Foundations of Computational Mathematics

, Volume 13, Issue 4, pp 479–516 | Cite as

Scaling Invariants and Symmetry Reduction of Dynamical Systems

  • Evelyne Hubert
  • George Labahn


Scalings form a class of group actions that have theoretical and practical importance. A scaling is accurately described by a matrix of integers. Tools from linear algebra over the integers are exploited to compute their invariants, rational sections (a.k.a. global cross-sections), and offer an algorithmic scheme for the symmetry reduction of dynamical systems. A special case of the symmetry reduction algorithm applies to reduce the number of parameters in physical, chemical or biological models.


Group actions Rational invariants Matrix normal form Model reduction Dimensional analysis Symmetry reduction Equivariant moving frame 

Mathematics Subject Classification

08-04 12-04 14L30 15-04 


  1. 1.
    I.M. Anderson, M.E. Fels, Exterior differential systems with symmetry, Acta Appl. Math. 87(1–3), 3–31 (2005). MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    R.L. Anderson, V.A. Baikov, R.K. Gazizov, W. Hereman, N.H. Ibragimov, F.M. Mahomed, S.V. Meleshko, M.C. Nucci, P.J. Olver, M.B. Sheftel, A.V. Turbiner, E.M. Vorob’ev, CRC Handbook of Lie Group Analysis of Differential Equations, vol. 3 (CRC Press, Boca Raton, 1996). Google Scholar
  3. 3.
    B. Beckermann, G. Labahn, G. Villard, Normal forms for general polynomial matrices, J. Symb. Comput. 41(6), 708–737 (2006). MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    G. Birkhoff, Hydrodynamics: A Study in Logic, Fact and Similitude (Princeton Univ. Press, Princeton, 1960). zbMATHGoogle Scholar
  5. 5.
    P. Bridgman, Dimensional Analysis (Yale Univ. Press, Yale, 1931). Google Scholar
  6. 6.
    P. Chossat, R. Lauterbach, Methods in Equivariant Bifurcations and Dynamical Systems. Advanced Series in Nonlinear Dynamics, vol. 15 (World Scientific, River Edge, 2000). zbMATHCrossRefGoogle Scholar
  7. 7.
    H. Cohen, A Course in Computational Algebraic Number Theory (Springer, Berlin, 1993). zbMATHCrossRefGoogle Scholar
  8. 8.
    M. Fels, P. Olver, Moving coframes. II. Regularization and theoretical foundations, Acta Appl. Math. 55(2), 127–208 (1999). MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    M.E. Fels, Integrating scalar ordinary differential equations with symmetry revisited, Found. Comput. Math. 7(4), 417–454 (2007). MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    K. Gatermann, Computer Algebra Methods for Equivariant Dynamical Systems. Lecture Notes in Mathematics, vol. 1728 (Springer, Berlin, 2000). zbMATHCrossRefGoogle Scholar
  11. 11.
    G. Havas, B. Majewski, K. Matthews, Extended gcd and Hermite normal form algorithms via lattice basis reduction, Exp. Math. 7(2), 125–136 (1998). MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    E. Hubert, Generation properties of Maurer–Cartan invariants, Preprint (2007).
  13. 13.
    E. Hubert, I. Kogan, Rational invariants of a group action. Construction and rewriting, J. Symb. Comput. 42(1–2), 203–217 (2007). MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    E. Hubert, I. Kogan, Smooth and algebraic invariants of a group action. Local and global constructions, Found. Comput. Math. 7(4), 455–493 (2007). MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    E. Hubert, G. Labahn, Rational invariants of scalings from Hermite normal forms, in Proceedings of International Symposium of Symbolic and Algebraic Computation, ISSAC’12 (ACM Press, New York, 2012), pp. 219–226. CrossRefGoogle Scholar
  16. 16.
    H.E. Huntley, Dimensional Analysis (Dover, New York, 1967). Google Scholar
  17. 17.
    Y. Ishida, Formula processing on physical systems, Complex Syst. 11(2), 141–160 (1997). MathSciNetzbMATHGoogle Scholar
  18. 18.
    G. Kemper, The computation of invariant fields and a new proof of a theorem by Rosenlicht, Transform. Groups 12, 657–670 (2007). MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    R. Khanin, Dimensional analysis in computer algebra, in Proceedings of International Symposium of Symbolic and Algebraic Computation, ISSAC’01 (ACM Press, New York, 2001), pp. 201–208. Google Scholar
  20. 20.
    F. Lemaire, A. Ürgüplü, A method for semi-rectifying algebraic and differential systems using scaling type Lie point symmetries with linear algebra, in Proceedings of International Symposium of Symbolic and Algebraic Computation, ISSAC’10 (ACM Press, New York, 2010), pp. 85–92. Google Scholar
  21. 21.
    C. Lin, L. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences. Society for Industrial and Applied Mathematics (1988). zbMATHCrossRefGoogle Scholar
  22. 22.
    E. Mansfield, A Practical Guide to the Invariant Calculus (Cambridge University Press, Cambridge, 2010). zbMATHCrossRefGoogle Scholar
  23. 23.
    J. Müller-Quade, T. Beth, Calculating generators for invariant fields of linear algebraic groups, in Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. LNCS, vol. 1719 (Springer, Berlin, 1999). Google Scholar
  24. 24.
    J.D. Murray, Mathematical Biology. Interdisciplinary Applied Mathematics, vol. 17 (Springer, Berlin, 2002). zbMATHGoogle Scholar
  25. 25.
    P.J. Olver, Applications of Lie Groups to Differential Equations. Graduate Texts in Mathematics, vol. 107 (Springer, New York, 1986). zbMATHCrossRefGoogle Scholar
  26. 26.
    V.L. Popov, E.B. Vinberg, Invariant Theory, in Algebraic Geometry. IV. Encyclopedia of Mathematical Sciences (Springer, Berlin, 1994). Google Scholar
  27. 27.
    A. Schrijver, Theory of Linear and Integer Programming (Wiley, New York, 1986). zbMATHGoogle Scholar
  28. 28.
    A. Sedoglavic, Reduction of algebraic parametric systems by rectification of their affine expanded Lie symmetries, in Algebraic Biology. LNCS, vol. 4545 (Springer, Berlin, 2007), pp. 277–291. CrossRefGoogle Scholar
  29. 29.
    A. Storjohann, Algorithms for matrix canonical forms. PhD thesis, Department of Computer Science, Swiss Federal Institute of Technology—ETH (2000). Google Scholar
  30. 30.
    A. Storjohann, G. Labahn, Asymptotically fast computation of Hermite Normal Forms of integer matrices, in Proceedings of International Symposium of Symbolic and Algebraic Computation, ISSAC’96 (ACM Press, New York, 1996), pp. 259–266. Google Scholar
  31. 31.
    B. Sturmfels, Gröbner Bases and Convex Polytopes (Am. Math. Soc., Providence, 1996). zbMATHGoogle Scholar

Copyright information

© SFoCM 2013

Authors and Affiliations

  1. 1.INRIA MéditerranéeSophia AntipolisFrance
  2. 2.Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada

Personalised recommendations