##
**Applied delay differential equations.**
*(English)*
Zbl 1201.34002

Surveys and Tutorials in the Applied Mathematical Sciences 3. New York, NY: Springer (ISBN 978-0-387-74371-4/pbk; 978-0-387-74372-1/ebook). xi, 204 p. (2009).

The book is devoted to different applications of delay differential equations (DDEs) in science and engineering. Special attention is paid to oscillatory instabilities and their asymptotic analysis.

The first chapter gives a short overview of elementary properties of DDEs which distinguish them from ordinary differential equations, introduces elementary DDE models for car following, economics, population dynamics, nonlinear optics, fluid dynamics, and mechanical engineering. Oscillatory behavior arising as a result of time delay is discussed. Two classes of DDEs exhibiting cyclic behavior are introduced: (i) delay requirement equations; and (ii) harmonic oscillations with delay.

The second chapter is devoted to the investigation of the stability of fixed point solutions of DDEs. For a simple DDE model, a transcendental characteristic equation is derived. An example of a Hopf bifurcation characterized by a pair of purely imaginary roots of the characteristic equation and leading to a self-pulsing behavior is presented. Linear stability analysis of fixed point solutions with respect to oscillatory instability is performed in simple DDE models of position control and sampling, payload oscillations, car following, and polarization switching in semiconductor lasers. Phenomena such as the multiplicity of Hopf instability curves, bistability between stationary and oscillating solutions, and metastable oscillations are described and discussed.

The third chapter presents some biological applications of DDEs. In particular, the generation of periodic cycles in population dynamics is exemplified using the so-called Nicholson blowflies equation. The appearance of Hopf bifurcation in this equation is analysed. Periodic crashes in circulating red blood cells are described using the Mackey-Glass equation with time delayed negative feedback. Analytical expressions for the extrema of the oscillations are derived. Following the paper by Milton and Longtin periodic oscillations of the pupil area in response to illumination change are described using a single DDE with piecewise constant feedback. The appearance of periodic breathing frequently observed in chronic heart failure is related to a Hopf bifurcation of a single DDE modeling human respiratory control. A Hopf bifurcation responsible for undamped pulsations of the mRNA and protein concentrations inthe oscillatory expression of the Hes1 protein is studied using a set of two DDEs. Finally, stability of stationary states with respect to a Hopf bifurcation is analyzed for two related problems, human postural control and inverted pendulum control.

In the fourth chapter, hydrodynamic applications of DDEs based on Bernoulli’s equation are discussed. A simple 2D nonlinear map is developed for the mathematical modeling of the clarinet. The thresholds corresponding to build-up and extinction of acoustical oscillations are calculated in this model and a hysteresis between the solutions with zero and nonzero acoustical pressure is demonstrated. A second order DDE is used to describe an oscillatory instability responsible for the appearance of such sleep disorders as snoring. Finally, a model of a liquid level control system based on a set of two DDEs is discussed. It is shown that delayed control can induce Hopf bifurcations characterized by different frequencies. Depending on the parameter values, the Hopf bifurcation lines in the parameter space may cross thus generating double Hopf bifurcation points. Near these points, the dynamical response of the system can be rich and includes jumps between periodic and quasiperiodic oscillations.

The fifth chapter is devoted to applications of DDE models for the description of chemical reactions. In the first paragraph a review of the results obtained by Zimmerman et al. on illuminated thermochemical reactions is given on the basis of a set of two DDEs governing the time evolution of the SO3F concentration and the temperature. Emergence of stable limit-cycle oscillations due to a Hopf bifurcation and the corresponding experimental results are discussed. In the next paragraph a first order equation for the evolution of the iodide concentration in the bistable iodate-arcenous acid reaction is considered. A procedure for the stabilization of an unstable steady state proposed by J.P. Lapante is described. The last paragraph of chapter 5 is devoted to the investigation of the effect of a weak delayed feedback term on the oscillatory solutions. Reduced equations that govern a slow evolution of the oscillation phase are derived and analyzed. Two situations are considered: (i) small amplitude oscillations close to a supercritical Hopf bifurcation; and (ii) a piecewise linear oscillator that exhibits strongly pulsating relaxation oscillations.

