In this paper we define an “indicator of tangent variation” that makes it possible to estimate the divergence between pathes starting from different initial conditions and/or parameters. First we work out theoretically this indicator; then we consider two examples: for the first one this indicator is well adapted. The second one corresponds to the case of an impact oscillator. In this case, grazing bifurcations can occur and prevent direct use of the defined indicator.
Ageno, A. and Sinopoli, A., 2005, “Lyapunov's exponents for non smooth dynamics with impacts: stability analysis of the rocking block” International Journal of Bifurcation and Chaos15(6), 2015—2040.
2.
Awrejcewicz, J. and Kudra, G., 2005, “Stability analysis and lyapunov exponents of a multi-body mechanical system with rigid unilateral constraints” Nonlinear Analysis, Theory, Methods and Applications63(5—7), e909—918.
3.
Azzoni, A., La Barbera, G., and Zaninetti, A., 1995, “Analysis and prediction of rockfalls using a mathematical model,” International Journal of Rock Mechanics and Mining Science andGeomechanics Abstracts32(7), 709—724.
4.
Bastien, J., 2000, Etude théorique et numérique d'inclusions différentielles maximales monotones. Applications à des modèles élastoplastiques. PhD thesis , Université LyonI.
5.
Bernardin, F., 2004, Equations différentielles stochastiques multivoques et modélisation de structures sous sollicitations aléatoires. PhD thesis, Université LyonI.
6.
Brach, R.M., 1984, “Friction, restitution, and energy loss in planar collisions,” Journal of Applied Mechanics51, 164—170.
7.
Brach, R.M., 1989, “Rigid body collisions,” Journal of Applied Mechanics56, 133—138.
8.
Brézis, H., 1973, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North Holland, Amsterdam.
Brogliato, B., Ten Dam, A.A., Paoli, L., Genot, F., and Abadie, M., 2002, “Numerical simulation of finite dimensional multibody nonsmooth mechanical systems,” ASME Applied Mechanics Reviews55(2), 107—150.
11.
Fredriksson, M. and Nordmark, A., 1997, “Bifurcations caused by grazing incidence in many degrees of freedom impact oscillators,” in Proceedings: Mathematical, Physical and Engineering Sciences453(1961), 1261— 1276.
12.
Glocker, C., 1999, “Formulation of spatial contact situations in rigid multibody systems,” Computer Methods in Applied Mechanics and Engineering177, 199—214.
13.
Guckenheimer, J. and Holmes, Ph., 1983, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, Springer Verlag, Berlin.
14.
Hogan, S., 1989, “On the dynamics of rigid-block motion under harmonic forcing,” Proceedings of the Royal Society, London A425, 441—475.
15.
Ivanov, A., 1988, “A constructive model of impact with friction,” Journal of Applied Mathematics and Mechanics52(6), 700—704.
16.
Ivanov, A., 1994, “The dynamics of systems near to grazing collision ,” Journal of Applied Mathematics and Mechanics58(3), 437—444.
17.
Janin, O., 2001, Contribution à l'identification, la modélisation et l'analyse de systèmes dynamiques à non-linéarités irrégulières. PhD thesis, Université LyonI.
18.
Leine, R.I., Brogliato, B., and Nijmeijer, H., 2002, “Periodic motion and bifurcations induced by the painleve paradox,” European Journal of Mechanics - A/Solids21(5), 869—896.
19.
Leine, R.I. and van Campen, D.H. , 2002, “Discontinuous bifurcations of periodic solutions” Mathematical and Computer Modelling36(3), 259—273.
20.
Moreau, J.J., 1994, “Some numerical methods in multibody dynamics: application to granular materials” European Journal of Mechanics A/Solids13(4), 93—114.
21.
Müller, P.C., 1995, “Calculation of Lyapunov exponents for dynamic systems with discontinuities,” Chaos, Solitons and Fractals5(9), 1671—1681.
22.
Oseledec, V., 1968, “A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems” Transactions of the Moscow Mathematics Society19, 197—231.
23.
Paoli, L., 1993, Analyse numérique de vibrations avec contraintes unilatérales. PhD thesis, Université LyonI.
24.
Paoli, L. and Schatzman, M., 1993, “Mouvement à un nombre fini de degrés de liberté avec contraintes unilatérales: cas avec perte d'énergie,” Mathematical Modelling and Numerical Analysis27(6), 673—717.
25.
Schatzman, M., 1978, “A class of nonlinear differential equations of second order in time,” Nonlinear Analysis2(3), 355—373.
26.
Souza, S. L. d. and Caldas, I.L., 2004, “Calculation of Lyapunov exponents in systems with impacts,” Chaos, Solitons and Fractals19, 569—579.
