×

Periodic solutions of a singularly perturbed delay differential equation. (English) Zbl 1167.34029

The singularly perturbed delay differential equation
\[ \varepsilon \dot x(t)=-x(t)+f(x(t-1),\lambda), \eqno{(1)} \]
serving as a model for physiological control systems and for the transmission of light through a ring cavity, exhibits slowly oscillating periodic solutions (SOPS) near the first period-doubling bifurcation point. In this article the authors consider the equation
\[ \varepsilon \dot x(t)=-x(t)+\lambda x(t-1)(1-x(t-1)), \eqno{(2)} \]
where the logistic nonlinearity is chosen useful at the study of periodic solutions to (1) in the neighborhood of the fixed point \(x_0, x_0=f(x_0,\lambda)\). Also any generic function \(f(x(t-1)\) can be approximated by a quadratic nonlinearity. Since the logistic map undergoes its first period-doubling bifurcation at \(\lambda=3\), it can be shown that it satisfies the conditions for the existence of SOPS to (2) in the neighborhood of \(\lambda=3\).
The square-wave SOPS of (2) near \(\lambda=3\) exhibit a peculiar symmetry. If one measures the time intervals between three successive crossing of the square-wave solution with the average of the period-\(2\) fixed points of the map, one of them turns out to be \(\sim 1+2\varepsilon,\) while the other is \(O(1)\). Taking into account this observation the authors formulate and solve the corresponding transition layer equations. Employing a two-parameter perturbation expansion in \(\varepsilon\) and \(\sigma=\sqrt{\lambda-3}\) and using an approximate scaling between \(\varepsilon\) and \(\sigma\) the authors obtain analytic expressions for the square-wave SOPS in the neighborhood of the first period-doubling bifurcation point of the map. These expressions are in good agreement with numerical solutions for a range of values of \(\varepsilon\) and \(\lambda\). Such approach is used then to obtain analytic expressions for other periodic solutions of (2) which can be considered as odd harmonics of the SOPS.

MSC:

34K13 Periodic solutions to functional-differential equations
34K18 Bifurcation theory of functional-differential equations
34K26 Singular perturbations of functional-differential equations

Software:

