Stability calculations for piecewise-smooth delay equations

International Journal of Bifurcation and Chaos

Volumn 19, Issue 2, 2009, Pages 639-650

  Barton, David A W ^a  

^a UNIVERSITY OF BRISTOL   (United Kingdom)

Author keywords

Delay differential equation; Nonsmooth; Numerical continuation; Stability

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; DIFFERENTIAL EQUATIONS; DYNAMICAL SYSTEMS; LINEARIZATION; NUMERICAL METHODS; STABILITY;

DELAY DIFFERENTIAL EQUATIONS; DISCONTINUITY MAPPINGS; INFINITE DIMENSIONAL; NONSMOOTH; NUMERICAL CONTINUATION; PIECEWISE SMOOTH DYNAMICAL SYSTEMS; PROPAGATION PROPERTIES; STABILITY CALCULATION;

SYSTEM STABILITY;

EID: 67449158199     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127409023263     Document Type: Article

References (49)

1. An der Heiden, U., Longtin, A., Mackey, M. & Milton, J. [1990] "Oscillatory modes in a nonlinear second-order differential equation with delay," J. Dyn. Diff. Eqs. 2, 423-449.
2. An der Heiden, U. & Reichard, K. [1990] "Multitude of oscillatory behaviour in a nonlinear second order differential-difference equation," Zeitschrift für Angewandte Mathematik und Mechanik 70, T621-T624.
3. Atay, F. [1998] "Van der Pol's oscillator under delayed feedback," J. Sound Vibr. 218, 333-339.
4. Barton, D., Krauskopf, B. & Wilson, R. [2006] "Periodic solutions and their bifurcations in a non-smooth second-order delay differential equation," Dyn. Syst. 21, 289-311.
5. Bayer, W. & an der Heiden, U. [1998] "Oscillation types and bifurcations of a nonlinear second-order differential-difference equation," J. Dyn. Diff. Eqs. 10, 303-326.
6. Boyd, J. [2001] Chebyshev and Fourier Spectral Methods, 2nd edition (Dover, NY).
7. Breakspear, M., Roberts, J., Terry, J., Rodrigues, S., Mahant, N. & Robinson, P. [2006] "A unifying explanation of primary generalized seizures through nonlinear brain modeling and bifurcation analysis," Cereb. Cort. 16, 1296-1313.
8. Colijn, C. & Mackey, M. [2007] "Bifurcation and bistability in a model of hematopoietic regulation," SIAM J. Appl. Dyn. Syst. 6, 378-394.
9. Colombo, A., di Bernardo, M., Hogan, S. & Kowalczyk, P. [2007] "Complex dynamics in a hysteretic relay feedback system with delay," J. Nonlin. Sci. 17, 85-108.
10. Coombes, S. & Osbaldestin, A. [2000] "Period-adding bifurcations and chaos in a periodically stimulated excitable neural relaxation oscillator," Phys. Rev. E 62, 4057-4066.
11. Dercole, F. & Kuznetsov, Y. [2005] "SlideCont: an Auto97 driver for bifurcation analysis of Filippov systems," ACM Trans. Math. Softw. 31, 95-119.
12. Dhooge, A., Govaerts, W., Kuznetsov, Y., Mestrom, W., Riet, A. & Sautois, B. [2006] MATCONT and CL MATCONT: Continuation Toolboxes in Matlab, http://www.matcont.ugent.be/.
13. Di Bernardo, M., Budd, C., Champneys, A. & Kowalczyk, P. [2007] Piecewise-Smooth Dynamical Systems: Theory and Applications, Applied Mathematical Sciences, Vol. 163 (Springer-Verlag).
14. Diekmann, O., van Gils, S., Lunel, S. & Walther, H.-O. [1995] Delay Equations: Functional, Complex, and Nonlinear Analysis, Applied Mathematical Sciences, Vol. 110 (Springer, NY).
15. Doedel, E., Champneys, A., Fairgrieve, T., Kuznetsov, Y., Sandstede, B. & Wang, X. [1998] "AUTO 97: continuation and bifurcation software for ordinary differential equations, http://indy.cs.concordia.ca/auto/.
16. Doedel, E., Keller, H. & Kernevez, J. [1991] "Numerical analysis and control of bifurcation problems, part II," Int. J. Bifurcation and Chaos 1, 745-772.
17. Dombóvári, Z. [2006] Bifurcation Analysis of a Cutting Process, Master's thesis, Budapest University of Technology and Economics, Department of Mechanical Engineering, Hungary.
18. Doole, S. & Hogan, S. [1996] "A piecewise linear suspension bridge model: nonlinear dynamics and orbit continuation," Dyn. Stab. Syst. 11, 19-47.
19. Edwards, C. & Spurgeon, S. [1998] Sliding Mode Control: Theory and Applications(CRC Press), Series in Systems and Control.
20. Engelborghs, K., Luzyanina, T., In 't Hout, K. J. & Roose, D. [2001a] "Collocation methods for the computation of periodic solutions of delay differential equations," SIAM J. Sci. Comput. 22, 1593-1609.
21. Engelborghs, K., Luzyanina, T. & Samaey, G. [2001b] DDE-BIFTOOL v. 2.00: A Matlab Package for Bifurcation Analysis of Delay Differential Equations, Technical Report TW330, Department of Computer Science, K. U. Leuven, Belgium.
22. Engelborghs, K. & Doedel, E. [2002] "Stability of piecewise polynomial collocation for computing periodic solutions of delay differential equations," Numerische Mathematik 91, 627-648.
