TC-HAT (TĈ): A novel toolbox for the continuation of periodic trajectories in hybrid dynamical systems

SIAM Journal on Applied Dynamical Systems

Volumn 7, Issue 4, 2008, Pages 1283-1322

  Thota, Phanikrishna ^a   Dankowicz, Harry ^b  

^a UNIVERSITY OF BRISTOL   (United Kingdom)

^b UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN   (United States)

Author keywords

Bifurcation analysis; Continuation software; Hybrid dynamical systems; TC HAT

Indexed keywords

BIFURCATION ANALYSES; BIFURCATION ANALYSIS; CONTINUATION SOFTWARE; HYBRID DYNAMICAL SYSTEMS; PERIODIC TRAJECTORIES; PROBLEM FORMULATIONS; SOFTWARE PROGRAMS; TC-HAT; TIME DYNAMICS;

BIFURCATION (MATHEMATICS); COMPRESSED AIR MOTORS; CONTINUOUS TIME SYSTEMS; DYNAMICAL SYSTEMS; KETONES; POSITION CONTROL; SWITCHING CIRCUITS; TRAJECTORIES;

SCHRODINGER EQUATION;

EID: 56449101230     PISSN: None     EISSN: 15360040     Source Type: Journal    
DOI: 10.1137/070703028     Document Type: Article

References (54)