Chapter 6 is devoted to the DDE models describing the feedback control of mechanical oscillations. The method of multiple scales is applied for the analysis of a damped harmonic oscillator with control force given by the control term in the form of a delayed first order derivative of the dependent variable. It is demonstrated that for certain parameter values this control force can destabilize the system. A more detailed study of this instability is performed by means of the so-called Minorsky equation which contains an additional nonlinear cubic delay term. It is demonstrated that, the instability of periodic oscillations results in the appearance of a solution quasiperiodic in time. The stability of this solution is studied with the help of a specially constructed two-dimensional map. Finally, a Ginzburg-Landau equation is derived to describe the dynamical behavior near a Hopf bifurcation in the limit of a very large delay. The fourth paragraph of chapter 6 is devoted to the investigation of the Johnson and Moon model – a nonlinear oscillator with delayed first derivative term. It is shown that instead of the usual square root law, this equation demonstrates an unusual \(1/4\) power law near the Hopf bifurcation point. The last two paragraphs of the chapter 6 deal with derivation of a second order DDE describing the chatter instability of a machine tool, the linear stability analysis of this equation, the weakly nonlinear analysis near a subcritical Hopf bifurcation point, and the discussion of experimental observations of the chatter instability.

Delay differential equations representing simplest models of semiconductor lasers with optoelectronic feedback, delayed incoherent, and delayed coherent optical feedback are studied in chapter 7. The first two paragraphs are devoted to the optoelectronic and incoherent feedback. It is shown that the optoelectronic feedback and incoherent feedback models can be described by the same equations in the limit of a small feedback rate. In this limit with the increase of the delay parameter, the stationary solution corresponding to time-independent laser intensity can exhibit Hopf bifurcations leading to the appearance of multiple stable limit cycle solutions. When the feedback rate is larger (moderate feedback), different Hopf bifurcations can interact leading to the appearance of resonant double Hopf bifurcation points. The dynamical behavior of the model equations near such degenerate points is studied analytically by calculating three successive bifurcations of the Hopf branch. The third paragraph is devoted to a study of a system of two delay differential equations proposed by Lang and Kobayashi to describe a single-mode semiconductor laser with coherent optical feedback. Solutions of these equations corresponding to a single mode and two modes of the external cavity as well as to mode beating are discussed. The existence of periodic solutions connecting Hopf bifurcation points of different branches of external cavity mode solutions is demonstrated numerically by using a continuation method. In the 4th paragraph of chapter 7, the imaging techniques based on the resonant dependence of a short cavity laser to optical feedback produced by ballistic photons retrodiffused from a target are discussed. The dependence of the laser relaxation oscillations frequency on the distance from the target is calculated analytically in the limit of low feedback rate. The results of these calculations are compared with the results of numerical simulations and experimental measurements. The last paragraph of this chapter provides a brief insight into the self-pulsing behavior of a system of two delay-differential equations which describes an opto-electronic oscillator incorporating a nonlinear modulator, an optical fiber delay line, and a radio-frequency amplifier.

The last (8th) chapter of the book discusses the synchronization of a pair of delayed coupled oscillators. In the case of weak coupling between the oscillators, their dynamics can be described by a delayed Adler equation for the oscillator phase difference. It is demonstrated that the “snaking” bifurcation diagram of this equation constructed analytically shows a good agreement with the experimental data obtained with a two delay coupled semiconductor laser. For the case when the coupling strength is strong enough, the phenomenon of “oscillator death” is demonstrated using two complex delay coupled differential equations describing the interaction between two weakly nonlinear oscillators near the Hopf instability threshold. Experimental observations of this phenomenon, which manifests itself as a delay induced stabilization of the steady state solution, are discussed.

