sergey kryzhevich - bcamâ¬ÂŠÂ · this is a space ð, consisting of ð¶1smooth pairs ð, ð....
TRANSCRIPT
Sergey Kryzhevich
BCAM, Bilbao , 13 February 2013
1) Vibro-impact systems: definitions and main properties. 2) Grazing bifurcation(s). 3) Estimates on eigenvalues 4) Homoclinic points. 5) Smale horseshoes.
1. Vibro-impact systems. Mathematical models: Newton, Hook, Peterka,74,
Pesin-Bunimovich-Sinai, 85 (for billiards), Schatzmann, 98 â the most general model, Babitsky, 98 review,âŠ
2. Grazing bifurcation. Nordmark, 91, Ivanov, 94, Dankowitz et al 02, Budd et al, 08,âŠ
Fig. 1. An example of single degree-of-freedom vibro-impact system
1 â point mass, 2 â spring, 3 - delimiter, 4 â damping element
ðœ = [0,ðâ] - values of parameter; ð ð¡, ð¥, ðŠ, ð :ð 3 à ðœ â ð - a ð¶2 smooth function, . ð ð¡ + ð, ð¥,ðŠ, ð â¡ ð(ð¡, ð¥,ðŠ, ð) Consider a system of 2 o.d.e.s ï¿œÌï¿œ = ðŠ; ï¿œÌï¿œ = ð(ð¡, ð¥,ðŠ, ð) (1) Denote the set of corresponding right hand sides by ðð = ðð(ðœ,ð,ð) Denote ð§ = (ð¥,ðŠ)
. . .
Eq. (1) is satisfied for ð¥ > 0. If ð¥ = 0 the following impact conditions take place: If ð¥ ð¡0 â 0 = 0 then ðŠ ð¡0 + 0 = âððŠ ð¡0 â 0 , ð¥ ð¡0 + 0 = ð¥(ð¡0 â 0) (2) . Here ð = ð ð â 0,1 is a ð¶2 smooth function. Denote by (*) the VIS defined by Eq. (1) and impact conditions (2) Let .
[ ] ( ) ( ){ }0,,,0,,0,0 111 >=âÃÃâ=Î yryyJ µµR
A function ð§ ð¡ = col(ð¥ ð¡ ,ðŠ(ð¡)) is called solution of the VIS (*) with a finite number of impacts on an interval (ð, ð) if the following conditions are satisfied: there exist instants ð¡0, ⊠, ð¡ð+1 where ð¡0 = ð, ð¡ð+1 = ð such that 1) The components ð¥ ð¡ is continuous, points ð¡1, ⊠, ð¡ð are all points of discontinuity for ðŠ(ð¡). 2) The function ð¥ ð¡ is non-negative and ð¡1, ⊠, ð¡ð are all its zeros. 3) ðŠ ð¡ð + 0 = âð ð ðŠ ð¡ð â 0 for all ð = 1, ⊠,ð 4) The function ð§ ð¡ is a solution of system (1) on every segment ð¡ð , ð¡ð+1 .
, , , .
Lemma 1. Let ð§ ð¡ = col(ð¥ ð¡ ,ðŠ(ð¡)) - be a solution of (*), corresponding to the value ð0 of the parameter and to the initial data (ð¡0, ð¥0,ðŠ0). Let ð¥02 + ðŠ02 > 0 and let the considered solution be defined on ð¡â, ð¡+ . Let the function ð¥(ð¡) have exactly ð zeros ð¡1, ⊠, ð¡ð on the interval (ð¡â, ð¡+). Suppose that ðŠ ð¡ð â 0 â 0 for all k. Then for any ð¡ â (ð¡â, ð¡+) â {ð¡1, ⊠, ð¡ð} there is a neighborhood ð of the point ð¡0, ð¥0,ðŠ0, ð0 such that the mapping ð§(ð¡, ð¡â², ð¥â²,ðŠâ²,ðð) is smooth with respect to (ð¡â², ð¥â²,ðŠâ², ðð) from ð. All corresponding solutions have exactly ð impacts ð¡ð(ð¡ð¡, ð¥ð¡,ðŠð¡,ðð¡) over (ð¡â, ð¡+). Impact instants ð¡ð(ð¡ð¡, ð¥ð¡,ðŠð¡, ðð¡) and corresponding velocities ðŠ(ð¡ð ð¡ð¡, ð¥ð¡,ðŠð¡, ðð¡ â 0) are smooth functions of ð¡ð¡, ð¥ð¡,ðŠð¡,ðð¡ in ð.
