Computer algebra methods for equivariant dynamical systems

Grobner bases.- Algorithms for the computation of invariants and equivariants.- Symmetric bifurcation theory.- 4. Orbit space reduction.

[1]  D. Hilbert.  Über die Theorie der algebraischen Formen , 1890 .

[2]  D. Hilbert,et al.  Ueber die vollen Invariantensysteme , 1893 .

[3]  E. Noether,et al.  Der Endlichkeitssatz der Invarianten endlicher Gruppen , 1915 .

[4]  F. S. Macaulay,et al.  The Algebraic Theory of Modular Systems , 1972 .

[5]  F. S. Macaulay.  Some Properties of Enumeration in the Theory of Modular Systems , 1927 .

[6]  Michael Francis Atiyah,et al.  Introduction to commutative algebra , 1969 .

[7]  J. Dieudonne,et al.  Invariant theory, old and new , 1971 .

[8]  Melvin Hochster,et al.  Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay , 1974 .

[9]  E. Bierstone.  Lifting isotopies from orbit spaces , 1975 .

[10]  Richard P. Stanley,et al.  Invariants of finite groups and their applications to combinatorics , 1979 .

[11]  David H. Sattinger,et al.  Group theoretic methods in bifurcation theory , 1979 .

[12]  Ferdinando Mora,et al.  An Algorithm to Compute the Equations of Tangent Cones , 1982, EUROCAM.

[13]  André Vanderbauwhede,et al.  Local bifurcation and symmetry , 1982 .

[14]  A. Meyer,et al.  The complexity of the word problems for commutative semigroups and polynomial ideals , 1982 .

[15]  H. Michael Möller,et al.  The computation of the Hilbert function , 1983, EUROCAL.

[16]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[17]  Louis Michel,et al.  Zeros of covariant vector fields for the point groups : invariant formulation , 1984 .

[18]  B. Buchberger,et al.  Grobner Bases : An Algorithmic Method in Polynomial Ideal Theory , 1985 .

[19]  M. Golubitsky,et al.  Singularities and groups in bifurcation theory , 1985 .

[20]  Dieter Armbruster.  Bifurcation Theory and Computer Algebra: An Initial Approach , 1985, European Conference on Computer Algebra.

[21]  M. F.,et al.  Bibliography , 1985, Experimental Gerontology.

[22]  C. Kelley.  Iterative Methods for Linear and Nonlinear Equations , 1987 .

[23]  Robert G. Cowell,et al.  CATFACT: Computer algebraic tools for applications of catastrophe theory , 1987, EUROCAL.

[24]  Volker Weispfenning,et al.  Admissible orders and linear forms , 1987, SIGS.

[25]  D. Armbruster,et al.  "Perturbation Methods, Bifurcation Theory and Computer Algebra" , 1987 .

[26]  H. Stetter,et al.  An Elimination Algorithm for the Computation of All Zeros of a System of Multivariate Polynomial Equations , 1988 .

[27]  Rüdiger Gebauer,et al.  On an Installation of Buchberger's Algorithm , 1988, J. Symb. Comput..

[28]  Alessandro Logar,et al.  A Computational Proof of the Noether Normalization Lemma , 1988, AAECC.

[29]  Leopoldo García Franquelo,et al.  An Algorithm for Symbolic Computation of Center Manifolds , 1988, ISSAC.

[30]  John Guckenheimer,et al.  Degenerate homoclinic cycles in perturbations of quadratic Hamiltonian systems , 1989 .

[31]  Winfried Neun,et al.  Symbolic solution of large stationary chemical kinetics problems , 1989, IMPACT Comput. Sci. Eng..

[32]  L. Billera,et al.  Gro¨bner basis methods for multivariate splines , 1989 .

[33]  D. Kapur,et al.  A Completion Procedure for Computing a Canonical Basis for a k-Subalgebra , 1989, Computers and Mathematics.

[34]  E. Kreuzer,et al.  NEWEUL — Software for the Generation of Symbolical Equations of Motion , 1990 .

[35]  Martin Krupa,et al.  Bifurcations of relative equilibria , 1990 .

[36]  J. Sanders,et al.  A Survey of Invariant Theory Applied to Normal Forms of Vectorfields with Nilpotent Linear Part , 1990 .

[37]  K. Gatemann.  Symbolic solution polynomial equation systems with symmetry , 1990, ISSAC '90.

[38]  A. I. Khibnik,et al.  LINLBF: A Program for Continuation and Bifurcation Analysis of Equilibria Up to Codimension Three , 1990 .

[39]  Noel G. Lloyd,et al.  REDUCE and the Bifurcation of Limit Cycles , 1990, J. Symb. Comput..

[40]  Carlo Traverso,et al.  “One sugar cube, please” or selection strategies in the Buchberger algorithm , 1991, ISSAC '91.

[41]  Bernd Sturmfels,et al.  Computing combinatorial decompositions of rings , 1991, Comb..

[42]  Karin Gatermann,et al.  Mixed symbolic-numeric solution of symmetrical nonlinear systems , 1991, ISSAC '91.