The first chapter gives a short overview of elementary properties of DDEs which distinguish them from ordinary differential equations, introduces elementary DDE models for car following, economics, population dynamics, nonlinear optics, fluid dynamics, and mechanical engineering. Oscillatory behavior arising as a result of time delay is discussed. Two classes of DDEs exhibiting cyclic behavior are introduced: (i) delay requirement equations; and (ii) harmonic oscillations with delay.

The second chapter is devoted to the investigation of the stability of fixed point solutions of DDEs. For a simple DDE model, a transcendental characteristic equation is derived. An example of a Hopf bifurcation characterized by a pair of purely imaginary roots of the characteristic equation and leading to a self-pulsing behavior is presented. Linear stability analysis of fixed point solutions with respect to oscillatory instability is performed in simple DDE models of position control and sampling, payload oscillations, car following, and polarization switching in semiconductor lasers. Phenomena such as the multiplicity of Hopf instability curves, bistability between stationary and oscillating solutions, and metastable oscillations are described and discussed.

The third chapter presents some biological applications of DDEs. In particular, the generation of periodic cycles in population dynamics is exemplified using the so-called Nicholson blowflies equation. The appearance of Hopf bifurcation in this equation is analysed. Periodic crashes in circulating red blood cells are described using the Mackey-Glass equation with time delayed negative feedback. Analytical expressions for the extrema of the oscillations are derived. Following the paper by Milton and Longtin periodic oscillations of the pupil area in response to illumination change are described using a single DDE with piecewise constant feedback. The appearance of periodic breathing frequently observed in chronic heart failure is related to a Hopf bifurcation of a single DDE modeling human respiratory control. A Hopf bifurcation responsible for undamped pulsations of the mRNA and protein concentrations inthe oscillatory expression of the Hes1 protein is studied using a set of two DDEs. Finally, stability of stationary states with respect to a Hopf bifurcation is analyzed for two related problems, human postural control and inverted pendulum control.

In the fourth chapter, hydrodynamic applications of DDEs based on Bernoulli’s equation are discussed. A simple 2D nonlinear map is developed for the mathematical modeling of the clarinet. The thresholds corresponding to build-up and extinction of acoustical oscillations are calculated in this model and a hysteresis between the solutions with zero and nonzero acoustical pressure is demonstrated. A second order DDE is used to describe an oscillatory instability responsible for the appearance of such sleep disorders as snoring. Finally, a model of a liquid level control system based on a set of two DDEs is discussed. It is shown that delayed control can induce Hopf bifurcations characterized by different frequencies. Depending on the parameter values, the Hopf bifurcation lines in the parameter space may cross thus generating double Hopf bifurcation points. Near these points, the dynamical response of the system can be rich and includes jumps between periodic and quasiperiodic oscillations.

The fifth chapter is devoted to applications of DDE models for the description of chemical reactions. In the first paragraph a review of the results obtained by Zimmerman et al. on illuminated thermochemical reactions is given on the basis of a set of two DDEs governing the time evolution of the SO3F concentration and the temperature. Emergence of stable limit-cycle oscillations due to a Hopf bifurcation and the corresponding experimental results are discussed. In the next paragraph a first order equation for the evolution of the iodide concentration in the bistable iodate-arcenous acid reaction is considered. A procedure for the stabilization of an unstable steady state proposed by J.P. Lapante is described. The last paragraph of chapter 5 is devoted to the investigation of the effect of a weak delayed feedback term on the oscillatory solutions. Reduced equations that govern a slow evolution of the oscillation phase are derived and analyzed. Two situations are considered: (i) small amplitude oscillations close to a supercritical Hopf bifurcation; and (ii) a piecewise linear oscillator that exhibits strongly pulsating relaxation oscillations.

