178, no. 2 (2002): 630. http://dx.doi.org/10.1006/jdeq.2001.4105.

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14

Rynne, Bryan P. "The asymptotic distribution of the eigenvalues of multiparameter Sturm–Liouville systems II." Proceedings of the Edinburgh Mathematical Society 37, no. 2 (1994): 301–16. http://dx.doi.org/10.1017/s0013091500006088.

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In a previous paper we studied the asymptotic distribution of the multiparameter eigenvalues of uniformly right definite multiparameter Sturm–Liouville eigenvalue problems. In this paper we extend the analysis to deal with multiparameter Sturm–Liouville problems satisfying uniform left definiteness, and non-uniform left and right definiteness.

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15

Cai, Guocai, Hongjing Pan, and Ruixiang Xing. "A Note on Parabolic Liouville Theorems and Blow-Up Rates for a Higher-Order Semilinear Parabolic System." International Journal of Differential Equations 2011 (2011): 1–9. http://dx.doi.org/10.1155/2011/896427.

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We improve some results of Pan and Xing (2008) and extend the exponent range in Liouville-type theorems for some parabolic systems of inequalities with the time variable onR. As an immediate application of the parabolic Liouville-type theorems, the range of the exponent in blow-up rates for the corresponding systems is also improved.

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16

Kibkalo, V., A. Fomenko, and I. Kharcheva. "Realizing integrable Hamiltonian systems by means of billiard books." Transactions of the Moscow Mathematical Society 82 (March 15, 2022): 37–64. http://dx.doi.org/10.1090/mosc/324.

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Fomenko’s conjecture that the topology of the Liouville foliations associated with integrable smooth or analytic Hamiltonian systems can be realized by means of integrable billiard systems is discussed. An algorithm of Vedyushkina and Kharcheva’s realizing 3-atoms by billiard books, which has been simplified significantly by formulating it in terms of f f -graphs, is presented. Note that, using another algorithm, Vedyushkina and Kharcheva have also realized an arbitrary type of the base of the Liouville foliation on the whole 3-dimensional isoenergy surface. This algorithm is illustrated graph

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17

Fu, Jing-Li, Lijun Zhang, Chaudry Khalique, and Ma-Li Guo. "Circulatory integral and Routh's equations of Lagrange systems with Riemann-Liouville fractional derivatives." Thermal Science 25, no. 2 Part B (2021): 1355–63. http://dx.doi.org/10.2298/tsci200520034f.

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In this paper, the circulatory integral and Routh?s equations of Lagrange systems are established with Riemann-Liouville fractional derivatives, and the circulatory integral of Lagrange systems is obtained by making use of the relationship between Riemann-Liouville fractional integrals and fractional derivatives. Thereafter, the Routh?s equations of Lagrange systems are given based on the fractional circulatory integral. Two examples are presented to illustrate the application of the results.

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18

Neamaty, A., and S. Mosazadeh. "On the Canonical Solution of the Sturm–Liouville Problem with Singularity and Turning Point of Even Order." Canadian Mathematical Bulletin 54, no. 3 (2011): 506–18. http://dx.doi.org/10.4153/cmb-2011-069-7.

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AbstractIn this paper, we are going to investigate the canonical property of solutions of systems of differential equations having a singularity and turning point of even order. First, by a replacement, we transform the system to the Sturm–Liouville equation with turning point. Using of the asymptotic estimates provided by Eberhard, Freiling, and Schneider for a special fundamental system of solutions of the Sturm–Liouville equation, we study the infinite product representation of solutions of the systems. Then we transform the Sturm–Liouville equation with turning point to the equation with s

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19

Smyrnelis, Panayotis. "Gradient estimates for semilinear elliptic systems and other related results." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 145, no. 6 (2015): 1313–30. http://dx.doi.org/10.1017/s0308210515000347.

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A periodic connection is constructed for a double well potential defined in the plane. This solution violates Modica's estimate as well as the corresponding Liouville theorem for general phase transition potentials. Gradient estimates are also established for several kinds of elliptic systems. They allow us to prove the Liouville theorem in some particular cases. Finally, we give an alternative form of the stress–energy tensor for solutions defined in planar domains. As an application, we deduce a (strong) monotonicity formula.

