January  2013, 33(1): 365-379. doi: 10.3934/dcds.2013.33.365

On Poisson's state-dependent delay

1. 

Mathematisches Institut, Universität Gießen, Arndtstr. 2, D 35392 Gießen, Germany

Received  July 2011 Revised  October 2011 Published  September 2012

In 1806 Poisson published one of the first papers on functional differential equations. Among others he studied an example with a state-dependent delay, which is motivated by a geometric problem. This example is not covered by recent results on initial value problems for differential equations with state-dependent delay. We show that the example generates a semiflow of differentiable solution operators, on a manifold of differentiable functions and away from a singular set. Initial data in the singular set produce multiple solutions.
Citation: Hans-Otto Walther. On Poisson's state-dependent delay. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 365-379. doi: 10.3934/dcds.2013.33.365
References:
[1]

O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H. O. Walther, "Delay Equations: Functional-, Complex- and Nonlinear Analysis," Springer, New York, 1995.

[2]

J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations," Springer, New York, 1993.

[3]

F. Hartung, Linearized stability for a class of neutral functional differential equations with state-dependent delays, Journal of Nonlinear Analysis: Theory, Methods and Applications, 69 (2008), 1629-1643.

[4]

F. Hartung, T. Krisztin, H. O. Walther and J. H. Wu, Functional differential equations with state-dependent delays: Theory and applications, in "Handbook of Differential Equations: Ordinary Differential Equations," Vol. III, 435-545, Elsevier/North-Holland, Amsterdam, 2006.

[5]

M. C. Irwin, "Smooth Dynamical Systems," Academic Press, London, 1980.

[6]

M. C. Mackey, Commodity price fluctuations:price-dependent delays and nonlinearities as explanatory factors, J. Economic Theory, 48 (1989), 497-509.

[7]

M. C. Mackey, personal communication, 2011.

[8]

S. D. Poisson, Sur les équations auxdifférences melées, Journal de l'Ecole Polytechnique, Tome, VI (1806), 126-147.

[9]

H. O. Walther, The solution manifold and $C^1$-smoothness of solution operators for differential equations with statedependent delay, J. Differential Eqs., 195 (2003), 46-65. doi: 10.1016/j.jde.2003.07.001.

[10]

H. O. Walther, Smoothness properties of semiflows for differential equations with state dependent delay, in "Proceedings of the International Conference on Differential and Functional Differential Equations, Moscow, 2002," 1 (2002), 40-55, Moscow State Aviation Institute (MAI), Moscow 2003, English version: Journal of the Mathematical Sciences, 124 (2004), 5193-5207. doi: 10.1023/B:JOTH.0000047253.23098.12.

[11]

H. O. Walther, Algebraic-delay differential systems, state-dependent delay, and temporal orderof reactions, J. Dynamics and Differential Eqs., 21 (2009), 195-232. doi: 10.1007/s10884-009-9129-6.

[12]

H. O. Walther, Semiflows for neutral equations with state-dependent delays,, Fields Inst. Communications, (). 

[13]

H. O. Walther, Linearized stability for semiflows generated by a class of neutral equations, with applications to state-dependent delays, Journal of Dynamics and Differential Equations, 22 (2010), 439-462. doi: 10.1007/s10884-010-9168-z.

[14]

H. O. Walther, Differential equations with locally bounded delay, J. Differential Equations, 252 (2012), 3001-3039.

show all references

References:
[1]

O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H. O. Walther, "Delay Equations: Functional-, Complex- and Nonlinear Analysis," Springer, New York, 1995.

[2]

J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations," Springer, New York, 1993.

[3]

F. Hartung, Linearized stability for a class of neutral functional differential equations with state-dependent delays, Journal of Nonlinear Analysis: Theory, Methods and Applications, 69 (2008), 1629-1643.

[4]

F. Hartung, T. Krisztin, H. O. Walther and J. H. Wu, Functional differential equations with state-dependent delays: Theory and applications, in "Handbook of Differential Equations: Ordinary Differential Equations," Vol. III, 435-545, Elsevier/North-Holland, Amsterdam, 2006.

[5]

M. C. Irwin, "Smooth Dynamical Systems," Academic Press, London, 1980.

[6]

M. C. Mackey, Commodity price fluctuations:price-dependent delays and nonlinearities as explanatory factors, J. Economic Theory, 48 (1989), 497-509.

[7]

M. C. Mackey, personal communication, 2011.

[8]

S. D. Poisson, Sur les équations auxdifférences melées, Journal de l'Ecole Polytechnique, Tome, VI (1806), 126-147.

[9]

H. O. Walther, The solution manifold and $C^1$-smoothness of solution operators for differential equations with statedependent delay, J. Differential Eqs., 195 (2003), 46-65. doi: 10.1016/j.jde.2003.07.001.

[10]

H. O. Walther, Smoothness properties of semiflows for differential equations with state dependent delay, in "Proceedings of the International Conference on Differential and Functional Differential Equations, Moscow, 2002," 1 (2002), 40-55, Moscow State Aviation Institute (MAI), Moscow 2003, English version: Journal of the Mathematical Sciences, 124 (2004), 5193-5207. doi: 10.1023/B:JOTH.0000047253.23098.12.

[11]

H. O. Walther, Algebraic-delay differential systems, state-dependent delay, and temporal orderof reactions, J. Dynamics and Differential Eqs., 21 (2009), 195-232. doi: 10.1007/s10884-009-9129-6.

[12]

H. O. Walther, Semiflows for neutral equations with state-dependent delays,, Fields Inst. Communications, (). 

[13]

H. O. Walther, Linearized stability for semiflows generated by a class of neutral equations, with applications to state-dependent delays, Journal of Dynamics and Differential Equations, 22 (2010), 439-462. doi: 10.1007/s10884-010-9168-z.

[14]

H. O. Walther, Differential equations with locally bounded delay, J. Differential Equations, 252 (2012), 3001-3039.

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