American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 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, (1995).   Google Scholar [2] J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations,", Springer, (1993).   Google Scholar [3] F. Hartung, Linearized stability for a class of neutral functional differential equations with state-dependent delays,, Journal of Nonlinear Analysis: Theory, 69 (2008), 1629.   Google Scholar [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, (2006), 435.   Google Scholar [5] M. C. Irwin, "Smooth Dynamical Systems,", Academic Press, (1980).   Google Scholar [6] M. C. Mackey, Commodity price fluctuations:price-dependent delays and nonlinearities as explanatory factors,, J. Economic Theory, 48 (1989), 497.   Google Scholar [7] M. C. Mackey, personal, communication, (2011).   Google Scholar [8] S. D. Poisson, Sur les équations auxdifférences melées,, Journal de l'Ecole Polytechnique, VI (1806), 126.   Google Scholar [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.  doi: 10.1016/j.jde.2003.07.001.  Google Scholar [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, 124 (2004), 5193.  doi: 10.1023/B:JOTH.0000047253.23098.12.  Google Scholar [11] H. O. Walther, Algebraic-delay differential systems, state-dependent delay, and temporal orderof reactions,, J. Dynamics and Differential Eqs., 21 (2009), 195.  doi: 10.1007/s10884-009-9129-6.  Google Scholar [12] H. O. Walther, Semiflows for neutral equations with state-dependent delays,, Fields Inst. Communications, ().   Google Scholar [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.  doi: 10.1007/s10884-010-9168-z.  Google Scholar [14] H. O. Walther, Differential equations with locally bounded delay,, J. Differential Equations, 252 (2012), 3001.   Google Scholar

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References:
 [1] O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H. O. Walther, "Delay Equations: Functional-, Complex- and Nonlinear Analysis,", Springer, (1995).   Google Scholar [2] J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations,", Springer, (1993).   Google Scholar [3] F. Hartung, Linearized stability for a class of neutral functional differential equations with state-dependent delays,, Journal of Nonlinear Analysis: Theory, 69 (2008), 1629.   Google Scholar [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, (2006), 435.   Google Scholar [5] M. C. Irwin, "Smooth Dynamical Systems,", Academic Press, (1980).   Google Scholar [6] M. C. Mackey, Commodity price fluctuations:price-dependent delays and nonlinearities as explanatory factors,, J. Economic Theory, 48 (1989), 497.   Google Scholar [7] M. C. Mackey, personal, communication, (2011).   Google Scholar [8] S. D. Poisson, Sur les équations auxdifférences melées,, Journal de l'Ecole Polytechnique, VI (1806), 126.   Google Scholar [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.  doi: 10.1016/j.jde.2003.07.001.  Google Scholar [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, 124 (2004), 5193.  doi: 10.1023/B:JOTH.0000047253.23098.12.  Google Scholar [11] H. O. Walther, Algebraic-delay differential systems, state-dependent delay, and temporal orderof reactions,, J. Dynamics and Differential Eqs., 21 (2009), 195.  doi: 10.1007/s10884-009-9129-6.  Google Scholar [12] H. O. Walther, Semiflows for neutral equations with state-dependent delays,, Fields Inst. Communications, ().   Google Scholar [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.  doi: 10.1007/s10884-010-9168-z.  Google Scholar [14] H. O. Walther, Differential equations with locally bounded delay,, J. Differential Equations, 252 (2012), 3001.   Google Scholar
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