
- Previous Article
- JGM Home
- This Issue
-
Next Article
A unifying approach for rolling symmetric spaces
Contact Hamiltonian and Lagrangian systems with nonholonomic constraints
1. | Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), C\Nicolás Cabrera, 13-15, Campus Cantoblanco, UAM, 28049 Madrid, Spain |
2. | Universidad de Alcalá (UAH) Campus Universitario. Ctra. Madrid-Barcelona, Km. 33, 600. 28805 Alcal´a de Henares, Madrid, Spain |
3. | Real Academia de Ciencias Exactas, Fisicas y Naturales C/de Valverde 22, 28004 Madrid, Spain |
In this article we develop a theory of contact systems with nonholonomic constraints. We obtain the dynamics from Herglotz's variational principle, by restricting the variations so that they satisfy the nonholonomic constraints. We prove that the nonholonomic dynamics can be obtained as a projection of the unconstrained Hamiltonian vector field. Finally, we construct the nonholonomic bracket, which is an almost Jacobi bracket on the space of observables and provides the nonholonomic dynamics.
References:
[1] |
R. Abraham and J. E. Marsden, Foundations of Mechanics, AMS Chelsea Publishing, Redwood City, CA, 1978.
doi: 10.1090/chel/364. |
[2] |
L. Bates and J. Śniatycki,
Nonholonomic reduction, Rep. Math. Phys., 32 (1993), 99-115.
doi: 10.1016/0034-4877(93)90073-N. |
[3] |
A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden and R. M. Murray,
Nonholonomic mechanical systems with symmetry, Arch. Rational Mech. Anal., 136 (1996), 21-99.
doi: 10.1007/BF02199365. |
[4] |
A. V. Borisov and I. S. Mamaev,
On the history of the development of the nonholonomic dynamics, Regul. Chaotic Dyn., 7 (2002), 43-47.
doi: 10.1070/RD2002v007n01ABEH000194. |
[5] |
A. Bravetti, Contact geometry and thermodynamics, Int. J. Geom. Methods Mod. Phys., 16 (2019), 51pp.
doi: 10.1142/S0219887819400036. |
[6] |
A. Bravetti, Contact Hamiltonian dynamics: The concept and its use, Entropy, 19 (2017), 12pp.
doi: 10.3390/e19100535. |
[7] |
A. Bravetti, M. de León, J. C. Marrero and E. Padrón, Invariant measures for contact Hamiltonian systems: Symplectic sandwiches with contact bread, J. Phys. A: Math. Theoret., 53 (2020).
doi: 10.1088/1751-8121/abbaaa. |
[8] |
A. Cannas da Silva and A. Weinstein, Geometric Models for Noncommutative Algebras, Berkeley Mathematics Lecture Notes, 10, American Mathematical Society, Providence, RI; Berkeley Center for Pure and Applied Mathematics, Berkeley, CA, 1999. |
[9] |
S. A. Chaplygin, Analysis of the Dynamics of Non-Holonomic Systems, Gostekhizdat, Mosow-Leningrad, 1949. Google Scholar |
[10] |
M. de León and D. M. de Diego,
A constraint algorithm for singular Lagrangians subjected to nonholonomic constraints, J. Math. Phys., 38 (1997), 3055-3062.
doi: 10.1063/1.532051. |
[11] |
M. de León and D. M. de Diego,
On the geometry of non-holonomic Lagrangian systems, J. Math. Phys., 37 (1996), 3389-3414.
doi: 10.1063/1.531571. |
[12] |
M. de León and D. M. de Diego,
Solving non-holonomic Lagrangian dynamics in terms of almost product structures, Extracta Math., 11 (1996), 325-347.
|
[13] |
M. de León and M. Lainz Valcázar, Contact Hamiltonian systems, J. Math. Phys., 60 (2019), 18pp.
doi: 10.1063/1.5096475. |
[14] |
M. de León and M. Lainz Valcázar, Infinitesimal symmetries in contact Hamiltonian systems, J. Geom. Phys., 153 (2020), 13pp.