This is a space ð, consisting of ð¶1smooth pairs ð, ð . The topology is minimal one there for any
pair (ð0, ð0) and any ð > 0 the set
ï¿œ ð, ð : supð¡,ð§,ð âð
( ð ð¡, ð§, ð â ð0 ð¡, ð§, ð + ï¿œðððð§
ð¡, ð§, ð
âðð0ðð§
ð¡, ð§, ð ï¿œ + ï¿œðððð
ð¡, ð§, ð
âðð0ðð
ð¡, ð§, ð ï¿œ + |ð ð â ð0(ð)|) < ð ï¿œ
is open.
âStiffâ model with a perturbation ï¿œÌï¿œ = ðŠ; ï¿œÌï¿œ = ð ð¡, ð¥,ðŠ + ð ð¡, ð¥,ðŠ ; ð ð¶1 < ð We assume conditions (2) take place. âSoftâ model with a perturbation. ï¿œÌï¿œ = ðŠ; ï¿œÌï¿œ = ð ð¡, ð¥,ðŠ + ð ð¡, ð¥,ðŠ + â(ð , ð¡, ð¥,ðŠ); ð ð¶1 < ð â ð , ð¡, ð¥,ðŠ = â2ðŒð ðâ ð¥ ðŠ â 1 + ðŒ2 ð 2ðâ ð¥ x, ðŒ = âlog ð/ð. Ï̲ =1 if x <0 and Ï̲ =0 if xâ¥0; addition of functions h replaces the conditions of impact. Consider stroboscopic (period shift) mappings for both
of these models. Call them ðð,ð and ðð,ð,ð Hyperbolic invariant sets persist for small ð and big ð and
small changes in ð.
1kÎŽ
Fig. 2. A «soft» model of impact, «sutured» from two linear systems. A delimiter is replaced with a very stiff spring.
Fig. 3. A grazing bifurcation. Here the case µ>0 corresponds to existence of a low-velocity impact, we can select µ equal to this velocity. µ=0 corresponds to a tangent motion (grazing), the case µ<0 corresponds to passage near delimiter without impact.
There exists a continuous family of Т â periodic solutions ð ð¡, ð = (ðð¥ ð¡,ð ,ððŠ ð¡, ð ) of system (*), satisfying following properties: 1) For any ð â ðœ the component ðð¥ ð¡,ð has exactly ð + 1 zeros ð¡0 ð , ⊠, ð¡ð(ð) over the period [0,ð). 2) Velocities ðð(ð) = ðŠ(ð¡ð(ð) â 0) are such that ðð ð â 0 for all ð â 0, ð0(0) = 0, ðð 0 â 0 for ð = 1, ⊠,ð. 3) Instants ð¡ð(ð) and impact velocities ðð(ð) continuously depend on the parameter ð â ðœ.
, ,
.
.
Condition 1.
Fig. 4. Curve Ð. Homoclinic points arizing due to bent of unstable (а) or stable (b) manifold. Here we consider case 1 â presence of periodic motions with a
low impact velocity.
(a) (b)
Period shift map ðð,ð(ð§0) = ð§(ð â ð,âð, ð§0, ð) Here Ξ > 0 is a small parameter, ð§ð,ð = ð(âð, ð) be the fixed point. Let
ðŽ = limð,ðâ 0
ðð§ðð§0
ð â ð,ð, ð§0,ð =ð11 ð12ð21 ð22
be the matrix, corresponding for motion out of grazing Condition 2. ð12 > 0 , Tr ðŽ < â1.
ðµð,ð =ðð§ðð§0
ð,âð, ð§0, ð
=âð 0
âð + 1 ð 0,0,0
ð0 ðâð (ðž + ð(µ+Ξ))
.
ð·ðð,ð ð§ð,ð = ðŽðµð,ð(ðž + 0(ð +\theta)) Trace is big if ð12 â 0, determinant is bounded. Eigenvalues ð+ â â as ð â 0 and ð â â 0 as ð â 0. Corresponding eigenvectors:
ð¢+ =ð21ð22 + O(ð + ð),
ð¢â = 01 + O(ð + ð).
Theorem 2. Let Conditions 1 and 2 be satisfied. Then there exist values ð0,ð0 such that for all ð â 0, ð0 ,ð â (0,ð0) there exist a natural number m and a compact set ðŸð,ð, invariant with respect to ðð,ð and such that the following statements are true. 1) There exists a neighborhood ðð,ð of the set ðŸð,ð such that the reduction ðð,ð|ðð,ð is a local diffeomorphism. The invariant set ðŸð,ð is hyperbolic 2) The invariant set ðŸð,ð of the diffeomorphism ðð,ð|ðð,ð is chaotic in Devaney sense, i.e. it is infinite, hyperbolic, topologically transitive (contains a dense orbit) and periodic points are dense there.
Fig .11. Description of the experiment and bifurcation diagrams, demonstrating presence of chaos in a
neighborhood of grazing
Molenaar et al. 2000
Ing et al., 2008; Wiercigroch & Sin, 1998 (see also Fig. 2)
Fig. 12. Bifurcation diagrams and chaotic dynamics on the set of central leaves in a neighborhood of a periodic point.