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20

M, Nandakumar, and K. S. Subrahamanian Moosath. "Rough Liouville Equivalence of Integrable Hamiltonian Systems." Advances in Dynamical Systems and Applications 15, no. 2 (2020): 153–69. http://dx.doi.org/10.37622/adsa/15.2.2020.153-169.

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21

Gallegos, Javier, and Manuel Duarte-Mermoud. "Asymptotic analysis of Riemann–Liouville fractional systems." Electronic Journal of Qualitative Theory of Differential Equations, no. 73 (2018): 1–16. http://dx.doi.org/10.14232/ejqtde.2018.1.73.

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22

FARINA, ALBERTO. "A LIOUVILLE PROPERTY FOR GINZBURG–LANDAU SYSTEMS." Analysis and Applications 05, no. 03 (2007): 285–90. http://dx.doi.org/10.1142/s0219530507000985.

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In this short paper, we consider solutions u ∈ C2(ℝN, ℝM) (with N,M ≥ 1) of the Ginzburg–Landau system Δu = u(|u|2 - 1). For N = 3 and M = 2, we prove that every solution satisfying ∫ℝ3 (|u|2 - 1)2 < +∞, is constant and of unit norm. We also give necessary and sufficient conditions, on the integers N and M, ensuring a Liouville property for finite potential energy solutions of the system under consideration.

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23

Wang, Weimin, and Li Hong. "Liouville-type theorems for semilinear elliptic systems." Nonlinear Analysis: Theory, Methods & Applications 75, no. 13 (2012): 5380–91. http://dx.doi.org/10.1016/j.na.2012.04.057.

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24

Qu, Changzheng, and Liu Chao. "Heterotic Liouville systems from the Bernoulli equation." Physics Letters A 199, no. 5-6 (1995): 349–52. http://dx.doi.org/10.1016/0375-9601(95)00147-u.

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25

Kiselev, A. V., and J. W. van de Leur. "Symmetry algebras of Lagrangian Liouville-type systems." Theoretical and Mathematical Physics 162, no. 2 (2010): 149–62. http://dx.doi.org/10.1007/s11232-010-0011-9.

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26

DʼAmbrosio, Lorenzo, and Enzo Mitidieri. "Liouville theorems for elliptic systems and applications." Journal of Mathematical Analysis and Applications 413, no. 1 (2014): 121–38. http://dx.doi.org/10.1016/j.jmaa.2013.11.052.

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27

Duviryak, A., and A. Nazarenko. "Liouville equation for the systems with constraints." Journal of Physical Studies 3, no. 4 (1999): 399–408. http://dx.doi.org/10.30970/jps.03.399.

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28

Gu, Yi, and Lei Zhang. "Degree counting theorems for singular Liouville systems." ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA - CLASSE DI SCIENZE 21, no. 2 (2020): 1103–35. http://dx.doi.org/10.2422/2036-2145.201812_007.

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29

Zhang, Zhengce. "Liouville-Type Theorems for Some Integral Systems." Applied Mathematics 01, no. 02 (2010): 94–100. http://dx.doi.org/10.4236/am.2010.12012.

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30

Barrett, Louis C., and Dennis N. Winslow. "Interlacing theorems for interface Sturm-Liouville systems." Journal of Mathematical Analysis and Applications 129, no. 2 (1988): 533–59. http://dx.doi.org/10.1016/0022-247x(88)90270-3.

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31

Battaglia, Luca. "Moser–Trudinger inequalities for singular Liouville systems." Mathematische Zeitschrift 282, no. 3-4 (2015): 1169–90. http://dx.doi.org/10.1007/s00209-015-1584-7.

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32

Jin, Jiaming. "Existence results for Liouville equations and systems." Journal of Mathematical Analysis and Applications 491, no. 2 (2020): 124325. http://dx.doi.org/10.1016/j.jmaa.2020.124325.

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33

Fomenko, A. T. "Topological invariants of Liouville integrable Hamiltonian systems." Functional Analysis and Its Applications 22, no. 4 (1989): 286–96. http://dx.doi.org/10.1007/bf01077420.

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34

Denton, Zachary, and Juan Diego Ramírez. "Quasilinearization method for finite systems of nonlinear RL fractional differential equations." Opuscula Mathematica 40, no. 6 (2020): 667–83. http://dx.doi.org/10.7494/opmath.2020.40.6.667.