doi: 10.1016/j.geomphys.2020.103651. |
[15] |
M. de León and M. Lainz Valcázar, Singular Lagrangians and precontact Hamiltonian systems, Int. J. Geom. Methods Mod. Phys., 16 (2019), 39pp.
doi: 10.1142/S0219887819501585. |
[16] |
M. de León, J. C. Marrero and D. M. de Diego,
Non-holonomic Lagrangian systems in jet manifolds, J. Phys. A, 30 (1997), 1167-1190.
doi: 10.1088/0305-4470/30/4/018. |
[17] |
M. de León and P. R. Rodrigues,
Higher-order mechanical systems with constraints, Internat. J. Theoret. Phys., 31 (1992), 1303-1313.
doi: 10.1007/BF00673930. |
[18] |
M. de León and P. R. Rodrigues, Methods of Differential Geometry in Analytical Mechanics, North-Holland Mathematics Studies, 158, North-Holland Publishing Co., Amsterdam, 1989. |
[19] |
M. A. de León,
A historical review on nonholomic mechanics, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 106 (2012), 191-284.
doi: 10.1007/s13398-011-0046-2. |
[20] |
J. Gaset, X. Gràcia, M. C. Muñoz-Lecanda, X. Rivas and N. Román-Roy, New contributions to the Hamiltonian and Lagrangian contact formalisms for dissipative mechanical systems and their symmetries, Int. J. Geom. Methods Mod. Phys., 17 (2020), 27pp.
doi: 10.1142/S0219887820500905. |
[21] |
F. Gay-Balmaz and H. Yoshimura,
A Lagrangian variational formulation for nonequilibrium thermodynamics. Part I: Discrete systems, J. Geom. Phys., 111 (2017), 169-193.
doi: 10.1016/j.geomphys.2016.08.018. |
[22] |
F. Gay-Balmaz and H. Yoshimura, From Lagrangian mechanics to nonequilibrium thermodynamics: A variational perspective, Entropy, 21 (2019).
doi: 10.3390/e21010008. |
[23] |
B. Georgieva, The variational principle of Hergloz and related resultst, in Geometry, Integrability and Quantization, Avangard Prima, Sofia, 2011, 214–225. |
[24] |
B. Georgieva, R. Guenther and T. Bodurov,
Generalized variational principle of Herglotz for several independent variables. First Noether-type theorem, J. Math. Phys., 44 (2003), 3911-3927.
doi: 10.1063/1.1597419. |
[25] |
H. Goldstein, C. P. Poole and J. L. Safko, Classical Mechanics, 2006. Google Scholar |
[26] |
G. Herglotz, Beruhrungstransformationen, in Lectures at the University of Gottingen, Gottingen, 1930. Google Scholar |
[27] |
A. Ibort, M. de León, G. Marmo and D. M. de Diego, Non-holonomic constrained systems as implicit differential equations. Geometrical structures for physical theories, I (Vietri, 1996), Rend. Sem. Mat. Univ. Politec. Torino, 54 (1996), 295–-317. |
[28] |
A. A. Kirillov,
Local Lie algebras, Uspehi Mat. Nauk, 31 (1976), 57-76.
doi: 10.1070/RM1976v031n04ABEH001556. |
[29] |
J. Koiller,
Reduction of some classical nonholonomic systems with symmetry, Arch. Rational Mech. Anal., 118 (1992), 113-148.
doi: 10.1007/BF00375092. |
[30] |
V. V. Kozlov,
On the integration theory of equations of nonholonomic mechanics, Regul. Chaotic Dyn., 7 (2002), 161-176.
doi: 10.1070/RD2002v007n02ABEH000203. |
[31] |
V. V. Kozlov,
Realization of nonintegrable constraints in classical mechanics, Dokl. Akad. Nauk SSSR, 272 (1983), 550-554.
|
[32] |
A. V. Kremnev and A. S. Kuleshov,
Nonlinear dynamics and stability of the skateboard, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 85-103.
doi: 10.3934/dcdss.2010.3.85. |
[33] |
A. S. Kuleshov,
A mathematical model of the snakeboard, Mat. Model., 18 (2006), 37-48.
