Switching to Nonhyperbolic Cycles from Codimension Two Bifurcations of Equilibria of Delay Differential Equations

In this paper we perform the parameter-dependent center manifold reduction near the generalized Hopf (Bautin), fold-Hopf, Hopf-Hopf and transcritical-Hopf bifurcations in delay differential equations (DDEs). This allows us to initialize the continuation of codimension one equilibria and cycle bifurcations emanating from these codimension two bifurcation points. The normal form coefficients are derived in the functional analytic perturbation framework for dual semigroups (sun-star calculus) using a normalization technique based on the Fredholm alternative. The obtained expressions give explicit formulas which have been implemented in the freely available numerical software package DDE-BifTool. While our theoretical results are proven to apply more generally, the software implementation and examples focus on DDEs with finitely many discrete delays. Together with the continuation capabilities of DDE-BifTool, this provides a powerful tool to study the dynamics near equilibria of such DDEs. The effectiveness is demonstrated on various models.

[1]  V. Flunkert,et al.  Pydelay - a python tool for solving delay differential equations , 2009 .

[2]  Sebastiaan G. Janssens,et al.  A class of abstract delay differential equations in the light of suns and stars , 2019, 1901.11526.

[3]  Weihua Jiang,et al.  Double Hopf bifurcation in active control system with delayed feedback: application to glue dosing processes for particleboard , 2016 .

[4]  Shangjiang Guo,et al.  Center manifolds theorem for parameterized delay differential equations with applications to zero si , 2011 .

[5]  W. Govaerts,et al.  Switching to nonhyperbolic cycles from codim 2 bifurcations of equilibria in ODEs , 2008 .

[6]  V. Sree Hari Rao,et al.  Complex dynamics of sexually reproductive generalist predator and gestation delay in a food chain model: double Hopf-bifurcation to Chaos , 2017 .

[7]  J. Hindmarsh,et al.  A model of the nerve impulse using two first-order differential equations , 1982, Nature.

[8]  Shangjiang Guo,et al.  Two-parameter bifurcations in a network of two neurons with multiple delays , 2008 .

[9]  Yingxiang Xu,et al.  Computation of double Hopf points for delay differential equations , 2015 .

[10]  M. Lichtner.  Variation of Constants Formula for Hyperbolic Systems , 2009 .

[11]  Odo Diekmann,et al.  Stability and Bifurcation Analysis of Volterra Functional Equations in the Light of Suns and Stars , 2007, SIAM J. Math. Anal..

[12]  H. Keller.  Lectures on Numerical Methods in Bifurcation Problems , 1988 .

[13]  Willy Govaerts,et al.  Numerical Methods for Two-Parameter Local Bifurcation Analysis of Maps , 2007, SIAM J. Sci. Comput..

[14]  Yu. A. Kuznetsov,et al.  On local bifurcations in neural field models with transmission delays , 2012, Journal of mathematical biology.

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

[16]  Giovanni Samaey,et al.  DDE-BIFTOOL Manual - Bifurcation analysis of delay differential equations , 2014, 1406.7144.

[17]  L. Magalhães,et al.  Normal Forms for Retarded Functional Differential Equations with Parameters and Applications to Hopf Bifurcation , 1995 .

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

[19]  Bart E. Oldeman,et al.  Numerical Bifurcation Analysis , 2022 .

[20]  Zhaosheng Feng,et al.  Fold-Hopf bifurcations of the Rose-Hindmarsh Model with Time Delay , 2011, Int. J. Bifurc. Chaos.

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

[22]  Zuolin Shen,et al.  Double Hopf bifurcation of coupled dissipative Stuart-Landau oscillators with delay , 2014, Appl. Math. Comput..

[23]  J. Hindmarsh,et al.  A model of neuronal bursting using three coupled first order differential equations , 1984, Proceedings of the Royal Society of London. Series B. Biological Sciences.

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

[26]  Weihua Jiang,et al.  Hopf-transcritical bifurcation in retarded functional differential equations , 2010 .

[27]  Jian Xu,et al.  Bursting Near Bautin bifurcation in a Neural Network with Delay Coupling , 2009, Int. J. Neural Syst..

[28]  Anca Veronica Ion.  "An example of Bautin-type bifurcation in a delay differential equation", plus errata , 2007 .

[29]  Xiaoqin Wu,et al.  Zero–Hopf bifurcation analysis of a Kaldor–Kalecki model of business cycle with delay , 2012 .

[30]  Willy Govaerts,et al.  Improved Homoclinic Predictor for Bogdanov-Takens Bifurcation , 2014, Int. J. Bifurc. Chaos.

