DDE-BIFTOOL v. 3.1.1 Manual — Bifurcation analysis of delay differential equations

5 Delay differential equations 6 5.1 Equations with constant delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 5.1.1 Steady states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 5.1.2 Periodic orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 5.1.3 Connecting orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 5.2 Equations with state-dependent delays . . . . . . . . . . . . . . . . . . . . . . . . . . 7

[1]  R. Bellman,et al.  Differential-Difference Equations , 1967 .

[2]  R. Seydel Practical bifurcation and stability analysis : from equilibrium to chaos , 1994 .

[3]  L. Shampine Solving Delay Differential Equations with dde 23 , 2000 .

[4]  Willy Govaerts,et al.  MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs , 2003, TOMS.

[5]  D. Roose,et al.  Continuation and Bifurcation Analysis of Delay Differential Equations , 2007 .

[6]  John Mallet-Paret,et al.  Stability of periodic solutions of state-dependent delay-differential equations , 2011 .

[7]  Dirk Roose,et al.  Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL , 2002, TOMS.

[8]  Frank Schilder,et al.  Recipes for Continuation , 2013, Computational science and engineering.

[9]  Dirk Roose,et al.  Numerical computation of stability and detection of Hopf bifurcations of steady state solutions of delay differential equations , 1999, Adv. Comput. Math..

[10]  Sue Ann Campbell,et al.  Stability, Bifurcation, and Multistability in a System of Two Coupled Neurons with Multiple Time Delays , 2000, SIAM J. Appl. Math..

[11]  R. Vermiglio,et al.  TRACE-DDE: a Tool for Robust Analysis and Characteristic Equations for Delay Differential Equations , 2009 .

[12]  John L. Casti,et al.  Introduction to the theory and application of differential equations with deviating arguments , 1973 .

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

[14]  J. Hale,et al.  Methods of Bifurcation Theory , 1996 .

[15]  Pak Chiu Leung,et al.  Numerical bifurcation analysis of delay differential equations , 1982 .

[16]  Koen Engelborghs,et al.  Stability of piecewise polynomial collocation for computing periodic solutions of delay differential equations , 2002, Numerische Mathematik.

[17]  J. Hale Theory of Functional Differential Equations , 1977 .

[18]  John Argyris,et al.  An Exploration of Chaos: An Introduction for Natural Scientists and Engineers , 1994 .

[19]  Giovanni Samaey,et al.  Numerical Computation of Connecting Orbits in Delay Differential Equations , 2002, Numerical Algorithms.

[20]  V. Kolmanovskii,et al.  Applied Theory of Functional Differential Equations , 1992 .

[21]  Jan Sieber,et al.  Finding periodic orbits in state-dependent delay differential equations as roots of algebraic equations , 2010, 1010.2391.

[22]  Christopher A. H. Paul,et al.  A User-Guide to Archi - An Explicit Runge-Kutta Code for Solving Delay and Neutral Differential Equa , 1995 .

[23]  J. A. Kuznecov Elements of applied bifurcation theory , 1998 .

[24]  Jianhong Wu,et al.  Introduction to Functional Differential Equations , 2013 .

[25]  Ernst Hairer,et al.  Stiff delay equations , 2007, Scholarpedia.

[26]  D. Sarafyan,et al.  DKLAG6: a code based on continuously imbedded sixth-order Runge-Kutta methods for the solution of state-dependent functional differential equations , 1997 .

[27]  K. ENGELBORGHS,et al.  On Stability of LMS Methods and Characteristic Roots of Delay Differential Equations , 2002, SIAM J. Numer. Anal..

[28]  R. D. Driver,et al.  Ordinary and Delay Differential Equations , 1977 .

[29]  Dirk Roose,et al.  Collocation Methods for the Computation of Periodic Solutions of Delay Differential Equations , 2000, SIAM J. Sci. Comput..

[30]  B. I. Wage,et al.  Normal form computations for Delay Differential Equations in DDE-BIFTOOL , 2014 .

[31]  Eusebius J. Doedel,et al.  Lecture Notes on Numerical Analysis of Nonlinear Equations , 2007 .

[32]  Gábor Stépán,et al.  Continuation of Bifurcations in Periodic Delay-Differential Equations Using Characteristic Matrices , 2006, SIAM J. Sci. Comput..

[33]  Willy Govaerts,et al.  Numerical methods for bifurcations of dynamical equilibria , 1987 .

[34]  Dirk Roose,et al.  Efficient computation of characteristic roots of delay differential equations using LMS methods , 2008 .

[35]  R. Eddie Wilson,et al.  Collocation schemes for periodic solutions of neutral delay differential equations , 2006 .

[36]  Dirk Roose,et al.  Numerical bifurcation Analysis of differential equations with State-Dependent Delay , 2001, Int. J. Bifurc. Chaos.

[37]  J. Kato Stability in functional differential equations , 1980 .

[38]  Wayne H. Enright,et al.  A delay differential equation solver based on a continuous Runge–Kutta method with defect control , 1997, Numerical Algorithms.

[39]  Dirk Roose,et al.  Numerical stability analysis and computation of Hopf bifurcation points for delay differential equations , 1996 .

[40]  Tian Hong-jiong,et al.  The numerical stability of linear multistep methods for delay differential equations with many delays , 1996 .