|
[34] |
P. Libermann and C.-M. Marle, Symplectic Geometry and Analytical Mechanics, Mathematics and Its Applications, 35, D. Reidel Publishing Co., Dordrecht, 1987.
doi: 10.1007/978-94-009-3807-6. |
[35] |
A. Lichnerowicz,
Les variétés de Jacobi et leurs algèbres de Lie associées, J. Math. Pures Appl. (9), 57 (1978), 453-488.
|
[36] |
Q. Liu, P. J. Torres and C. Wang,
Contact Hamiltonian dynamics: Variational principles, invariants, completeness and periodic behavior, Ann. Physics, 395 (2018), 26-44.
doi: 10.1016/j.aop.2018.04.035. |
[37] |
N. K. Moshchuk,
On the motion of Chaplygin's sledge, J. Appl. Math. Mech., 51 (1987), 426-430.
doi: 10.1016/0021-8928(87)90079-7. |
[38] |
J. I. Ne${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$mark and N. A. Fufaev, Dynamics of Nonholonomic Systems, Translations of Mathematical Monographs, 33, American Mathematical Society, Providence, RI, 1972.
doi: 10.1090/mmono/033. |
[39] |
V. V. Rumiantsev,
On Hamilton's principle for nonholonomic systems, Prikl. Mat. Mekh., 42 (1978), 387-399.
|
[40] |
V. V. Rumyantsev,
Variational principles for systems with unilateral constraints, J. Appl. Math. Mech., 70 (2006), 808-818.
doi: 10.1016/j.jappmathmech.2007.01.002. |
[41] |
A. A. Simoes, D. M. de Diego, M. de León and M. L. Valcázar, On the geometry of discrete contact mechanics, preprint, arXiv: 2003.11892. Google Scholar |
[42] |
A. A. Simoes, M. de León, M. L. Valcázar and D. M. de Diego,
Contact geometry for simple thermodynamical systems with friction, Proc. A, 476 (2020), 244-259.
doi: 10.1098/rspa.2020.0244. |
[43] |
I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Progress in Mathematics, 118, Birkhäuser Verlag, Basel, 1994.
doi: 10.1007/978-3-0348-8495-2. |
[44] |
A. van der Schaft, Classical thermodynamics revisited: A systems and control perspective, preprint, arXiv: 2010.04213. Google Scholar |
[45] |
M. Vermeeren, A. Bravetti and M. Seri, Contact variational integrators, J. Phys. A, 52 (2019), 28pp.
doi: 10.1088/1751-8121/ab4767. |
[46] |
A. M. Vershik and L. D. Faddeev, Differential geometry and Lagrangian mechanics with constraints, Soviet Physics. Doklady, 17 (1972), 34-36. Google Scholar |
show all references
References:
[1] |
R. Abraham and J. E. Marsden, Foundations of Mechanics, AMS Chelsea Publishing, Redwood City, CA, 1978.
doi: 10.1090/chel/364. |
[2] |
L. Bates and J. Śniatycki,
Nonholonomic reduction, Rep. Math. Phys., 32 (1993), 99-115.
doi: 10.1016/0034-4877(93)90073-N. |
[3] |
A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden and R. M. Murray,
Nonholonomic mechanical systems with symmetry, Arch. Rational Mech. Anal., 136 (1996), 21-99.
doi: 10.1007/BF02199365. |
[4] |
A. V. Borisov and I. S. Mamaev,
On the history of the development of the nonholonomic dynamics, Regul. Chaotic Dyn., 7 (2002), 43-47.
doi: 10.1070/RD2002v007n01ABEH000194. |
[5] |
A. Bravetti, Contact geometry and thermodynamics, Int. J. Geom. Methods Mod. Phys., 16 (2019), 51pp.
doi: 10.1142/S0219887819400036. |
[6] |
A. Bravetti, Contact Hamiltonian dynamics: The concept and its use, Entropy, 19 (2017), 12pp.
doi: 10.3390/e19100535. |
[7] |
A. Bravetti, M. de León, J. C. Marrero and E. Padrón, Invariant measures for contact Hamiltonian systems: Symplectic sandwiches with contact bread, J. Phys. A: Math. Theoret., 53 (2020).
