Time delay in the swing equation: a variety of bifurcations. (English) Zbl 1491.34091

Summary: The present paper addresses the swing equation with additional delayed damping as an example for pendulumlike systems. In this context, it is proved that recurring sub- and supercritical Hopf bifurcations occur if time delay is increased. To this end, a general formula for the first Lyapunov coefficient in second order systems with additional delayed damping and delay-free nonlinearity is given. Insofar, the paper extends the results about the stability switching of equilibria in linear time delay systems from Cooke and Grossman. In addition to the analytical results, periodic solutions are numerically dealt with. The numerical results demonstrate how a variety of qualitative behaviors are generated in the simple swing equation by only introducing time delay in a damping term.
©2019 American Institute of Physics


34K60 Qualitative investigation and simulation of models involving functional-differential equations
34K18 Bifurcation theory of functional-differential equations
34K13 Periodic solutions to functional-differential equations
34K20 Stability theory of functional-differential equations
34K21 Stationary solutions of functional-differential equations
Full Text: DOI arXiv


[1] Khalil, H. K., Nonlinear Systems (2002)
[2] Schäfer, B.; Matthiae, M.; Timme, M.; Witthaut, D., Decentral smart grid control, New J. Phys., 17, 059502 (2015)
[3] Schaeffer, D. G.; Cain, J. W., Ordinary Differential Equations: Basics and Beyond (2016) · Zbl 1364.34003
[4] Andronov, A. A.; Vitt, A. A.; Khaikin, S. E., Theory of Oscillators: Adiwes International Series in Physics (1966) · Zbl 0188.56304
[5] Leonov, G. A.; Burkin, I. M.; Shepeljavyi, A. I., Frequency Methods in Oscillation Theory (1996) · Zbl 0844.34005
[6] Manik, D.; Witthaut, D.; Schäfer, B.; Matthiae, M.; Sorge, A.; Rohden, M.; Katifori, E.; Timme, M., Supply networks: Instabilities without overload, Eur. Phys. J. Spec. Top., 223, 2527-2547 (2014)
[7] Levi, M.; Hoppensteadt, F. C.; Miranker, W. L., Dynamics of the Josephson junction, Quart. Appl. Math., 36, 167-198 (1978)
[8] Diekmann, O.; Verduyn Lunel, S. M.; Gils, S. A.; Walther, H.-O., Delay Equations: Functional-, Complex-, and Nonlinear Analysis (1995) · Zbl 0826.34002
[9] Hale, J. K.; Verduyn Lunel, S. M., Introduction to Functional Differential Equations (1993) · Zbl 0787.34002
[10] Cooke, K. L.; Grossman, Z., Discrete delay, distributed delay and stability switches, J. Math. Anal. Appl., 86, 592-627 (1982) · Zbl 0492.34064
[11] Gilliam, D. S., Shubov, V. I., Byrnes, C. I., and Vugrin, E. D., “Output regulation for delay systems: Tracking and disturbance rejection for an oscillator with delayed damping,” in Proceedings of the International Conference on Control Applications (IEEE, 2002), Vol. 1, pp. 554-558.
[12] Sherman, S., A note on stability calculations and time lag, Quart. Appl. Math., 5, 92-97 (1947) · Zbl 0029.12702
[13] Ansoff, H. I.; Krumhansl, J., A general stability criterion for linear oscillating systems with constant time lag, Quart. Appl. Math., 6, 337-341 (1948) · Zbl 0033.18101
[14] Pinney, E., Ordinary Difference-Differential Equations (1958) · Zbl 0091.07901
[15] Minorsky, N., Self-excited oscillations in dynamical systems possessing retarded action, J. Appl. Mech., 9, 65-71 (1942)
[16] Minorsky, N., Self-excited mechanical oscillations, J. Appl. Phys., 19, 332-338 (1948)
[17] Erneux, T., Applied Delay Differential Equations (2009) · Zbl 1201.34002
