×

Merging homoclinic solutions due to state-dependent delay. (English) Zbl 1326.34107

This paper has as a starting point the equation \[ x'(t)=-ax(t-1),\, a\in \left(\frac{\pi}{2}, \frac{5\pi}{2}\right), \] which is known to be unstable and hyperbolic. In a recent previous paper [Adv. Differ. Equ. 19, No. 9–10, 911–946 (2014; Zbl 1300.34162)] the author, for \(a\) in a small interval \((a_1, \frac{5\pi}{2})\), constructed a state dependent delay \(d_Q(\phi)\) on an open set \(Q\) in the space \(C[-2,0])\), so that the equation \[ x'(t)=-ax(t-d_Q(x_t))\enskip\quad (\ast) \] has a homoclinic solution \(h:\mathbb R\to\mathbb R\) with \(h\neq 0\) and \(h(t)\to 0\) as \(| t| \to \infty\). Here, among others, the main result is that the equation \((\ast)\) has a one-parameter family of pairwise different homoclinic solutions \(c^[r]:\mathbb R\to\mathbb R\), \(| r| <r_U\), which merge at some \(t=t_y >0\), so that \(c^[r](t)=c^[0](t)\) for all \(r\in (-r_U,r_U )\) and \(t\geq t_y\).
An extensive and very enlightening introduction is given in the beginning of the paper about the considered problem.

MSC:

34K10 Boundary value problems for functional-differential equations
34K05 General theory of functional-differential equations

Citations:

Zbl 1300.34162
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Diekmann, O.; van Gils, S. A.; Verduyn Lunel, S. M.; Walther, H. O., Delay Equations: Functional-, Complex- and Nonlinear Analysis (1995), Springer: Springer New York · Zbl 0826.34002
[2] Hale, J. K.; Verduyn Lunel, S. M., Introduction to Functional Differential Equations (1993), Springer: Springer New York · Zbl 0787.34002
[3] Hartung, F.; Krisztin, T.; Walther, H. O., Functional differential equations with state-dependent delays: theory and applications, (Canada, A.; Drabek, P.; Fonda, A., Handbook of Differential Equations, Ordinary Differential Equations, vol. 3 (2006), Elsevier Science B.V., North-Holland), 435-545 · Zbl 1107.34002
[4] Krisztin, T.; Walther, H. O.; Wu, J., Shape Smoothness, and Invariant Stratification of an Attracting Set for Delayed Monotone Positive Feedback, Fields Inst. Monogr., vol. 11 (1999), AMS: AMS Providence · Zbl 1004.34002
[5] Lani-Wayda, B.; Walther, H. O., A Shilnikov phenomenon due to variable delay, by means of the fixed point index, J. Dynam. Differential Equations (2015), in press
[6] Mallet-Paret, J., Morse decompositions for differential delay equations, J. Differential Equations, 72, 270-315 (1988) · Zbl 0648.34082
[7] Mallet-Paret, J.; Nussbaum, R. D., Boundary layer phenomena for differential-delay equations with state-dependent time-lags: I, Arch. Ration. Mech. Anal., 120, 99-146 (1992) · Zbl 0763.34056
[8] Mallet-Paret, J.; Nussbaum, R. D., Boundary layer phenomena for differential-delay equations with state-dependent time-lags: II, J. Reine Angew. Math., 477, 129-197 (1996) · Zbl 0854.34072
[9] Mallet-Paret, J.; Nussbaum, R. D.; Paraskevopoulos, P., Periodic solutions for functional differential equations with multiple state-dependent time lags, Topol. Methods Nonlinear Anal., 3, 101-162 (1994) · Zbl 0808.34080
[10] Shilnikov, L. P., The existence of a denumerable set of periodic motions in four-dimensional space in an extended neighbourhood of a saddle-focus, Soviet Math. Dokl., 8, 54-58 (1967) · Zbl 0155.41805
[11] Shilnikov, L. P.; Shilnikov, A., Shilnikov bifurcation, Scholarpedia, 2, 8, 1891 (2007)
[12] Walther, H. O., The solution manifold and \(C^1\)-smoothness of solution operators for differential equations with state dependent delay, J. Differential Equations, 195, 46-65 (2003) · Zbl 1045.34048
[13] Walther, H. O., Smoothness properties of semiflows for differential equations with state dependent delay, (Proceedings of the International Conference on Differential and Functional Differential Equations, vol. 1. Proceedings of the International Conference on Differential and Functional Differential Equations, vol. 1, Moscow, 2002 (2003), Moscow State Aviation Institute (MAI): Moscow State Aviation Institute (MAI) Moscow). (Proceedings of the International Conference on Differential and Functional Differential Equations, vol. 1. Proceedings of the International Conference on Differential and Functional Differential Equations, vol. 1, Moscow, 2002 (2003), Moscow State Aviation Institute (MAI): Moscow State Aviation Institute (MAI) Moscow), J. Math. Sci., 124, 5193-5207 (2004), (in Russian); English version: · Zbl 1069.37015
[14] Walther, H. O., A homoclinic loop generated by variable delay, J. Dynam. Differential Equations (2015), in press · Zbl 1339.34077
[15] Walther, H. O., Complicated histories close to a homoclinic loop generated by variable delay, Adv. Differential Equations, 19, 911-946 (2014) · Zbl 1300.34162
[16] Walther, H. O., Topics in delay differential equations, Jahresber. Dtsch. Math.-Ver., 116, 87-114 (2014) · Zbl 1295.34003
[17] Winston, E., Uniqueness of solutions of state dependent delay differential equations, J. Math. Anal. Appl., 47, 620-625 (1974) · Zbl 0286.34112
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.