1. J. ADOLFSSON, H. DANKOWICZ, AND A. B. NORDMARK, 3D passive walkers: Stability analysis in the presence of discontinuities, Nonlinear Dynam., 24 (2001), pp. 205-229.
2. E. L. ALLONGER AND K. GEORG, Numerical Continuation Methods: An Introduction, Springer-Verlag, New York, 1990.
3. D. A. W. BARTON, B. KRAUSKOPF, AND R. E. WILSON, Collocation schemes for periodic solutions of neutral delay differential equations, j. Differ. Equations Appl., 12 (2006), pp. 1087-1101.
4. W. J. BEYN, A. CHAMPNEYS, E. DOEDEL, W. GOVAERTS, Y. A. KUZNETSOV, AND B. SANDSTEDE, Numerical continuation and computation of normal forms, in Handbook of Dynamical Systems, Vol. 2, B. Fiedler, G. Iooss, and N. Kopell, eds., North-Holland, Amsterdam, 2002, pp. 149-219.
5. D. S. BINDEL, J.W. DEMEL, M. J. FRIEDMAN, W. J. GOVAERTS, AND Y. A. KUZNETSOV, Bifurcation analysis of large equilibrium, systems in MATLAB, in Proceedings of the International Conference on Computational Science (ICCS), Lecture Notes in Comput. Sci. 3514, Springer-Verlag, Berlin, 2005, pp. 50-57.
6. A. R. CHAMPNEYS, Y. A. KUZNETSOV, AND B. SANDSTEDE, A numerical toolbox for homoclinic bifurcation analysis, Internat. j. Bifur. Chaos Appl. Sci. Engrg., 6 (1996), pp. 867-887.
7. A. R. CHAMPNEYS AND Y. A. KUZNETSOV, Numerical detection and continuation of codimension-two homoclinic orbits, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 4 (1994), pp. 785-822.
8. H. DANKOWICZ AND A. B. NORDMARK, On the origin and bifurcations of stick-slip oscillations, Phys. D, 136 (1999), pp. 280-302.
9. H. DANKOWICZ AND X. ZHAO, Local analysis of co-dimension-one and co-dimension-two grazing bifurcations in impact microactuators, Phys. D, 202 (2005), pp. 238-257.
10. M. DELLNITZ, Computational bifurcation of periodic, solutions in systems with symmetry, IMA J. Numer. Anal., 12 (1992), pp. 429-455.
11. F. DERCOLE AND Y. A. KUZNETSOV, SLIDECONT: An AUTO 97 driver for bifurcation analysis of Filippov systems, ACM Trans. Math. Software, 31 (2005), pp. 95-119.
12. P. DEUFLHARD, Computation of periodic solutions of nonlinear ODE's, BIT, 24 (1984), pp. 456-466.
13. A. DHOOGE,W. GOVAERTS, AND Y. A. KUZNETSOV, MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs, ACM Trans. Math. Software, 29 (2003), pp. 141-164.
14. M. DI BERNARDO, P. KOWALCZYK, AND A. NORDMARK, Bifurcations of dynamical systems with sliding Derivation of normal-form mappings, Phys. D, 170 (2002), pp. 175-205.
15. E. J. DOEDEL AND M. J. FRIEDMAN, Numerical computation of heteroclinic orbits, J. Comput. Appl. Math., 26 (1989), pp. 155-170.
16. E. J. DOEDEL, H. B. KHELLER, AND J. P. KERNEVEZ, Numerical analysis and control of bifurcation problems, part II: Bifurcation in infinite dimensions, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 1 (1991), pp. 745-772.
17. E. J. DOEDEL, R. C. PAFFENROTH, H. B. KELLER, D. J. DICHMANN, J. GALAN-VIOQUE, and A. V ANDERBAUWHEDE, Computation of periodic solutions of conservative systems with application to the 3-body problem, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), pp. 1353-1381.
18. E. J. DOEDEL, W. GAOVAERTS AND YU. A. KUZNETSOV, Computation of periodic solution bifurcations in ODEs using bordered systems, SIAM J. Numer. Anal., 41 (2003), pp. 401-435.
19. E. .J. DOEDEL, W. GOVAERTS, Y. A. KUZNETSOV, AND A. DHOOGE, Numerical continuation of branch points of equilibria and periodic orbits, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), pp. 841-860.
20. K. ENGELBORGHS, T. LUZYANINA, AND D. ROOSE, Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL, ACM Trans. Math. Software, 28 (2002), pp. 1-21.
21. K, ENGELBORGHS, T, LUZYANINA, K. J. IN'T HOUT, AND D. ROOSE, Collocation methods for the computation of periodic solutions of delay differential equations, SIAM J. Sci. Comput., 22 (2000), pp. 1593-1609.
22. K. ENGELBORGHS AND E. J. DOEDEL, Stability of piecewise. polynomial collocation for computing periodic solutions of delay differential equations, Numer. Math., 91 (2002), pp. 627-648.
23. S. FOALE, Analytical determination of bifurcations in an impact oscillator, Proc. R. Soc. Lond. Ser. A, 347 (1994), pp. 373-364.
24. S. FOALE AND R. BISHOP, Bifurcations in impacting systems, Nonlinear Dynam., 6 (1994), pp. 285-299.
25. M. FRIEDMAN, W. GOVAERTS, Y. A. KUZNETSOV, AND B. SAUTOIS, Continuation of homoclinic orbits in MATLAB, in Proceedings of the International Conference on Computational Science (ICCS), Lecture Notes in Comput. Sci. 3514, Springer-Verlag, Berlin, 2005, pp. 263-270.