[31]  Benoit Dionne,et al.  Zero-Hopf bifurcation in the Van der Pol oscillator with delayed position and velocity feedback , 2014, 1402.5866.

[32]  Jian Peng,et al.  Bifurcation analysis in active control system with time delay feedback , 2013, Appl. Math. Comput..

[33]  Odo Diekmann,et al.  Perturbation theory for dual semigroups. IV. The intertwining formula and the canonical pairing , 1986 .

[34]  Willy Govaerts,et al.  Numerical Periodic Normalization for Codim 1 Bifurcations of Limit Cycles , 2005, SIAM J. Numer. Anal..

[35]  Pei Yu,et al.  Double Hopf bifurcation in a container crane model with delayed position feedback , 2013, Appl. Math. Comput..

[36]  Odo Diekmann,et al.  Perturbation theory for dual semigroups. I. The sun-reflexive case , 1986 .

[37]  Yu. A. Kuznetsov,et al.  Pitchfork–Hopf bifurcations in 1D neural field models with transmission delays , 2015 .

[38]  Redouane Qesmi,et al.  Double Hopf bifurcation in delay differential equations , 2014 .

[39]  Weihua Jiang,et al.  Double Hopf bifurcation and chaos in Liu system with delayed feedback , 2011 .

[40]  E. A. Spiegel,et al.  Amplitude Equations for Systems with Competing Instabilities , 1983 .

[41]  Willy Govaerts,et al.  Numerical Periodic Normalization for Codim 2 Bifurcations of Limit Cycles: Computational Formulas, Numerical Implementation, and Examples , 2013, SIAM J. Appl. Dyn. Syst..

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

[43]  JUHONG GE,et al.  An Efficient Method for Studying fold-Hopf bifurcation in Delayed Neural Networks , 2011, Int. J. Bifurc. Chaos.

[44]  L. Magalhães,et al.  Normal Forms for Retarded Functional Differential Equations and Applications to Bogdanov-Takens Singularity , 1995 .

[45]  Sebastiaan G. Janssens,et al.  On a normalization technique for codimension two bifurcations of equilibria of delay differential equations , 2010 .

[46]  DILEEP MENON,et al.  AN INTRODUCTION TO FUNCTIONAL ANALYSIS , 2010 .

[47]  Jian Xu,et al.  Fold-Hopf bifurcation Analysis for a Coupled FitzHugh-Nagumo Neural System with Time Delay , 2010, Int. J. Bifurc. Chaos.

[48]  Anthony Francis Ruston.  Fredholm Theory in Banach Spaces , 1986 .

[49]  Jan Sieber,et al.  Local bifurcations in differential equations with state-dependent delay. , 2017, Chaos.

[50]  Odo Diekmann,et al.  Perturbation theory for dual semigroups , 1987 .

[51]  Anca Veronica Ion,et al.  Bautin bifurcation in a delay differential equation modeling leukemia , 2012 .

[52]  R. Nagel,et al.  A Short Course on Operator Semigroups , 2006 .

[53]  Willy Govaerts,et al.  Analysis of bifurcations of limit cycles with Lyapunov exponents and numerical normal forms , 2014 .

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

[55]  Odo Diekmann,et al.  Perturbation theory for dual semigroups II. Time-dependent perturbations in the sun-reflexive case , 1988, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[56]  Kwok-wai Chung,et al.  Effects of time delayed position feedback on a van der Pol–Duffing oscillator , 2003 .

[57]  Yu. A. Kuznetsov,et al.  Numerical Normalization Techniques for All Codim 2 Bifurcations of Equilibria in ODE's , 1999 .

[58]  Tonghua Zhang,et al.  Delay-Induced Double Hopf Bifurcations in a System of Two Delay-Coupled van der Pol-Duffing Oscillators , 2015, Int. J. Bifurc. Chaos.

[59]  Jian Xu,et al.  Bautin bifurcation analysis for synchronous solution of a coupled FHN neural system with delay , 2010 .

[60]  R. Nagel,et al.  One-parameter semigroups for linear evolution equations , 1999 .

[61]  Odo Diekmann,et al.  Perturbation theory for dual semigroups. III. Nonlinear Lipschitz continuous perturbations in the sun-reflexive [case] , 1986 .

[62]  Yong Wang,et al.  Hopf-transcritical bifurcation in toxic phytoplankton–zooplankton model with delay , 2014 .

[63]  O. Diekmann,et al.  Abstract Delay Equations Inspired by Population Dynamics , 2007 .

[64]  Odo Diekmann,et al.  Equations with infinite delay: blending the abstract and the concrete , 2012 .