23. Fossas, E. & Olivar, G. [1996] "Study of chaos in the buck converter," Circuits Syst.-I: Fund. Th. Appl. 43, 13-25.
24. Hale, J. & Verduyn Lunel, S. [1993] Introduction to Functional Differential Equations, Applied Mathematical Sciences, Vol. 99 (Springer).
25. Herrmann, G. [2001] "A robust delay adaptation scheme for Pyragas' chaos control method," Phys. Lett. A 287, 245-256.
26. Kalmár-Nagy, T., Pratt, J., Davies, M. & Kennedy, M. [1999] "Experimental and analytical investigation of the subcritical instability in metal cutting," in Proc. DETC'99, Las Vegas, Nevada, USA.
27. Kalmár-Nagy, T., Stepan, G. & Moon, F. [2001] "Subcritical Hopf bifurcation in the delay equation model for machine tool vibrations," Nonlin. Dyn. 26, 121-142.
28. Kang, W., Thota, P., Wilcox, B. & Dankowicz, H. [2007] "Bifurcation analysis of a microactuator using a new toolbox for continuation of hybrid system trajectories," in Proc. ASME IDETC/CIE, Las Vegas, USA.
29. Kollár, L., Stépán, G. & Turi, J. [2001] "Dynamics of delayed piecewise linear systems," in Proc. of the Fifth Mississippi State Conf. Differential Equations and Computational Simulations, Starkville,MS, USA.
30. Kuznetsov, Y., Rinaldi, S. & Gragnani, A. [2003] "Oneparameter bifurcations in planar Filippov systems," Int. J. Bifurcation and Chaos 13, 2157-2188.
31. Luzyanina, T., Engelborghs, K., Lust, K. & Roose, D. [1997] "Computation, continuation and bifurcation analysis of periodic solutions of delay differential equations," Int. J. Bifurcation and Chaos 7, 2547-2560.
32. Luzyanina, T., Engelborghs, K. & Roose, D. [2001] "Numerical bifurcation analysis of differential equations with state-dependent delay," Int. J. Bifurcation and Chaos 11, 737-753.
33. Luzyanina, T. & Engelborghs, K. [2002] "Computing floquet multipliers for functional differential equations," Int. J. Bifurcation and Chaos 12, 2977-2989.
34. Matsumoto, T. [1984] "A chaotic attractor from Chua's circuit," IEEE Trans. Circuits Syst. 31, 1055-1058.
35. Nordmark, A. [1991] "Non-periodic motion caused by grazing incidence in an impact oscillator," J. Sound Vib. 145, 279-297.
36. Piiroinen, P. [2002] Recurrent Dynamics of Nonsmooth Systems with Application to Human Gait, PhD thesis, Royal Institute of Technology, Department of Mechanics, Stockholm, Sweden.
37. Pyragas, K. [1992] "Continuous control of chaos by self-controlling feedback," Phys. Lett. A 170, 421-428.
38. Pyragas, K. [2002] "Analytical properties and optimization of time-delayed feedback control," Phys. Rev. E 66, 026207.
39. Seydel, R. [1994] Practical Bifurcation and Stability Analysis(Springer, NY).
40. Sieber, J. & Krauskopf, B. [2008] "Control based bifurcation analysis for experiments," Nonlin. Dyn. 51, 365-377.
41. Stépán, G. [1989] Retarded Dynamical Systems(Longman, London).
42. Stépán, G. & Kalmar-Nagy, T. [1997] "Nonlinear regenerative machine tool vibrations," in Proc. ASME Design Engineering Technical Conference, Sacramento, California. Stépán, G. [2001] "Modelling nonlinear regenerative effects in metal cutting," Philos. Trans. Roy. Soc. 359, 739-757.
43. Thompson, S. & Shampine, L. [2004] "A friendly Fortran DDE solver," in Proc. 3rd Int. Conf. Numerical Solution of Volterra and Delay Equations, Arizona State University, AZ, USA.
44. Thota, P. & Dankowicz, H. [2008] "TC-HAT (-TC): A novel toolbox for the continuation of periodic trajectories in hybrid dynamical systems," SIAM J. Appl. Dyn. Syst. 7, 1283-1322.
45. Trefethen, L. [2000] Spectral Methods in Matlab(SIAM). Tyson, J. & Novák, N. [2002] "Cell cycle controls," Computational Cell Biology(Springer Science + Business Media), pp. 261-284.
46. Wahi, P. [2005] A Study of Delay Differential Equations with Applications to Machine Tool Vibrations, PhD thesis, Department of Mechanical Engineering, Indian Institute of Science, Bangalore, India.
47. Wiercigroch, M. & de Kraker, B. (eds.) [2000] Applied Nonlinear Dynamics and Chaos of Mechanical Systems with Discontinuities, World Scientific Series on Nonlinear Science: Series A, Vol. 28 (World Scientific, Singapore).
48. Young, K., Utkin, V. & Ozguner, U. [1999] "A control engineer's guide to sliding mode control," IEEE Trans. Contr. Syst. Technol. 7, 328-342.
49. Zhusubaliyev, Z. & Mosekilde, E. [2003] Bifurcations and Chaos in Piecewise-Smooth Dynamical Systems, World Scientific Series on Nonlinear Science: Series A, Vol. 44 (World Scientific, Singapore).