RADAR5
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] (Bessel Functions, Part II. Bessel Functions, Part II, Mathematical Tables, vol. v.X (1952), Cambridge University Press)
[2] Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (1981), National Bureau of Standards: National Bureau of Standards Washington D.C. · Zbl 0515.33001
[3] M.H. Adhikari, Numerical and asymptotic studies of delay-differential equations, Doctoral dissertation, University of New Mexico, Albuquerque, NM, USA, Dec 2007; M.H. Adhikari, Numerical and asymptotic studies of delay-differential equations, Doctoral dissertation, University of New Mexico, Albuquerque, NM, USA, Dec 2007
[4] Bellen, A.; Zennaro, M., Numerical Methods for Delay Differential Equations (2003), Springer-Verlag: Springer-Verlag New York · Zbl 0749.65042
[5] Byrd, P.; Friedman, M., Handbook of Elliptic Integrals for Engineers and Physicists (1954), Springer: Springer Berlin · Zbl 0055.11905
[6] Chow, S. N.; Hale, J. K.; Huang, W., From sine waves to square waves in delay equations, Proc. Roy. Soc. Edinburgh Sect. A-Math., 120, 223-229 (1992) · Zbl 0764.34048
[7] S.N. Chow, J. Mallet-Paret, Singularly pertubed delay differential equations, in: Coupled Nonlinear Oscillators, in: Proceedings of the Joint U.S. Army - Center for Nonlinear Studies Workshop, Los Alamos, New Mexico, 1983, pp. 7-12; S.N. Chow, J. Mallet-Paret, Singularly pertubed delay differential equations, in: Coupled Nonlinear Oscillators, in: Proceedings of the Joint U.S. Army - Center for Nonlinear Studies Workshop, Los Alamos, New Mexico, 1983, pp. 7-12
[8] Coutsias, E. A.; Hagstrom, T.; Hesthaven, J.; Torres, D.; Ilin, Andrew V.; Ridgway Scott, L., Integration preconditioners for differential operators in spectral \(\tau \)-methods, Proceedings of ICOSAHOM 3. Proceedings of ICOSAHOM 3, Houston Journal of Mathematics, 21-38 (1996), (special issue)
[9] Coutsias, E. A.; Hagstrom, T.; Torres, D., An efficient spectral method for ordinary differential equations with rational function coefficients, Math. Comput., 65, 214, 611-635 (1996) · Zbl 0846.65037
[10] Erneux, T.; Larger, L.; Lee, M. W.; Goedgebuer, J. P., Ikeda hopf bifurcation revisited, Physica D, 194, 1-2, 49-64 (2004) · Zbl 1099.34065
[11] Gautschi, W., Computational aspects of three-term recurrence relations, SIAM Rev., 9, 1, 24-82 (1967) · Zbl 0168.15004
[12] Guglielmi, N.; Hairer, E., Implementing radau iia methods for stiff delay differential equations, Computing, 67, 1, 1-12 (2001) · Zbl 0986.65069
[13] Hale, J. K.; Huang, W. Z., Period-doubling in singularly perturbed delay equations, J. Differential Equations, 114, 1, 1-23 (1994) · Zbl 0817.34040
[14] Hale, J. K.; Huang, W. Z., Periodic solutions of singularly perturbed delay equations, Z. Angew. Math. Phys., 47, 1, 57-88 (1996) · Zbl 0841.34080
[15] Ikeda, K., Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system, Opt. Commun., 30, 2, 257-261 (1979)
[16] Ikeda, K., Chaos and optical bistability: Bifurcation structure, (Mandel, L.; Wolf, E., Coherence and Quantum Optics, vol. V (1984)), 875-882
[17] Ikeda, K.; Akimoto, O., Successive bifurcations and dynamical multi-stability in a bistable optical-system — a detailed study of the transition to chaos, Appl. Phys. B, 28, 2-3, 170-171 (1982)
[18] K. Ikeda, O. Akimoto, Optical turbulence, in: Y. Kuramoto (Ed.), Chaos and Statistical Methods: Proceedings of the Sixth Kyoto Summer Institute, Kyoto, Japan, 1983, pp. 249-257; K. Ikeda, O. Akimoto, Optical turbulence, in: Y. Kuramoto (Ed.), Chaos and Statistical Methods: Proceedings of the Sixth Kyoto Summer Institute, Kyoto, Japan, 1983, pp. 249-257
[19] Ikeda, K.; Daido, H.; Akimoto, O., Optical turbulence - chaotic behavior of transmitted light from a ring cavity, Phys. Rev. Lett., 45, 9, 709-712 (1980)
[20] Ikeda, K.; Kondo, K.; Akimoto, O., Successive higher-harmonic bifurcations in systems with delayed feedback, Phys. Rev. Lett., 49, 20, 1467-1470 (1982)
[21] Ikeda, K.; Matsumoto, K., High-dimensional chaotic behavior in systems with time-delayed feedback, Physica D, 29, 1-2, 223-235 (1987) · Zbl 0626.58014
[22] Mackey, M. C.; Glass, L., Oscillation and chaos in physiological control-systems, Science, 197, 4300, 287-288 (1977) · Zbl 1383.92036
[23] Mallet-Paret, J.; Nussbaum, R., A bifurcation gap for a singularly perturbed delay equation, (Barnsley, M. F.; Demko, S. G., Chaotic Dynamics and Fractals (1986)), 263-286
[24] Mallet-Paret, J.; Nussbaum, R. D., Global continuation and asymptotic behavior for periodic solutions of a differential-delay equation, Ann. Mat. Pura Appl., 145, 33-128 (1986) · Zbl 0617.34071
[25] Olver, F., Error analysis of miller’s recurrence algorithm, Math. Comput., 18, 85, 65-74 (1964) · Zbl 0115.34502
[26] M.R. Semak, Asymptotic and numerical studies of a differential-delay system, Doctoral dissertation, University of New Mexico, Albuquerque, NM, USA, Jul 2000; M.R. Semak, Asymptotic and numerical studies of a differential-delay system, Doctoral dissertation, University of New Mexico, Albuquerque, NM, USA, Jul 2000
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.