Chapter 6 is devoted to the DDE models describing the feedback control of mechanical oscillations. The method of multiple scales is applied for the analysis of a damped harmonic oscillator with control force given by the control term in the form of a delayed first order derivative of the dependent variable. It is demonstrated that for certain parameter values this control force can destabilize the system. A more detailed study of this instability is performed by means of the so-called Minorsky equation which contains an additional nonlinear cubic delay term. It is demonstrated that, the instability of periodic oscillations results in the appearance of a solution quasiperiodic in time. The stability of this solution is studied with the help of a specially constructed two-dimensional map. Finally, a Ginzburg-Landau equation is derived to describe the dynamical behavior near a Hopf bifurcation in the limit of a very large delay. The fourth paragraph of chapter 6 is devoted to the investigation of the Johnson and Moon model – a nonlinear oscillator with delayed first derivative term. It is shown that instead of the usual square root law, this equation demonstrates an unusual \(1/4\) power law near the Hopf bifurcation point. The last two paragraphs of the chapter 6 deal with derivation of a second order DDE describing the chatter instability of a machine tool, the linear stability analysis of this equation, the weakly nonlinear analysis near a subcritical Hopf bifurcation point, and the discussion of experimental observations of the chatter instability.

Delay differential equations representing simplest models of semiconductor lasers with optoelectronic feedback, delayed incoherent, and delayed coherent optical feedback are studied in chapter 7. The first two paragraphs are devoted to the optoelectronic and incoherent feedback. It is shown that the optoelectronic feedback and incoherent feedback models can be described by the same equations in the limit of a small feedback rate. In this limit with the increase of the delay parameter, the stationary solution corresponding to time-independent laser intensity can exhibit Hopf bifurcations leading to the appearance of multiple stable limit cycle solutions. When the feedback rate is larger (moderate feedback), different Hopf bifurcations can interact leading to the appearance of resonant double Hopf bifurcation points. The dynamical behavior of the model equations near such degenerate points is studied analytically by calculating three successive bifurcations of the Hopf branch. The third paragraph is devoted to a study of a system of two delay differential equations proposed by Lang and Kobayashi to describe a single-mode semiconductor laser with coherent optical feedback. Solutions of these equations corresponding to a single mode and two modes of the external cavity as well as to mode beating are discussed. The existence of periodic solutions connecting Hopf bifurcation points of different branches of external cavity mode solutions is demonstrated numerically by using a continuation method. In the 4th paragraph of chapter 7, the imaging techniques based on the resonant dependence of a short cavity laser to optical feedback produced by ballistic photons retrodiffused from a target are discussed. The dependence of the laser relaxation oscillations frequency on the distance from the target is calculated analytically in the limit of low feedback rate. The results of these calculations are compared with the results of numerical simulations and experimental measurements. The last paragraph of this chapter provides a brief insight into the self-pulsing behavior of a system of two delay-differential equations which describes an opto-electronic oscillator incorporating a nonlinear modulator, an optical fiber delay line, and a radio-frequency amplifier.

The last (8th) chapter of the book discusses the synchronization of a pair of delayed coupled oscillators. In the case of weak coupling between the oscillators, their dynamics can be described by a delayed Adler equation for the oscillator phase difference. It is demonstrated that the “snaking” bifurcation diagram of this equation constructed analytically shows a good agreement with the experimental data obtained with a two delay coupled semiconductor laser. For the case when the coupling strength is strong enough, the phenomenon of “oscillator death” is demonstrated using two complex delay coupled differential equations describing the interaction between two weakly nonlinear oscillators near the Hopf instability threshold. Experimental observations of this phenomenon, which manifests itself as a delay induced stabilization of the steady state solution, are discussed.

Reviewer: A. G. Vladimirov (Berlin)

### MSC:

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

34K60 | Qualitative investigation and simulation of models involving functional-differential equations |

34K35 | Control problems for functional-differential equations |

34C15 | Nonlinear oscillations and coupled oscillators for ordinary differential equations |

65P30 | Numerical bifurcation problems |

34K18 | Bifurcation theory of functional-differential equations |

34K20 | Stability theory of functional-differential equations |

34K13 | Periodic solutions to functional-differential equations |

34K26 | Singular perturbations of functional-differential equations |