Direct computation of Hopf bifurcation points in differential-algebraic equations

Abstract In this work, we address the bifurcation theory for differential-algebraic equations (DAEs). Our main subject is the direct computation of Hopf bifurcation points, for which we present a novel methodology. In order to achieve this goal, the theory of ordinary differential equations (ODEs) was extended to formulate a concise stability criterion and provide a procedure for the computation of characteristic polynomials of DAEs. Hopf bifurcation points of DAE models of any index and with any number of parameters can now be easily handled. The methodology is tested for the calculation of Hopf bifurcation points of a benchmark model in chemical engineering in a fast and accurate way.

[1]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[2]  Fouad Teymour,et al.  Dynamics and stability of polymerization process flow sheets , 1995 .

[3]  P. Rabier.  The Hopf bifurcation theorem for quasilinear differential-algebraic equations , 1999 .

[4]  Chukwudi Anyakoha,et al.  A review of particle swarm optimization. Part I: background and development , 2007, Natural Computing.

[5]  Achim Kienle,et al.  Nonlinear computation in DIVA — methods and applications , 2000 .

[6]  C. W. Gear,et al.  Simultaneous Numerical Solution of Differential-Algebraic Equations , 1971 .

[7]  Andreas Griewank,et al.  The Calculation of Hopf Points by a Direct Method , 1983 .

[8]  R. Beardmore,et al.  A Hopf bifurcation theorem for singular differential-algebraic equations , 2008, Math. Comput. Simul..

[9]  M. Mrozek,et al.  Finding Eigenvalues of Self-maps with the Kronecker Canonical Form , 2015 .

[10]  K. Ikramov.  Matrix pencils: Theory, applications, and numerical methods , 1993 .

[11]  Achim Kienle,et al.  Nonlinear Analysis of gPROMS Models Using DIVA as via a CAPE ESO interface , 2002 .

[12]  Gustaf Söderlind,et al.  Remarks on the stability of high-index DAEs with respect to parametric perturbations , 1992, Computing.

[13]  S. Campbell.  Linearization of DAEs along trajectories , 1995 .

[14]  Kody M. Powell,et al.  Nonlinear modeling, estimation and predictive control in APMonitor , 2014, Comput. Chem. Eng..

[15]  Argimiro Resende Secchi,et al.  Direct initialisation and solution of high-index DAE systems , 2005 .

[16]  Vu Hoang Linh,et al.  Robust Stability of Differential-Algebraic Equations , 2013 .

[17]  Roswitha März,et al.  Practical Lyapunov stability criteria for differential algebraic equations , 1994 .

[18]  C. D. Perttunen,et al.  Lipschitzian optimization without the Lipschitz constant , 1993 .

[19]  Bernd Simeon,et al.  On the History of Differential-Algebraic Equations , 2017 .

[20]  Evaristo C. Biscaia,et al.  Stability analysis of differential-algebraic equations in AUTO_DAE , 2006 .

[21]  Linda R. Petzold,et al.  Numerical solution of initial-value problems in differential-algebraic equations , 1996, Classics in applied mathematics.

[22]  Wei-Min Liu,et al.  Criterion of Hopf Bifurcations without Using Eigenvalues , 1994 .

[23]  Tatjana Stykel,et al.  On criteria for asymptotic stability of differential-algebraic equations , 2002 .

[24]  Sebastian Reich.  On the local qualitative behavior of differential-algebraic equations , 1995 .