Resonance Phenomena in a Scalar Delay Differential Equation with Two State-Dependent Delays

We study a scalar delay differential equation (DDE) with two delayed feedback terms that depend linearly on the state. The associated constant-delay DDE, obtained by freezing the state dependence, is linear and without recurrent dynamics. With state-dependent delay terms, on the other hand, the DDE shows very complicated dynamics. To investigate this, we perform a bifurcation analysis of the system and present its bifurcation diagram in the plane of the two feedback strengths. It is organized by Hopf-Hopf bifurcation points that give rise to curves of torus bifurcation and associated two-frequency dynamics in the form of invariant tori and resonance tongues. We numerically determine the type of the Hopf-Hopf bifurcation points by computing the normal form on the center manifold; this requires the expansion of the functional defining the state-dependent DDE in a power series whose terms up to order three contain only constant delays. We implemented this expansion and the computation of the normal form coef...

[1]  Claire M. Postlethwaite,et al.  Stabilization of Long-Period Periodic Orbits Using Time-Delayed Feedback Control , 2008, SIAM J. Appl. Dyn. Syst..

[2]  Claire M. Postlethwaite,et al.  Time-delayed feedback control of unstable periodic orbits near a subcritical Hopf bifurcation , 2010, 1006.3479.

[3]  Dimitri Breda,et al.  Pseudospectral Discretization of Nonlinear Delay Equations: New Prospects for Numerical Bifurcation Analysis , 2016, SIAM J. Appl. Dyn. Syst..

[4]  Hal L. Smith,et al.  An introduction to delay differential equations with applications to the life sciences / Hal Smith , 2010 .

[5]  Philipp Hövel,et al.  Time-delayed feedback in neurosystems , 2008, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[6]  J. Mallet-Paret,et al.  Boundary layer phenomena for differential-delay equations with state-dependent time lags, I. , 1992 .

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

[8]  Bernd Krauskopf,et al.  Unlocking Dynamical Diversity: Optical Feedback Effects on Semiconductor Lasers , 2005 .

[9]  Foss,et al.  Multistability and delayed recurrent loops. , 1996, Physical review letters.

[10]  V Flunkert,et al.  Refuting the odd-number limitation of time-delayed feedback control. , 2006, Physical review letters.

[11]  Y. N. Kyrychko,et al.  distributed-delay coupling Amplitude and phase dynamics in oscillators with , 2013 .

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

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

[14]  M. Waurick,et al.  A differential equation with state-dependent delay from cell population biology , 2014, 1411.3097.

[15]  John Mallet-Paret,et al.  Boundary layer phenomena for differential-delay equations with state-dependent time lags: III , 2003 .

[16]  John Mallet-Paret,et al.  Superstability and rigorous asymptotics in singularly perturbed state-dependent delay-differential equations , 2011 .

[17]  F. Hartung,et al.  On the Exponential Stability of a State-Dependent DelayEquationIstv , 2007 .

[18]  Kestutis Pyragas.  Continuous control of chaos by self-controlling feedback , 1992 .

[19]  Victor G. LeBlanc,et al.  Realizability of the normal form for the triple-zero nilpotency in a class of delayed nonlinear oscillators , 2012, 1205.5045.

[20]  E. Schöll,et al.  Amplitude and phase dynamics in oscillators with distributed-delay coupling , 2012, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[21]  Ferenc Hartung,et al.  Chapter 5 Functional Differential Equations with State-Dependent Delays: Theory and Applications , 2006 .

[22]  A. R. Humphries,et al.  Dynamics of a delay differential equation with multiple state-dependent delays , 2012 .

[23]  Ferenc Hartung,et al.  Nonlinear Variation of Constants Formula for Differential Equations with State-Dependent Delays , 2016 .

[24]  A. R. Humphries,et al.  Periodic Solutions of a Singularly Perturbed Delay Differential Equation with Two State-Dependent Delays , 2014, 1411.6060.

[25]  K. Pyragas,et al.  Adaptive modification of the delayed feedback control algorithm with a continuously varying time delay , 2011 .

[26]  Ferenc Hartung,et al.  Exponential stability of a state-dependent delay system , 2007 .

[27]  T. Erneux,et al.  Singular Hopf bifurcation in a differential equation with large state-dependent delay , 2014, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[28]  A. R. Humphries,et al.  A Mathematical Model of Granulopoiesis Incorporating the Negative Feedback Dynamics and Kinetics of G-CSF/Neutrophil Binding and Internalization , 2015, Bulletin of mathematical biology.

[29]  Yuan Yuan,et al.  Threshold dynamics in an SEIRS model with latency and temporary immunity , 2014, Journal of mathematical biology.

[30]  Redouane Qesmi,et al.  Center-stable manifolds for differential equations with state-dependent delays , 2008 .

[31]  Jianhong Wu,et al.  Bifurcation Theory of Functional Differential Equations , 2013 .

[32]  Hans Kaper,et al.  Mathematics and Climate , 2013 .

[33]  G. Stépán,et al.  State-dependent delay in regenerative turning processes , 2006 .

[34]  Gregg M. Gallatin,et al.  Nonlinear laser dynamics : from quantum dots to cryptography , 2011 .