[43]  Richard Liska,et al.  Algorithms for difference schemes construction on non-orthogonal logically rectangular meshes , 1991, ISSAC '91.

[44]  Victor G. Ganzha,et al.  Stability analysis of difference schemes by the catastrophe theory methods and by means of computer algebra , 1991, ISSAC '91.

[45]  Combined Analytical — Numerical Analysis of Nonlinear Dynamical Systems , 1991 .

[46]  Lauren L. Rose,et al.  A dimension series for multivariate splines , 1991, Discret. Comput. Geom..

[47]  Andreas Hohmann,et al.  Symbolic exploitation of symmetry in numerical pathfollowing , 1991, IMPACT Comput. Sci. Eng..

[48]  G. Sartori.  Geometric invariant theory: a model-independent approach to spontaneous symmetry and/or supersymmetry breaking , 1991 .

[49]  Y. N. Lakshman.  A Single Exponential Bound on the Complexity of Computing Gröbner Bases of Zero Dimensional Ideals , 1991 .

[50]  Enrique Ponce,et al.  Normal Forms for Planar Systems With Nilpotent Linear Part , 1991 .

[51]  W. Kleczka,et al.  Systematic computer-aided analysis of dynamic systems , 1991, ISSAC '91.

[52]  Michael Eugene Stillman,et al.  Computation of Hilbert Functions , 1992, J. Symb. Comput..

[53]  Hopf Bifurcation with Z4 × T2 Symmetry , 1992 .

[54]  Karin Gatermann,et al.  Computation of Bifurcation Graphs , 1992 .

[55]  Daniel Lazard,et al.  Solving Zero-Dimensional Algebraic Systems , 1992, J. Symb. Comput..

[56]  Peter Gritzmann,et al.  Minkowski Addition of Polytopes: Computational Complexity and Applications to Gröbner Basis , 1993, SIAM J. Discret. Math..

[57]  A. Fässler,et al.  Group Theoretical Methods and Their Applications , 1992 .

[58]  Keith O. Geddes,et al.  Algorithms for computer algebra , 1992 .

[59]  Jürgen Menck.  Analyse nichthyperbolischer Gleichgewichtspunkte in dynamischen Systemen unter Ausnutzung von Symmetrien, mit Anwendungen von Computeralgebra , 1992 .

[60]  Philip Holmes,et al.  Heteroclinic cycles and modulated travelling waves in a system with D 4 symmetry , 1992 .

[61]  David A. Cox,et al.  Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics) , 2007 .

[62]  Bernd Sturmfels,et al.  Finding sparse systems of parameters , 1993 .

[63]  S. Collart.  The Grr Obner Walk , 1993 .

[64]  S. Walcher.  On Transformations into Normal Form , 1993 .

[65]  Jean-Charles Faugère,et al.  Efficient Computation of Zero-Dimensional Gröbner Bases by Change of Ordering , 1993, J. Symb. Comput..

[66]  P Chossat,et al.  Forced reflectional symmetry breaking of an O(2)-symmetric homoclinic cycle , 1993 .

[67]  Willy Hereman,et al.  Review of Symbolic Software for the Computation of Lie Symmetries of Differential Equations , 1994 .

[68]  Patrick A. Worfolk,et al.  Zeros of Equivariant Vector Fields: Algorithms for an Invariant Approach , 1994, J. Symb. Comput..

[69]  B. Broer.  Classification of Cohen-Macaulay modules of covariants for systems of binary forms , 1994 .

[70]  Mark Sofroniou,et al.  Symbolic Derivation of Runge-Kutta Methods , 1994, J. Symb. Comput..

[71]  Hamiltonian Structure of the Reversible Nonsemisimple 1:1 Resonance , 1994 .

[72]  P. Ashwin,et al.  Symmetry groups of attractors , 1994 .

[73]  Bernd Sturmfels.  On the number of real roots of a sparse polynomial system , 1994 .

[74]  W. Langford,et al.  Symmetry-breaking bifurcations on multidimensional fixed point subspaces , 1994 .

[75]  Karin Gatermann,et al.  A Remark on the Detection of Symmetry of Attractors , 1994 .

[76]  Richard H. Rand,et al.  Topics in Nonlinear Dynamics with Computer Algebra , 1994 .

[77]  I. Melbourne.  Instantaneous Symmetry and Symmetry on Average in the Couette-Taylor and Faraday Experiments , 1994 .

[78]  Edgar E. Enochs,et al.  On Cohen-Macaulay rings , 1994 .

[79]  N. Sri Namachchivaya,et al.  Normal form for generalized Hopf bifurcation with non-semisimple 1 ∶ 1 resonance , 1994 .

[80]  D. Eisenbud.  Commutative Algebra: with a View Toward Algebraic Geometry , 1995 .

[81]  B. Sturmfels.  Gröbner bases and convex polytopes , 1995 .

[82]  D. Gruntz.  Symbolic Computation of Explicit Runge-Kutta Formulas , 1998 .

[83]  Willi-Hans Steeb,et al.  DIFFERENTIAL EQUATIONS AND COMPUTER ALGEBRA , 1995 .