Akhmet, M. U. (2009). Li-Yorke chaos in systems with impacts. Journal of Mathematical Analysis & Applications, 351(2), 804-810.
di Bernardo, M., Budd C. J., Champneys A. R. & Kowalczyk P. (2007) Bifurcations and Chaos in Piecewise-Smooth Dynamical Systems: Theory and Applications. New York, Springer.
Babitsky, V. I. (1998) Theory of Vibro-Impact Systems and Applications, Berlin, Germany, Springer.
Banerjee, S., Yorke, J. A. & Grebogi, C. (1998) Robust chaos. Physical Review Letters, 80(14), 3049-3052.
Budd, C. (1995) Grazing in impact oscillators. In: Branner B. & Hjorth P. (Ed.) Real and Complex Dynamical Systems (pp. 47-64). Kluwer Academic Publishers.
Bunimovich L.A., Pesin Ya. G., Sinai Ya. G. & Jacobson M. V. (1985) Ergodic theory of smooth dynamical systems. Modern problems of mathematics. Fundamental trends, 2, 113-231.
Chernov N., Markarian R. (2001) Introduction to the ergodic theory of chaotic billiards. IMCA, Lima.
Chillingworth, D.R.J. (2010) Dynamics of an impact oscillator near a degenerate graze. Nonlinearity, 23, 2723-2748.
Chin W., Ott E., Nusse, H. E. & Grebogi, C. (1995) Universal behavior of impact oscillators near grazing incidence. Physics Letters A 201(2), 197-204.
Devaney, R. L. (1987) An Introduction to Chaotic Dynamical Systems. Redwood City, CA: Addison-Wesley.
Fredriksson, M. H. & Nordmark, A. B. (1997) Bifurcations caused by grazing incidence in many degrees of freedom impact oscillators. Proceedings of the Royal Society, London. Ser. A. 453, 1261-1276.
ÐПÑбОкПв С. Ð. & ÐеМÑÑеМОМа Ð.Ð. (2005) ÐОÑÑÑкаÑОÑ, пÑОвПЎÑÑÐ°Ñ Ðº Ñ Ð°ÐŸÑОÑеÑкОЌ ЎвОжеМОÑÐŒ в ЎОМаЌОÑеÑÐºÐžÑ ÑОÑÑÐµÐŒÐ°Ñ Ñ ÑЎаÑÐœÑЌО взаОЌПЎейÑÑвОÑЌО. ÐОÑÑеÑеМÑ. ÑÑавМеМОÑ, 41(8), 1046-1052.
Holmes, P. J. (1982) The dynamics of repeated impacts with a sinusoidally vibrating table. Journal of Sound and Vibration 84(2), 173-189.
Ing, J., Pavlovskaia E., Wiercigroch M, Banerjee S. (2008) Experimental study of an impact oscillator with a one-side elastic constraint near grazing. Physica D 239, 312-321.
Ivanov, A. P. (1996) Bifurcations in impact systems. Chaos, Solitons & Fractals 7(10), 1615-1634.
ÐПзлПв Ð.Ð., ТÑеÑев Ð.Ð. (1991) ÐОллОаÑÐŽÑ. ÐеМеÑОÑеÑкПе ввеЎеМОе в ÐŽÐžÐœÐ°ÐŒÐžÐºÑ ÑОÑÑеЌ Ñ ÑЎаÑаЌО. Ð.: ÐзЎ-вП ÐÐУ. 1991. 168 c.
Molenaar J, van de Water W. & de Wegerand J. (2000) Grazing impact oscillations. Physical Review E, 62(2), 2030-2041.
Nordmark, A. B. (1991) Non-periodic motion caused by grazing incidence in an impact oscillator. Journal of Sound & Vibration, 145(2), 279-297.
Palis, J., di Melo, W. (1982) Geometric Theory of Dynamical Systems. Springer-Verlag, 1982.
Pavlovskaia, E. ,Wiercigroch, M. (2004) Analytical drift reconstruction for visco-elastic impact oscillators operating in periodic and chaotic regimes. Chaos, Solitons & Fractals 19(1), 151-161.
Smale, S. (1965) Diffeomorfisms with many periodic points. Differential & Combinatorial. Topology. Princeton: University Press, 63-81.
Thomson, J. M. T., Ghaffari, R. (1983) Chaotic dynamics of an impact oscillator. Physical Review A, 27(3), 1741-1743.
Wiercigroch, M., Sin, V.W.T. (1998) Experimental study of a symmetrical piecewise base-excited oscillator, J. Appl. Mech.65, 657-663.
Whiston, G. S. (1987) Global dynamics of a vibro-impacting linear oscillator. Journal of Sound & Vibration, 118(3), 395-429.