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In this paper the quasilinearization method is extended to finite systems of Riemann-Liouville fractional differential equations of order \(0\lt q\lt 1\). Existence and comparison results of the linear Riemann-Liouville fractional differential systems are recalled and modified where necessary. Using upper and lower solutions, sequences are constructed that are monotonic such that the weighted sequences converge uniformly and quadratically to the unique solution of the system. A numerical example illustrating the main result is given.

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35

Luo, Lin, and Engui Fan. "Finite-Dimensional Hamiltonian Systems from Li Spectral Problem by Symmetry Constraints." Zeitschrift für Naturforschung A 62, no. 7-8 (2007): 399–405. http://dx.doi.org/10.1515/zna-2007-7-808.

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A hierarchy associated with the Li spectral problem is derived with the help of the zero curvature equation. It is shown that the hierarchy possesses bi-Hamiltonian structure and is integrable in the Liouville sense. Moreover, the mono- and binary-nonlinearization theory can be successfully applied in the spectral problem. Under the Bargmann symmetry constraints, Lax pairs and adjoint Lax pairs are nonlineared into finite-dimensional Hamiltonian systems (FDHS) in the Liouville sense. New involutive solutions for the Li hierarchy are obtained.

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36

Pan, Xue, Xiuwen Li, and Jing Zhao. "Solvability and Optimal Controls of Semilinear Riemann-Liouville Fractional Differential Equations." Abstract and Applied Analysis 2014 (2014): 1–11. http://dx.doi.org/10.1155/2014/216919.

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We consider the control systems governed by semilinear differential equations with Riemann-Liouville fractional derivatives in Banach spaces. Firstly, by applying fixed point strategy, some suitable conditions are established to guarantee the existence and uniqueness of mild solutions for a broad class of fractional infinite dimensional control systems. Then, by using generally mild conditions of cost functional, we extend the existence result of optimal controls to the Riemann-Liouville fractional control systems. Finally, a concrete application is given to illustrate the effectiveness of our

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37

Wang, Zhongqian, Xuejun Zhang, and Mingliang Song. "Three nonnegative solutions for Sturm-Liouville BVP and application to the complete Sturm-Liouville equations." AIMS Mathematics 8, no. 3 (2023): 6543–58. http://dx.doi.org/10.3934/math.2023330.

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<abstract><p>The main purpose of this manuscript is to investigate the Sturm-Liouville BVP for non-autonomous Lagrangian systems. Under the suitable assumptions, we establish an existence theorem for three nonnegative solutions via Bonanno-Candito's three critical point theory. As an application in the complete Sturm-Liouville equations with Sturm-Liouville BVC, we get an existence theorem of three nonnegative solutions. Meanwhile, we give three examples to show the correctness of our results.</p></abstract>

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38

van der Kamp, Peter H., Theodoros E. Kouloukas, G. R. W. Quispel, Dinh T. Tran, and Pol Vanhaecke. "Integrable and superintegrable systems associated with multi-sums of products." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 470, no. 2172 (2014): 20140481. http://dx.doi.org/10.1098/rspa.2014.0481.

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39

Wang, Fei, and Yongqing Yang. "Correction: FRACTIONAL ORDER BARBALAT’S LEMMA AND ITS APPLICATIONS IN THE STABILITY OF FRACTIONAL ORDER NONLINEAR SYSTEMS." Mathematical Modelling and Analysis 22, no. 4 (2017): 503–13. http://dx.doi.org/10.3846/13926292.2017.1329755.

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This paper investigates fractional order Barbalat’s lemma and its applications for the stability of fractional order nonlinear systems with Caputo fractional derivative at first. Then, based on the relationship between Caputo fractional derivative and Riemann-Liouville fractional derivative, fractional order Barbalat’s lemma with Riemann-Liouville derivative is derived. Furthermore, according to these results, a set of new formulations of Lyapunov-like lemmas for fractional order nonlinear systems are established. Finally, an example is presented to verify the theoretical results in this paper

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40

Fomenko, Anatoly Timofeevich, and Viktoriya Viktorovna Vedyushkina. "Billiards and integrable systems." Russian Mathematical Surveys 78, no. 5 (2023): 881–954. http://dx.doi.org/10.4213/rm10100e.

