×

Stability calculations for piecewise-smooth delay equations. (English) Zbl 1170.34350

Summary: This paper describes a new method for computing the stability of nonsmooth periodic orbits of piecewise-smooth dynamical systems with delay. Stability computations for piecewise-smooth dynamical systems without delay have previously been performed using discontinuity mappings to \`\` correct” the linearized period map. However, this approach is less convenient for systems with delays due to the infinite dimensional nature of the problem. Additional problems arise due to the discontinuity propagation properties of delay differential equations. The method proposed is based around a multi-point boundary value solver, which allows the correct linearized period map to be constructed directly. We present numerical examples showing the rapid convergence of the method and also illustrate its use as part of a numerical bifurcation study.

MSC:

34K20 Stability theory of functional-differential equations
34K05 General theory of functional-differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1007/BF01054042 · Zbl 0725.34073 · doi:10.1007/BF01054042
[2] an der Heiden U., Zeitschrift für Angewandte Mathematik und Mechanik 70 pp T621–
[3] DOI: 10.1006/jsvi.1998.1843 · Zbl 1235.70142 · doi:10.1006/jsvi.1998.1843
[4] DOI: 10.1080/14689360500539363 · Zbl 1121.34071 · doi:10.1080/14689360500539363
[5] DOI: 10.1023/A:1022670017537 · Zbl 0911.34064 · doi:10.1023/A:1022670017537
[6] Boyd J., Chebyshev and Fourier Spectral Methods (2001) · Zbl 0994.65128
[7] DOI: 10.1093/cercor/bhj072 · doi:10.1093/cercor/bhj072
[8] DOI: 10.1137/050640072 · Zbl 1210.37033 · doi:10.1137/050640072
[9] DOI: 10.1007/s00332-005-0745-y · Zbl 1170.34033 · doi:10.1007/s00332-005-0745-y
[10] DOI: 10.1103/PhysRevE.62.4057 · doi:10.1103/PhysRevE.62.4057
[11] DOI: 10.1145/1055531.1055536 · Zbl 1073.65070 · doi:10.1145/1055531.1055536
[12] di Bernardo M., Applied Mathematical Sciences 163, in: Piecewise-Smooth Dynamical Systems: Theory and Applications (2007)
[13] DOI: 10.1007/978-1-4612-4206-2 · doi:10.1007/978-1-4612-4206-2
[14] DOI: 10.1142/S0218127491000555 · Zbl 0876.65060 · doi:10.1142/S0218127491000555
[15] DOI: 10.1080/02681119608806215 · Zbl 0855.34041 · doi:10.1080/02681119608806215
[16] Edwards C., Series in Systems and Control, in: Sliding Mode Control: Theory and Applications (1998)
[17] DOI: 10.1137/S1064827599363381 · Zbl 0981.65082 · doi:10.1137/S1064827599363381
[18] DOI: 10.1007/s002110100313 · Zbl 1002.65089 · doi:10.1007/s002110100313
[19] DOI: 10.1109/81.481457 · doi:10.1109/81.481457
[20] DOI: 10.1007/978-1-4612-4342-7 · doi:10.1007/978-1-4612-4342-7
[21] DOI: 10.1016/S0375-9601(01)00473-X · Zbl 0971.93080 · doi:10.1016/S0375-9601(01)00473-X
[22] DOI: 10.1023/A:1012990608060 · Zbl 1005.70019 · doi:10.1023/A:1012990608060
[23] DOI: 10.1142/S0218127403007874 · Zbl 1079.34029 · doi:10.1142/S0218127403007874
[24] DOI: 10.1142/S0218127497001709 · Zbl 0910.34057 · doi:10.1142/S0218127497001709
[25] DOI: 10.1142/S0218127401002407 · Zbl 1090.65551 · doi:10.1142/S0218127401002407
[26] DOI: 10.1142/S0218127402006291 · Zbl 1048.65126 · doi:10.1142/S0218127402006291
[27] DOI: 10.1109/TCS.1984.1085459 · Zbl 0551.94020 · doi:10.1109/TCS.1984.1085459
[28] DOI: 10.1016/0022-460X(91)90592-8 · doi:10.1016/0022-460X(91)90592-8
[29] DOI: 10.1016/0375-9601(92)90745-8 · doi:10.1016/0375-9601(92)90745-8
[30] DOI: 10.1103/PhysRevE.66.026207 · doi:10.1103/PhysRevE.66.026207
[31] Seydel R., Practical Bifurcation and Stability Analysis (1994) · Zbl 0806.34028
[32] DOI: 10.1007/s11071-007-9217-2 · Zbl 1170.70399 · doi:10.1007/s11071-007-9217-2
[33] Stépán G., Retarded Dynamical Systems (1989) · Zbl 0686.34044
[34] DOI: 10.1098/rsta.2000.0753 · Zbl 1169.74431 · doi:10.1098/rsta.2000.0753
[35] DOI: 10.1137/070703028 · Zbl 1192.34004 · doi:10.1137/070703028
[36] DOI: 10.1137/1.9780898719598 · Zbl 0953.68643 · doi:10.1137/1.9780898719598
[37] J. Tyson and N. Novák, Computational Cell Biology (Springer Science + Business Media, 2002) pp. 261–284.
[38] DOI: 10.1142/9789812796301 · doi:10.1142/9789812796301
[39] DOI: 10.1109/87.761053 · doi:10.1109/87.761053
[40] DOI: 10.1142/9789812564436 · doi:10.1142/9789812564436
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.