[35]  H. I. Freedman,et al.  Analysis of a model representing stage-structured population growth with state-dependent time delay , 1992 .

[36]  Nicola Guglielmi,et al.  Recent trends in the numerical solution of retarded functional differential equations* , 2009, Acta Numerica.

[37]  Bernd Krauskopf.  The Bifurcation Set for the 1: 4 Resonance Problem , 1994, Exp. Math..

[38]  Hans-Otto Walther.  Complicated histories close to a homoclinic loop generated by variable delay , 2014 .

[39]  Qingwen Hu,et al.  Global Hopf bifurcation for differential equations with state-dependent delay , 2010 .

[40]  Bernd Krauskopf,et al.  A Global Bifurcation Analysis of the Subcritical Hopf Normal Form Subject to Pyragas Time-Delayed Feedback Control , 2014, SIAM J. Appl. Dyn. Syst..

[41]  Rafael de la Llave,et al.  Construction of quasi-periodic solutions of state-dependent delay differential equations by the parameterization method II: Analytic case , 2016 .

[42]  Silviu-Iulian Niculescu,et al.  Stability of Traffic Flow Behavior with Distributed Delays Modeling the Memory Effects of the Drivers , 2007, SIAM J. Appl. Math..

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

[44]  Tibor Krisztin,et al.  A local unstable manifold for differential equations with state-dependent delay , 2003 .

[45]  Eugen Stumpf,et al.  On a Differential Equation with State-Dependent Delay , 2010 .

[46]  Bernd Krauskopf,et al.  Bifurcation sequences at 1:4 resonance: an inventory , 1994 .

[47]  Kirk Green,et al.  Computing unstable manifolds of periodic orbits in delay differential equations , 2003 .

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

[49]  Markus Eichmann,et al.  A local Hopf Bifurcation Theorem for difierential equations with state - dependent delays , 2006 .

[50]  H. Walther,et al.  Smoothness Properties of Semiflows for Differential Equations with State-Dependent Delays , 2004 .

[51]  Sue Ann Campbell,et al.  A closed NPZ model with delayed nutrient recycling , 2014, Journal of mathematical biology.

[52]  Bernd Krauskopf,et al.  Tracking oscillations in the presence of delay-induced essential instability , 2008 .

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

[54]  Huaiping Zhu,et al.  The Bifurcation Study of 1:2 Resonance in a Delayed System of Two Coupled Neurons , 2013 .

[55]  Bernd Krauskopf,et al.  Bifurcation analysis of delay-induced resonances of the El-Niño Southern Oscillation , 2011, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

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

[58]  G. Stépán.  Retarded dynamical systems : stability and characteristic functions , 1989 .

[59]  V Flunkert,et al.  Beyond the odd number limitation: a bifurcation analysis of time-delayed feedback control. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[60]  L. Glass,et al.  Oscillation and chaos in physiological control systems. , 1977, Science.

[61]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[62]  A. Bellen,et al.  Numerical methods for delay differential equations , 2003 .

[63]  Kirk Green,et al.  Bistability and torus break-up in a semiconductor laser with phase-conjugate feedback , 2002 .

[64]  S. Lunel,et al.  Delay Equations. Functional-, Complex-, and Nonlinear Analysis , 1995 .

[65]  Sue Ann Campbell,et al.  Approximating the stability region of a neural network with a general distribution of delays , 2010, Neural Networks.

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

[67]  Y. Kuznetsov.  Elements of Applied Bifurcation Theory , 2023, Applied Mathematical Sciences.

[68]  Toru Ohira,et al.  Balancing with positive feedback: the case for discontinuous control , 2009, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

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

[70]  D. Aronson,et al.  Bifurcations from an invariant circle for two-parameter families of maps of the plane: A computer-assisted study , 1982 .

[71]  Richard Bellman,et al.  Differential-Difference Equations , 1967 .

[72]  Gábor Stépán,et al.  Acceleration feedback improves balancing against reflex delay , 2013, Journal of The Royal Society Interface.

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

[74]  John Mallet-Paret,et al.  Periodic solutions for functional differential equations with multiple state-depend time lags , 1994 .

[75]  Odo Diekmann,et al.  UvA-DARE ( Digital Academic Repository ) Daphnia revisited : local stability and bifurcation theory for physiologically structured population models explained by way of an example , 2010 .

[76]  Judy M Rorison.  Fundamental Issues of Nonlinear Laser Dynamics , 2000 .

[77]  John Mallet-Paret,et al.  Boundary layer phenomena for differential-delay equations with state dependent time lags: II. , 1996 .

[78]  Rafael de la Llave,et al.  Construction of Quasi-periodic Solutions of State-Dependent Delay Differential Equations by the Parameterization Method I: Finitely Differentiable, Hyperbolic Case , 2017 .

[79]  Frederic Guichard,et al.  Synchronization in ecological systems by weak dispersal coupling with time delay , 2012, Theoretical Ecology.

[80]  Antonio Politi,et al.  Electromagnetic two-body problem: recurrent dynamics in the presence of state-dependent delay , 2010 .

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

[82]  Sue Ann Campbell,et al.  Stability and Bifurcations of Equilibria in a Multiple-Delayed Differential Equation , 1994, SIAM J. Appl. Math..