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SIGMA, 2012, Volume 8, 106, 21 pages (Mi sigma783)  

This article is cited in 17 scientific papers (total in 17 papers)

Nonlocal Symmetries, Telescopic Vector Fields and $\lambda$-Symmetries of Ordinary Differential Equations

Concepción Muriel, Juan Luis Romero

Department of Mathematics, University of Cádiz, 11510 Puerto Real, Spain

Abstract: This paper studies relationships between the order reductions of ordinary differential equations derived by the existence of $\lambda$-symmetries, telescopic vector fields and some nonlocal symmetries obtained by embedding the equation in an auxiliary system. The results let us connect such nonlocal symmetries with approaches that had been previously introduced: the exponential vector fields and the $\lambda$-coverings method. The $\lambda$-symmetry approach let us characterize the nonlocal symmetries that are useful to reduce the order and provides an alternative method of computation that involves less unknowns. The notion of equivalent $\lambda$-symmetries is used to decide whether or not reductions associated to two nonlocal symmetries are strictly different.

Keywords: nonlocal symmetries; $\lambda$-symmetries; telescopic vector fields; order reductions; differential invariants


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MSC: 34A05; 34A34
Received: July 9, 2012; in final form December 19, 2012; Published online December 28, 2012

Citation: Concepción Muriel, Juan Luis Romero, “Nonlocal Symmetries, Telescopic Vector Fields and $\lambda$-Symmetries of Ordinary Differential Equations”, SIGMA, 8 (2012), 106, 21 pp.

Citation in format AMSBIB
\by Concepci\'on~Muriel, Juan~Luis~Romero
\paper Nonlocal Symmetries, Telescopic Vector Fields and $\lambda$-Symmetries of Ordinary Differential Equations
\jour SIGMA
\yr 2012
\vol 8
\papernumber 106
\totalpages 21

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    1. Cicogna G., Gaeta G., Walcher S., “Dynamical Systems and SIGMA-Symmetries”, J. Phys. A-Math. Theor., 46:23 (2013), 235204  crossref  mathscinet  zmath  adsnasa  isi  scopus
    2. Cicogna G., “Generalized Notions of Symmetry of Odes and Reduction Procedures”, Math. Meth. Appl. Sci., 37:12 (2014), 1819–1827  crossref  mathscinet  zmath  isi  scopus
    3. Muriel C., Romero J.L., “Lambda-Symmetries of Some Chains of Ordinary Differential Equations”, Nonlinear Anal.-Real World Appl., 16 (2014), 191–201  crossref  mathscinet  zmath  isi  scopus
    4. Muriel C., Romero J.L., “The Lambda-Symmetry Reduction Method and Jacobi Last Multipliers”, Commun. Nonlinear Sci. Numer. Simul., 19:4 (2014), 807–820  crossref  mathscinet  adsnasa  isi  scopus
    5. Mohanasubha R., Chandrasekar V.K., Senthilvelan M., Lakshmanan M., “Interplay of Symmetries, Null Forms, Darboux Polynomials, Integrating Factors and Jacobi Multipliers in Integrable Second-Order Differential Equations”, Proc. R. Soc. A-Math. Phys. Eng. Sci., 470:2163 (2014), 20130656  crossref  mathscinet  zmath  isi  scopus
    6. Gaeta G., “Simple and Collective Twisted Symmetries”, J. Nonlinear Math. Phys., 21:4 (2014), 593–627  crossref  mathscinet  isi  scopus
    7. Gaeta G., “Symmetry and Lie-Frobenius Reduction of Differential Equations”, J. Phys. A-Math. Theor., 48:1 (2015), 015202  crossref  mathscinet  zmath  adsnasa  isi  scopus
    8. Senthilvelan M., Chandrasekar V.K., Mohanasubha R., “Symmetries of Nonlinear Ordinary Differential Equations: the Modified Emden Equation as a Case Study”, Pramana-J. Phys., 85:5 (2015), 755–787  crossref  adsnasa  isi  scopus
    9. Carinena J.F., de Lucas J., Ranada M.F., “Jacobi Multipliers, Non-Local Symmetries, and Nonlinear Oscillators”, J. Math. Phys., 56:6 (2015), 063505  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    10. Mohanasubha R., Chandrasekar V.K., Senthilvelan M., Lakshmanan M., “Interplay of symmetries and other integrability quantifiers in finite-dimensional integrable nonlinear dynamical systems”, Proc. R. Soc. A-Math. Phys. Eng. Sci., 472:2190 (2016), 20150847  crossref  mathscinet  zmath  isi  scopus
    11. Preston, S., Non-commuting variations in mathematics and physics: A survey, Interaction of Mechanics and Mathematics, Springer, 2016  crossref  mathscinet  zmath  scopus
    12. R. Mohanasubha, M. Senthilvelan, “On the symmetries of a nonlinear non-polynomial oscillator”, Commun. Nonlinear Sci. Numer. Simul., 43 (2017), 111–117  crossref  isi  elib  scopus
    13. J. Mendoza, C. Muriel, “Exact solutions and Riccati-type first integrals”, J. Nonlinear Math. Phys., 24:1, SI (2017), 75–89  crossref  mathscinet  isi  scopus
    14. D. I. Sinelshchikov, N. A. Kudryashov, “On the Jacobi last multipliers and Lagrangians for a family of Lienard-type equations”, Appl. Math. Comput., 307 (2017), 257–264  crossref  mathscinet  isi  scopus
    15. G. G. Polat, T. Ozer, “New conservation laws, Lagrangian forms, and exact solutions of modified Emden equation”, J. Comput. Nonlinear Dyn., 12:4, SI (2017), 041001  crossref  isi  scopus
    16. R. Mohanasubha, V. K. Chandrasekar, M. Senthilvelan, M. Lakshmanan, “On the interconnections between various analytic approaches in coupled first-order nonlinear differential equations”, Commun. Nonlinear Sci. Numer. Simul., 62 (2018), 213–228  crossref  mathscinet  isi  scopus
    17. Avellar J., Cardoso M.S., Duarte L.G.S., da Mota L. A. C. P., “Dealing With Rational Second Order Ordinary Differential Equations Where Both Darboux and Lie Find It Difficult: the S-Function Method”, Comput. Phys. Commun., 234 (2019), 302–314  crossref  mathscinet  isi  scopus
  • Symmetry, Integrability and Geometry: Methods and Applications
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