doi: 10.1088/1751-8121/abbaaa. |
[8] |
A. Cannas da Silva and A. Weinstein, Geometric Models for Noncommutative Algebras, Berkeley Mathematics Lecture Notes, 10, American Mathematical Society, Providence, RI; Berkeley Center for Pure and Applied Mathematics, Berkeley, CA, 1999. |
[9] |
S. A. Chaplygin, Analysis of the Dynamics of Non-Holonomic Systems, Gostekhizdat, Mosow-Leningrad, 1949. Google Scholar |
[10] |
M. de León and D. M. de Diego,
A constraint algorithm for singular Lagrangians subjected to nonholonomic constraints, J. Math. Phys., 38 (1997), 3055-3062.
doi: 10.1063/1.532051. |
[11] |
M. de León and D. M. de Diego,
On the geometry of non-holonomic Lagrangian systems, J. Math. Phys., 37 (1996), 3389-3414.
doi: 10.1063/1.531571. |
[12] |
M. de León and D. M. de Diego,
Solving non-holonomic Lagrangian dynamics in terms of almost product structures, Extracta Math., 11 (1996), 325-347.
|
[13] |
M. de León and M. Lainz Valcázar, Contact Hamiltonian systems, J. Math. Phys., 60 (2019), 18pp.
doi: 10.1063/1.5096475. |
[14] |
M. de León and M. Lainz Valcázar, Infinitesimal symmetries in contact Hamiltonian systems, J. Geom. Phys., 153 (2020), 13pp.
doi: 10.1016/j.geomphys.2020.103651. |
[15] |
M. de León and M. Lainz Valcázar, Singular Lagrangians and precontact Hamiltonian systems, Int. J. Geom. Methods Mod. Phys., 16 (2019), 39pp.
doi: 10.1142/S0219887819501585. |
[16] |
M. de León, J. C. Marrero and D. M. de Diego,
Non-holonomic Lagrangian systems in jet manifolds, J. Phys. A, 30 (1997), 1167-1190.
doi: 10.1088/0305-4470/30/4/018. |
[17] |
M. de León and P. R. Rodrigues,
Higher-order mechanical systems with constraints, Internat. J. Theoret. Phys., 31 (1992), 1303-1313.
doi: 10.1007/BF00673930. |
[18] |
M. de León and P. R. Rodrigues, Methods of Differential Geometry in Analytical Mechanics, North-Holland Mathematics Studies, 158, North-Holland Publishing Co., Amsterdam, 1989. |
[19] |
M. A. de León,
A historical review on nonholomic mechanics, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 106 (2012), 191-284.
doi: 10.1007/s13398-011-0046-2. |
[20] |
J. Gaset, X. Gràcia, M. C. Muñoz-Lecanda, X. Rivas and N. Román-Roy, New contributions to the Hamiltonian and Lagrangian contact formalisms for dissipative mechanical systems and their symmetries, Int. J. Geom. Methods Mod. Phys., 17 (2020), 27pp.
doi: 10.1142/S0219887820500905. |
[21] |
F. Gay-Balmaz and H. Yoshimura,
A Lagrangian variational formulation for nonequilibrium thermodynamics. Part I: Discrete systems, J. Geom. Phys., 111 (2017), 169-193.
doi: 10.1016/j.geomphys.2016.08.018. |
[22] |
F. Gay-Balmaz and H. Yoshimura, From Lagrangian mechanics to nonequilibrium thermodynamics: A variational perspective, Entropy, 21 (2019).
doi: 10.3390/e21010008. |
[23] |
B. Georgieva, The variational principle of Hergloz and related resultst, in Geometry, Integrability and Quantization, Avangard Prima, Sofia, 2011, 214–225. |
[24] |
B. Georgieva, R. Guenther and T. Bodurov,
Generalized variational principle of Herglotz for several independent variables. First Noether-type theorem, J. Math. Phys., 44 (2003), 3911-3927.