26. M. GOLUBITSKY AND D. G. SCHAEFFER, Singularities and Groups in Bifurcation Theory, Springer-Verlag, New York, 1985.
27. W. GOVAERTS, Y. A. KUZNETSOV, AND A. DHOOGE, Numerical continuation of bifurcations of limit cycles in MATLAB, SIAM J. Sci. Comput., 27 (2005), pp. 231-252.
28. W. GOVAERTS, Numerical bifurcation analysis for ODEs, J. Comput. Appl. Math., 125 (2000), pp. 57-68.
29. J. GUCKENHEIMER, Computer simulation and beyond-for the 21st century. Notices Amer. Math. Soc, 45 (1998), pp. 1120-1123.
30. A. P. IVANOV, Bifurcations in impact systems, Chaos Solitons Fractals, 7 (1996), pp. 1615-1634.
31. J. D. JANSEN, Nonlinear rotor dynamics as applied to oilwell drillstring vibrations, J. Sound Vibration, 147 (1991), pp. 115-135.
32. W. KANG, P. THOTA, B. WLLCOX, AND H. DANKOWICZ, Bifurcation analysis of a microactuator using a new toolbox for continuation of hybrid system, trajectories, ASME J. Comput. Nonlinear Dyn., to appear.
33. K. KARAGIANNIS AND F. PFEIFFER, Theoretical and experimental investigations of gear-rattling, Nonlinear Dynam., 2 (1991), pp. 367-387.
34. H. B. KELLER, Lectures on Numerical Methods in Bifurcation Problems, Springer-Verlag, New York, 1987.
35. R. I. LEINE, D. H. VAN CAMPEN, AND W. J. G. KEULT.JES, Stick-slip whirl interaction in drillstring dynamics, J. Vibration Acoustics Transactions of the ASME, 124 (2002), pp. 209-220.
36. A. C. J. LUO AND B. C. GEGG, On the mechanism of stick and nonstick, periodic motions in a periodically forced, linear oscillator with dry friction, J. Vibration Acoustics Transactions of the ASME, 128 (2006), pp. 399-418.
37. K. LUST, D. ROOSE, A. SPENCE, AND A. R. CHAMPNEYS, An adaptive Newton-Picard algorithm with subspace iteration for computing periodic solutions, SIAM J. Sci. Comput., 19 (1998), pp. 1188-1209.
38. T. LUZYANINA, K. ENGELBORGHS, AND D. ROOSE, Numerical bifurcation analysis of differential equations with state-dependent delay, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), pp. 737-753.
39. P. C. MÜLLER, Calculation of Lyapunov exponents for dynamics systems with discontinuities, Chaos Solitons Fractals, 5 (1995), pp. 1671-1681.
40. F. J. MUNOZ-ALMARAZ, E. FREIRE, J. GALAN, E. DOEDEL, AND A. VANDERBAUWHEDE, Continuation of periodic orbits in conservative and Hamiltonian systems, Phys. D, 181 (2003), pp. 1-38.
41. A. B. NORDMARK AND P. KOWALCZYK, A codimension-two scenario of sliding solutions in grazing-sliding bifurcations, Nonlinearity, 19 (2006), pp. 1-26.
42. A. B. NORDMARK, Existence of periodic orbits in grazing bifurcations of impacting mechanical oscillators, Nonlinearity, 14 (2001), pp. 1517-1542.
43. A. B. NORDMARK, Personal communication.
44. D. ROOSE, K. LUST, A. CHAMPNEYS, AND A. SPENCE, A Newton-Picard shooting method for computing periodic solutions of large-scale dynamical systems, Chaos Solitons Fractals, 5 (1995), pp. 1913-1925.
45. A. G. SALINGER, E. A. BURROUGHS, R. P. PAWLOWSKI E. T. PHIPPS, AND L. A. ROMERO, Bifurcation tracking algorithms and software for large scale applications, Internat. J. Bifur. Chaos Appl. Sci.Engrg., 15 (2005), pp. 1015-1032.
46. G. S.AMAEY, K. ENGELBORGHS, AND D. ROOSE, Numerical computation of connecting orbits in delay differential equations, Numer. Algorithms, 30 (2002), pp. 335-352.
47. F. SVAHN AND H. DANKOWICZ, Energy transfer in vibratory systems with friction exhibiting low-velocity collisions, J. Vib. Control, 14 (2008), pp. 255-284.
48. R. SZALAI, G. STÉPÁN, AND S. J. HOGAN, Continuation of bifurcations in periodic delay-differential equations using characteristic matrices, SIAM J. Sci. Comput., 28 (2006), pp. 1301-1317.
49. P. THOTA, Analytical and Computational Tools for the Study of Grazing Bifurcations of Periodic Orbits and Invariant Tori, Ph.D. dissertation, Virginia Polytechnic Institute and State University, Blacks-burg, VA, 2007.
50. P. THOTA, X. ZHAO, AND H. DANKOWICZ, Codimension-two grazing bifurcation points in single-degree-of-freedom impact oscillators, J. Computational Nonlinear Dynam., 1 (2006), pp. 328-335.
51. J. J. TYSON AND B. NOVÁK Cell cycle controls, in Computational Cell Biology, C. P. Fall, E. S. Marland, J. M. Wagner, and J. J. Tyson, eds., Springer Science+Business Media, New York, 2002, pp. 261-284.
52. V. I. UTKIN, Sliding Modes in Control Optimization, Springer-Verlag, Berlin, 1992.
53. C. WLJLFF AND A. SCHEBESCH, Numerical continuation of symmetric periodic orbits, SIAM J. Appl. Dyn. Syst., 5 (2006), pp. 435-475.
54. X. ZHAO, C. K. REDDY, AND A. H. NAYFEH, Nonlinear dynamics of an electrically driven impact microactuator, Nonlinear Dynam., 40 (2005), pp. 227-239.