Summer 2016
Seminar: Delay Equations
Prof. Dr. Bernold Fiedler,
Isabelle Schneider
Schedule, Summer 2016
 Seminar:
 Thursday 10.0012.00, Seminarraum 130, Arnimallee 3
Description
In this seminar, we want to learn about the theory of delay equations and apply the
theory to some examples such as models in control theory, epidemics, or the growth of a
sunflower.
Topics
Introduction to delay equations 1  Linear equations, April 21, 2016
 Existence/uniqueness
 Characteristic equation
 Laplace transform
 Fundamental solution
 Maybe: variation of constants, if time allows
 References: [6], [7]
Introduction to delay equations 2  General theory, April 28, 2016
 Existence/uniqueness
 Continuous dependence
 Continuation of solutions
 Differentiability of solutions
 References: [6], [7]
Introduction to delay equations 3  Small solutions, May 12, 2016
 The solution map
 Small solutions
 If time allows: compactness
 References: [6], [7]
The zeronumber  A Lyapunov function for scalar delaydifferential
equations, May 19, 2016
 A dropping lemma for delay equations
 Theorem A of [17], statement and proof
 If time allows: talk shortly about consequences like Morse decomposition
 References: [17]
The delayed Mathieu equation, May 26, 2016
 Problem formulation
 Hill's infinite matrix
 Calculation of the stability boundaries
 References: [9], [10]
The global attractor of the sunflower equation, June 2, 2016
 Short introduction to the model equation [11]
 Characterization of the global attractor [15]
 References: [11], [15], [16]
Periodic solutions of a delay equation in epidemics, June 9, 2016
 Explain the model equation
 State and prove Theorem 1 [1]
 Discuss shortly the biological consequences
 References: [1], [20]
Stability of a nuclear reactor with delay, June 23, 2016
 Introduce the model
 Characteristic equations
 Stability in the small region, stability in the large only if time permits
 References: [4], [5]
Pyragas control: Delay stabilization of periodic orbits, June 30, 2016
 Explain Pyragas control
 Show that stabilization near Hopf bifurcation is successful
 References: [2], [3], [12], [19]
Pyragas control: The oddnumber limitation, July 7, 2016
 Statements of the two theorems and proofs
 Compare autonomous and nonautonomous case
 References: [8], [18]
Lyapunov matrices for timedelay systems
 LyapunovKrasovskii functional
 Theorem 4 [14]
 Depending on time: The onedelay case or small examples
 References: [13], [14]
References
 [1] Kenneth L Cooke and James L Kaplan. A periodicity threshold theorem for epidemics
and population growth. Mathematical Biosciences 31(1), pp. 87
104 (1976).
 [2] Bernold Fiedler et al. Beyond the odd number limitation of timedelayed feedback
control. Handbook of chaos control. John Wiley & Sons, pp. 7384 (2008).
 [3] Bernold Fiedler et al. Refuting the oddnumber limitation of timedelayed feedback
control. Physical Review Letters 98(11), p. 114101 (2007).
 [4] VD Goryachenko. Stability of a nuclear power generation plant with circulating
fuel. Soviet Atomic Energy 21(1), pp. 613617 (1966).
 [5] VD Goryachenko and Yu F Trunin. Stability of a nuclear reactor as an object with
a time delay. Atomic Energy 23(6), pp. 12651269 (1967).
 [6] Jack K Hale. Theory of functional differential equations. Vol. 3. Springer, 1977.
 [7] Jack K Hale and Sjoerd M Verduyn Lunel. Introduction to functional differential
equations. Vol. 99. Springer, 1993.
 [8] Edward W Hooton and Andreas Amann. Analytical limitation for timedelayed
feedback control in autonomous systems. Physical Review Letters 109(15),
p. 154101 (2012).
 [9] Tamàs Insperger. Stability analysis of periodic delaydifferential equations modeling
machine tool chatter. PhD dissertation. Budapest University of Technology
and Economics, 2002.
 [10] Tamàs Insperger and Gabor Stépán. Stability chart for the delayed Mathieu equation.
Proceedings of the Royal Society of London A: Mathematical, Physical
and Engineering Sciences. Vol. 458, pp. 19891998 (2002).
 [11] D Israelsson and A Johnsson. A theory for circumnutations in Helianthus annuus.
Physiologia Plantarum 20(4), pp. 957976 (1967).
 [12] W Just et al. Beyond the odd number limitation: a bifurcation analysis of timedelayed
feedback control. Physical Review E 76(2), p. 026210 (2007).
 [13] VL Kharitonov and AP Zhabko. LyapunovKrasovskii approach to the robust
stability analysis of timedelay systems. Automatica 39(1), pp. 1520 (2003).
 [14] Vladimir L Kharitonov and E Plischke. Lyapunov matrices for timedelay systems.
Systems & Control Letters 55(9), pp. 697706 (2006).
 [15] Marcos Lizana. Existence and partial characterization of the global attractor for
the sunflower equation. Journal of mathematical analysis and applications 190(1), pp. 111 (1995).
 [16] Marcos Lizana. Global analysis of the sunflower equation with small delay.
Nonlinear Analysis: Theory, Methods & Applications 36(6), pp. 697706 (1999).
 [17] John MalletParet. Morse decompositions for delaydifferential equations.
Journal of Differential Equations 72(2), pp. 270315 (1988).
 [18] Hiroyuki Nakajima. On analytical properties of delayed feedback control of chaos.
In: Physics Letters A 232(3), pp. 207210 (1997).
 [19] Kestutis Pyragas. Continuous control of chaos by selfcontrolling feedback.
Physics Letters A 170(6), pp. 421428 (1992).
 [20] Hal L Smith. On periodic solutions of a delay integral equation modelling epidemics.
Journal of Mathematical Biology 4(1) , pp. 6980 (1977).