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The survey is devoted to the class of integrable Hamiltonian systems and the class of integrable billiard systems and to the recent results of the authors and their students on the problem of comparison of these classes from the point of view of leafwise homeomorphy of their Liouville foliations. The key tool here are billiards on piecewise planar CW-complexes - topological billiards and billiard books - introduced by Vedyushkina. A construction of the class of evolutionary (force) billiards, introduced recently by Fomenko, is presented, enabling one to model a system in several non-singular e

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41

Chartbupapan, Watcharin, Ovidiu Bagdasar, and Kanit Mukdasai. "A Novel Delay-Dependent Asymptotic Stability Conditions for Differential and Riemann-Liouville Fractional Differential Neutral Systems with Constant Delays and Nonlinear Perturbation." Mathematics 8, no. 1 (2020): 82. http://dx.doi.org/10.3390/math8010082.

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The novel delay-dependent asymptotic stability of a differential and Riemann-Liouville fractional differential neutral system with constant delays and nonlinear perturbation is studied. We describe the new asymptotic stability criterion in the form of linear matrix inequalities (LMIs), using the application of zero equations, model transformation and other inequalities. Then we show the new delay-dependent asymptotic stability criterion of a differential and Riemann-Liouville fractional differential neutral system with constant delays. Furthermore, we not only present the improved delay-depend

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42

Codesido, Santiago, and F. Adrián F. Tojo. "A Liouville’s Formula for Systems with Reflection." Mathematics 9, no. 8 (2021): 866. http://dx.doi.org/10.3390/math9080866.

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In this work, we derived an Abel–Jacobi–Liouville identity for the case of two-dimensional linear systems of ODEs (ordinary differential equations) with reflection. We also present a conjecture for the general case and an application to coupled harmonic oscillators.

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43

Tuan Duong, Anh, and Quoc Hung Phan. "A Liouville-type theorem for cooperative parabolic systems." Discrete & Continuous Dynamical Systems - A 38, no. 2 (2018): 823–33. http://dx.doi.org/10.3934/dcds.2018035.

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44

D'Ambrosio, Lorenzo, and Enzo Mitidieri. "Hardy-Littlewood-Sobolev systems and related Liouville theorems." Discrete & Continuous Dynamical Systems - S 7, no. 4 (2014): 653–71. http://dx.doi.org/10.3934/dcdss.2014.7.653.

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45

Liu, Jian. "Path integral Liouville dynamics for thermal equilibrium systems." Journal of Chemical Physics 140, no. 22 (2014): 224107. http://dx.doi.org/10.1063/1.4881518.

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46

Littlejohn, Lance L., and Allan M. Krall. "Orthogonal polynomials and singular Sturm-Liouville Systems, I." Rocky Mountain Journal of Mathematics 16, no. 3 (1986): 435–80. http://dx.doi.org/10.1216/rmj-1986-16-3-435.

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47

Hile, G. N., and Chiping Zhou. "Liouville theorems for elliptic systems of arbitrary order." Applicable Analysis 73, no. 1-2 (1999): 115–30. http://dx.doi.org/10.1080/00036819908840768.

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48

LI, YUQIANG. "RIEMANN–LIOUVILLE PROCESSES ARISING FROM BRANCHING PARTICLE SYSTEMS." Stochastics and Dynamics 13, no. 03 (2013): 1250022. http://dx.doi.org/10.1142/s0219493712500220.

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It is proved in this paper that Riemann–Liouville processes can arise from the temporal structures of the scaled occupation time fluctuation limits of the site-dependent (d, α, σ(x)) branching particle systems in the case of 1 = d < α < 2 and ∫ℝ σ(x) d x < ∞.

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49

Annaby, M. H., J. Bustoz, and M. E. H. Ismail. "On sampling theory and basic Sturm–Liouville systems." Journal of Computational and Applied Mathematics 206, no. 1 (2007): 73–85. http://dx.doi.org/10.1016/j.cam.2006.05.024.

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50

Gorovoy, O., and E. Ivanov. "Superfield actions for N = 4 WZNW-Liouville systems." Nuclear Physics B 381, no. 1-2 (1992): 394–412. http://dx.doi.org/10.1016/0550-3213(92)90653-s.

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