doi: 10.1063/1.1597419. |
[25] |
H. Goldstein, C. P. Poole and J. L. Safko, Classical Mechanics, 2006. Google Scholar |
[26] |
G. Herglotz, Beruhrungstransformationen, in Lectures at the University of Gottingen, Gottingen, 1930. Google Scholar |
[27] |
A. Ibort, M. de León, G. Marmo and D. M. de Diego, Non-holonomic constrained systems as implicit differential equations. Geometrical structures for physical theories, I (Vietri, 1996), Rend. Sem. Mat. Univ. Politec. Torino, 54 (1996), 295–-317. |
[28] |
A. A. Kirillov,
Local Lie algebras, Uspehi Mat. Nauk, 31 (1976), 57-76.
doi: 10.1070/RM1976v031n04ABEH001556. |
[29] |
J. Koiller,
Reduction of some classical nonholonomic systems with symmetry, Arch. Rational Mech. Anal., 118 (1992), 113-148.
doi: 10.1007/BF00375092. |
[30] |
V. V. Kozlov,
On the integration theory of equations of nonholonomic mechanics, Regul. Chaotic Dyn., 7 (2002), 161-176.
doi: 10.1070/RD2002v007n02ABEH000203. |
[31] |
V. V. Kozlov,
Realization of nonintegrable constraints in classical mechanics, Dokl. Akad. Nauk SSSR, 272 (1983), 550-554.
|
[32] |
A. V. Kremnev and A. S. Kuleshov,
Nonlinear dynamics and stability of the skateboard, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 85-103.
doi: 10.3934/dcdss.2010.3.85. |
[33] |
A. S. Kuleshov,
A mathematical model of the snakeboard, Mat. Model., 18 (2006), 37-48.
|
[34] |
P. Libermann and C.-M. Marle, Symplectic Geometry and Analytical Mechanics, Mathematics and Its Applications, 35, D. Reidel Publishing Co., Dordrecht, 1987.
doi: 10.1007/978-94-009-3807-6. |
[35] |
A. Lichnerowicz,
Les variétés de Jacobi et leurs algèbres de Lie associées, J. Math. Pures Appl. (9), 57 (1978), 453-488.
|
[36] |
Q. Liu, P. J. Torres and C. Wang,
Contact Hamiltonian dynamics: Variational principles, invariants, completeness and periodic behavior, Ann. Physics, 395 (2018), 26-44.
doi: 10.1016/j.aop.2018.04.035. |
[37] |
N. K. Moshchuk,
On the motion of Chaplygin's sledge, J. Appl. Math. Mech., 51 (1987), 426-430.
doi: 10.1016/0021-8928(87)90079-7. |
[38] |
J. I. Ne${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$mark and N. A. Fufaev, Dynamics of Nonholonomic Systems, Translations of Mathematical Monographs, 33, American Mathematical Society, Providence, RI, 1972.
doi: 10.1090/mmono/033. |
[39] |
V. V. Rumiantsev,
On Hamilton's principle for nonholonomic systems, Prikl. Mat. Mekh., 42 (1978), 387-399.
|
[40] |
V. V. Rumyantsev,
Variational principles for systems with unilateral constraints, J. Appl. Math. Mech., 70 (2006), 808-818.
doi: 10.1016/j.jappmathmech.2007.01.002. |
[41] |
A. A. Simoes, D. M. de Diego, M. de León and M. L. Valcázar, On the geometry of discrete contact mechanics, preprint, arXiv: 2003.11892. Google Scholar |
[42] |
A. A. Simoes, M. de León, M. L. Valcázar and D. M. de Diego,
Contact geometry for simple thermodynamical systems with friction, Proc. A, 476 (2020), 244-259.
doi: 10.1098/rspa.2020.0244. |
[43] |
I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Progress in Mathematics, 118, Birkhäuser Verlag, Basel, 1994.