[84]  J. A. van Hulzen,et al.  The Application of Computer Algebra for the Discretization and Coding of the Navier-Stokes Equations , 1995 .

[85]  P. Worfolk,et al.  An integrable normal form for water waves in infinite depth , 1995 .

[86]  Manfred Göbel,et al.  Computing Bases for Rings of Permutation-Invariant Polynomials , 1995, J. Symb. Comput..

[87]  M. Bergh.  Modules of Covariants , 1995 .

[88]  J. Verschelde,et al.  Symmetrical Newton Polytopes for Solving Sparse Polynomial Systems , 1995 .

[89]  H. M. Möller,et al.  Multivariate polynomial equations with multiple zeros solved by matrix eigenproblems , 1995 .

[90]  K. Gatermann.  Automatic Classiication of Normal Forms , 1995 .

[91]  Pascal Chossat,et al.  The 1:2 resonance withO(2) symmetry and its applications in hydrodynamics , 1995 .

[92]  M. Ziegler.  Volume 152 of Graduate Texts in Mathematics , 1995 .

[93]  John G. Milton,et al.  Limit cycles, tori, and complex dynamics in a second-order differential equation with delayed negative feedback , 1995 .

[94]  K. Gatermann.  Testing for Sn-Symmetry with a Recursive Detective , 1996 .

[95]  Hans J. Stetter.  Analysis of zero clusters in multivariate polynomial systems , 1996, ISSAC '96.

[96]  CARLO TRAVERSO,et al.  Hilbert Functions and the Buchberger Algorithm , 1996, J. Symb. Comput..

[97]  M. Field.  Lectures on bifurcations, dynamics and symmetry , 1996 .

[98]  Massimo Caboara,et al.  Multigraded Hilbert functions and Buchberger algorithm , 1996, ISSAC '96.

[99]  Bodo Werner,et al.  Secondary Hopf Bifurcation Caused by Steady-state Steady-state Mode Interaction , 1996 .

[100]  Ernst W. Mayr,et al.  An optimal algorithm for constructing the reduced Gröbner basis of binomial ideals , 1996, ISSAC '96.

[101]  Noel G. Lloyd,et al.  Algorithmic Derivation of Centre Conditions , 1996, SIAM Rev..

[102]  M. MacCallum.  Computer algebra and differential equations , 1996 .

[103]  Gerhard Pfister,et al.  Advances and improvements in the theory of standard bases and syzygies , 1996 .

[104]  Gregor Kemper,et al.  Calculating Invariant Rings of Finite Groups over Arbitrary Fields , 1996, J. Symb. Comput..

[105]  Ronald Cools,et al.  Mixed-volume computation by dynamic lifting applied to polynomial system solving , 1996, Discret. Comput. Geom..

[106]  Franz Winkler,et al.  Polynomial Algorithms in Computer Algebra , 1996, Texts and Monographs in Symbolic Computation.

[107]  Hans J. Stetter,et al.  Detection and Validation of Clusters of Polynomial Zeros , 1997, J. Symb. Comput..

[108]  T. K. Callahan,et al.  Symmetry-breaking bifurcations on cubic lattices , 1997 .

[109]  Jan A. Sanders,et al.  Bifurcation analysis for spherically symmetric systems using invariant theory , 1997 .

[110]  Muriel Kœnig.  Linearization of vector fields on the orbit space of the action of a compact Lie group , 1997 .

[111]  Antoine Colin,et al.  Solving a system of algebraic equations with symmetries , 1997 .

[112]  Willy Hereman,et al.  Review of symbolic software for lie symmetry analysis , 1997 .

[113]  Victor F. Edneral,et al.  Computer evaluation of cyclicity in planar cubic system , 1997, ISSAC.

[114]  Wolmer V. Vasconcelos,et al.  Computational methods in commutative algebra and algebraic geometry , 1997, Algorithms and computation in mathematics.

[115]  Thomas Sauer.  Polynomial Interpolation of Minimal Degree and Grr Obner Bases , 1998 .

[116]  Gert Vegter,et al.  Resonances in a spring-pendulum: algorithms for equivariant singularity theory , 1998 .

[117]  Frédéric Guyard,et al.  Gröbner Bases, Invariant Theory and Equivariant Dynamics , 2022 .

[118]  Harm Derksen,et al.  Computation of Invariants for Reductive Groups , 1999 .

[119]  Gregor Kemper,et al.  Some Algorithms in Invariant Theory of Finite Groups , 1999 .

[120]  P. Chossat,et al.  Generalized Heteroclinic Cycles in Spherically Invariant Systems and Their Perturbations , 1999 .

[121]  Willy Govaerts,et al.  Numerical detection of symmetry breaking bifurcation points with nonlinear degeneracies , 1999, Math. Comput..

[122]  DEGENERATE BIFURCATIONS WITH Z2⊕Z2-SYMMETRY , 1999 .

[123]  A. Newell.  Detecting the symmetry of attractors , 2002 .

[124]  Effective algebraic geometry and normal forms of reversible mappings , 2002 .

[125]  D. Pasechnik.  On the Nullcone and Hironaka Decompositions of the Invariants of a Torus , 2022 .