[18] Campbell, S. A.; Bélair, J.; Ohira, T.; Milton, J., Limit cycles, tori, and complex dynamics in a second-order differential equation with delayed negative feedback, J. Dyn. Differ. Equ., 7, 213-236 (1995) · Zbl 0816.34048
[19] Mithulananthan, N., Canizares, C. A., and Reeve, J., “Indices to detect Hopf bifurcation in power systems,” in Proceedings of NAPS-2000 (IEEE, 2000), pp. 15-23.
[20] Ji, W.; Venkatasubramanian, V., Dynamics of a minimal power system: Invariant tori and quasi-periodic motions, IEEE Trans. Circuits Syst. I, 42, 981-1000 (1995)
[21] Tan, C.-W.; Varghese, M.; Varaiya, P.; Wu, F. F., Bifurcation, chaos, and voltage collapse in power systems, Proc. IEEE, 83, 1484-1496 (1995)
[22] Bellman, R.; Cooke, K. L., Differential-Difference Equations (1963) · Zbl 0105.06402
[23] Hahn, W., Stability of Motion (1967)
[24] Grainger, J. J.; Stevenson, W. D., Power System Analysis (1994)
[25] Bergen, A. R.; Vittal, V., Power Systems Analysis (2000)
[26] Anderson, P. M.; Fouad, A.-A. A., Power System Control and Stability (1982)
[27] Power System Stability and Control, edited by P. S. Kundur and N. J. Balu (McGraw-Hill, New York, NY, 1994).
[28] Michiels, W.; Niculescu, S.-I., Stability, Control, and Computation for Time-Delay Systems: An Eigenvalue-Based Approach (2014)
[29] Stépán, G., Retarded Dynamical Systems: Stability and Characteristic Functions (1989) · Zbl 0686.34044
[30] Neimark, J., D-subdivisions and spaces of quasi-polynomials, Priklad. Mat. Mekh., 13, 349-380 (1949)
[31] Insperger, T.; Stépán, G., Semi-Discretization for Time-Delay Systems: Stability and Engineering Applications (2011) · Zbl 1245.93004
[32] Kuznetsov, Y. A., Elements of Applied Bifurcation Theory (1998) · Zbl 0914.58025
[33] Hassard, B. D.; Kazarinoff, N. D.; Wan, Y.-H., Theory and Applications of Hopf Bifurcation (1981) · Zbl 0474.34002
[34] Kazarinoff, N. D.; Wan, Y.-H.; van den Driessche, P., Hopf bifurcation and stability of periodic solutions of differential-difference and integro-differential equations, IMA J. Appl. Math., 21, 461-477 (1978) · Zbl 0379.45021
[35] Engelborghs, K.; Luzyanina, T.; Roose, D., Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL, ACM Trans. Math. Softw., 28, 1-21 (2002) · Zbl 1070.65556
[36] Boese, F. G., The stability chart for the linearized Cushing equation with a discrete delay and with gamma-distributed delays, J. Math. Anal. Appl., 140, 510-536 (1989) · Zbl 0677.92015
[37] Abdallah, C., Dorato, P., Benites-Read, J., and Byrne, R., “Delayed positive feedback can stabilize oscillatory systems,” in American Control Conference (IEEE, 1993), pp. 3106-3107.
[38] Menck, P. J.; Heitzig, J.; Marwan, N.; Kurths, J., How basin stability complements the linear-stability paradigm, Nat. Phys., 9, 89-92 (2013)
[39] Leng, S.; Lin, W.; Kurths, J., Basin stability in delayed dynamics, Sci. Rep., 6, 21449 (2016)
[40] Scholl, T. H.; Hagenmeyer, V.; Gröll, L.
[41] Roose, D. and Szalai, R., “Continuation and bifurcation analysis of delay differential equations,” in Numerical Continuation Methods for Dynamical Systems, edited by B. Krauskopf, H. M. Osinga, and J. Galán-Vioque (Springer, Dordrecht, 2007), pp. 359-399. · Zbl 1132.34001
[42] Breda, D.; Maset, S.; Vermiglio, R., Stability of Linear Delay Differential Equations: A Numerical Approach with MATLAB (2015) · Zbl 1315.65059
[43] Bosschaert, M., Wage, B., and Kuznetsov, Y., Description of the Extension ddebiftool_nmfm (2015), see .
[44] Wage, B., Normal form computations for delay differential equations in DDE-BIFTOOL (2014)
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