doi: 10.1007/978-3-0348-8495-2. |
[44] |
A. van der Schaft, Classical thermodynamics revisited: A systems and control perspective, preprint, arXiv: 2010.04213. Google Scholar |
[45] |
M. Vermeeren, A. Bravetti and M. Seri, Contact variational integrators, J. Phys. A, 52 (2019), 28pp.
doi: 10.1088/1751-8121/ab4767. |
[46] |
A. M. Vershik and L. D. Faddeev, Differential geometry and Lagrangian mechanics with constraints, Soviet Physics. Doklady, 17 (1972), 34-36. Google Scholar |
[1] |
Alberto Bressan, Ke Han, Franco Rampazzo. On the control of non holonomic systems by active constraints. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3329-3353. doi: 10.3934/dcds.2013.33.3329 |
[2] |
Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087 |
[3] |
F.J. Herranz, J. de Lucas, C. Sardón. Jacobi--Lie systems: Fundamentals and low-dimensional classification. Conference Publications, 2015, 2015 (special) : 605-614. doi: 10.3934/proc.2015.0605 |
[4] |
Francisco Braun, Jaume Llibre, Ana Cristina Mereu. Isochronicity for trivial quintic and septic planar polynomial Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5245-5255. doi: 10.3934/dcds.2016029 |
[5] |
Marian Gidea, Rafael de la Llave, Tere M. Seara. A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold. Discrete & Continuous Dynamical Systems - A, 2020, 40 (12) : 6795-6813. doi: 10.3934/dcds.2020166 |
[6] |
Wenmin Gong, Guangcun Lu. On coupled Dirac systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4329-4346. doi: 10.3934/dcds.2017185 |
[7] |
Haiyan Wang. Existence and nonexistence of positive radial solutions for quasilinear systems. Conference Publications, 2009, 2009 (Special) : 810-817. doi: 10.3934/proc.2009.2009.810 |
[8] |
Tuvi Etzion, Alexander Vardy. On $q$-analogs of Steiner systems and covering designs. Advances in Mathematics of Communications, 2011, 5 (2) : 161-176. doi: 10.3934/amc.2011.5.161 |
[9] |
Lekbir Afraites, Abdelghafour Atlas, Fahd Karami, Driss Meskine. Some class of parabolic systems applied to image processing. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1671-1687. doi: 10.3934/dcdsb.2016017 |
[10] |
Graziano Crasta, Philippe G. LeFloch. Existence result for a class of nonconservative and nonstrictly hyperbolic systems. Communications on Pure & Applied Analysis, 2002, 1 (4) : 513-530. doi: 10.3934/cpaa.2002.1.513 |
[11] |
Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1717-1746. doi: 10.3934/dcdss.2020451 |
[12] |
Khosro Sayevand, Valeyollah Moradi. A robust computational framework for analyzing fractional dynamical systems. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021022 |
[13] |
Marcelo Messias. Periodic perturbation of quadratic systems with two infinite heteroclinic cycles. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1881-1899. doi: 10.3934/dcds.2012.32.1881 |
[14] |
Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189 |
[15] |
Valery Y. Glizer. Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 307-320. doi: 10.3934/naco.2020027 |
[16] |
Longxiang Fang, Narayanaswamy Balakrishnan, Wenyu Huang. Stochastic comparisons of parallel systems with scale proportional hazards components equipped with starting devices. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2021004 |
[17] |
Emma D'Aniello, Saber Elaydi. The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195 |
[18] |
Peter Benner, Jens Saak, M. Monir Uddin. Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 1-20. doi: 10.3934/naco.2016.6.1 |
[19] |
Rabiaa Ouahabi, Nasr-Eddine Hamri. Design of new scheme adaptive generalized hybrid projective synchronization for two different chaotic systems with uncertain parameters. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2361-2370. doi: 10.3934/dcdsb.2020182 |
[20] |
John Leventides, Costas Poulios, Georgios Alkis Tsiatsios, Maria Livada, Stavros Tsipras, Konstantinos Lefcaditis, Panagiota Sargenti, Aleka Sargenti. Systems theory and analysis of the implementation of non pharmaceutical policies for the mitigation of the COVID-19 pandemic. Journal of Dynamics & Games, 2021 doi: 10.3934/jdg.2021004 |
2019 Impact Factor: 0.649
Tools
Metrics
Other articles
by authors
[Back to Top]