Parametric Qualitative Analysis of Ordinary Differential Equations: Computer Algebra Methods for Excluding Oscillations (Extended Abstract) (Invited Talk)

Investigating oscillations for parametric ordinary differential equations (ODEs) has many applications in science and engineering but is a very hard problem. Already for two dimensional polynomial systems this question is related to Hilbert’s 16th problem, which is still unsolved [1].

[1]  Benjamin Weitz,et al.  An inverse problem in reaction kinetics , 2011 .

[2]  Thomas Sturm,et al.  New Domains for Applied Quantifier Elimination , 2006, CASC.

[3]  James H. Davenport,et al.  Real Quantifier Elimination is Doubly Exponential , 1988, J. Symb. Comput..

[4]  Adam W. Strzebonski,et al.  Solving Systems of Strict Polynomial Inequalities , 2000, J. Symb. Comput..

[5]  Andreas Weber,et al.  Symbolic Equilibrium Point Analysis in Parameterized Polynomial Vector Fields , 2002 .

[6]  Thomas Sturm,et al.  Algorithmic Global Criteria for Excluding Oscillations , 2011, Bulletin of mathematical biology.

[7]  V. Weispfenning.  A New Approach to Quantifier Elimination for Real Algebra , 1998 .

[8]  B. F. Caviness,et al.  Quantifier Elimination and Cylindrical Algebraic Decomposition , 2004, Texts and Monographs in Symbolic Computation.

[9]  Volker Weispfenning,et al.  The Complexity of Linear Problems in Fields , 1988, Journal of symbolic computation.

[10]  Ricardo Riaza,et al.  Differential-Algebraic Systems: Analytical Aspects and Circuit Applications , 2008 .

[11]  Thomas Sturm,et al.  Investigating Algebraic and Logical Algorithms to Solve Hopf Bifurcation Problems in Algebraic Biology , 2009, Math. Comput. Sci..

[12]  A. Goriely.  Integrability and Nonintegrability of Dynamical Systems , 2001 .

[13]  Zhikun She,et al.  A semi-algebraic approach for asymptotic stability analysis ☆ , 2009 .

[14]  Michal Fečkan,et al.  A generalization of Bendixson’s criterion , 2001 .

[15]  Markus Eiswirth,et al.  Toric ideals and graph theory to analyze Hopf bifurcations in mass action systems , 2005, J. Symb. Comput..

[16]  Thomas Sturm,et al.  Investigating Generic Methods to Solve Hopf Bifurcation Problems in Algebraic Biology , 2008, AB.

[17]  Werner M. Seiler,et al.  Involution - The Formal Theory of Differential Equations and its Applications in Computer Algebra , 2009, Algorithms and computation in mathematics.

[18]  I. Bendixson.  Sur les courbes définies par des équations différentielles , 1901 .

[19]  François Lemaire,et al.  Applying a Rigorous Quasi-Steady State Approximation Method for Proving the Absence of Oscillations in Models of Genetic Circuits , 2008, AB.

[20]  J. Tóth,et al.  Bendixon‐Type Theorems with Applications , 1987 .

[21]  Thomas Sturm,et al.  Weak Integer Quantifier Elimination Beyond the Linear Case , 2007, CASC.

[22]  Wei Niu,et al.  Algebraic Approaches to Stability Analysis of Biological Systems , 2008, Math. Comput. Sci..

[23]  Thomas Sturm,et al.  REDLOG: computer algebra meets computer logic , 1997, SIGS.

[24]  M'hammed El Kahoui,et al.  Deciding Hopf Bifurcations by Quantifier Elimination in a Software-component Architecture , 2000, J. Symb. Comput..

[25]  Thomas Sturm,et al.  Weak quantifier elimination for the full linear theory of the integers , 2007, Applicable Algebra in Engineering, Communication and Computing.

[26]  Hoon Hong,et al.  Testing Stability by Quantifier Elimination , 1997, J. Symb. Comput..

[27]  Volker Weispfenning,et al.  Quantifier Elimination for Real Algebra — the Quadratic Case and Beyond , 1997, Applicable Algebra in Engineering, Communication and Computing.

[28]  J. Craggs.  Applied Mathematical Sciences , 1973 .

[29]  Adam W. Strzebonski,et al.  Cylindrical Algebraic Decomposition using validated numerics , 2006, J. Symb. Comput..

[30]  James S. Muldowney,et al.  Compound matrices and ordinary differential equations , 1990 .

[31]  François Lemaire,et al.  On Proving the Absence of Oscillations in Models of Genetic Circuits , 2007, AB.

[32]  Y. Il'yashenko.  Centennial History of Hilbert’s 16th Problem , 2002 .

[33]  W. Rheinboldt,et al.  Theoretical and numerical analysis of differential-algebraic equations , 2002 .

[34]  Thomas Sturm,et al.  Simplification of Quantifier-Free Formulae over Ordered Fields , 1997, J. Symb. Comput..

[35]  Volker Weispfenning,et al.  Simulation and Optimization by Quantifier Elimination , 1997, J. Symb. Comput..

[36]  Henry C Tuckwell,et al.  On the behavior of solutions in viral dynamical models. , 2004, Bio Systems.

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

[38]  M A Nowak,et al.  Human immunodeficiency virus drug therapy and virus load , 1997, Journal of virology.

[39]  Andreas Dolzmann,et al.  Generic Hermitian Quantifier Elimination , 2004, AISC.

[40]  Christopher W. Brown.  QEPCAD B: a system for computing with semi-algebraic sets via cylindrical algebraic decomposition , 2004, SIGS.