sergey kryzhevich - bcamโฌยฆย ยท this is a space ๐, consisting of ๐ถ1smooth pairs ๐, ๐....
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 โ the most general model, Babitsky, 98 review,โฆ
2. Grazing bifurcation. Nordmark, 91, Ivanov, 94, Dankowitz et al 02, Budd et al, 08,โฆ
Fig. 1. An example of single degree-of-freedom vibro-impact system
1 โ point mass, 2 โ spring, 3 - delimiter, 4 โ damping element
๐ฝ = [0,๐โ] - values of parameter; ๐ ๐ก, ๐ฅ, ๐ฆ, ๐ :๐ 3 ร ๐ฝ โ ๐ - a ๐ถ2 smooth function, . ๐ ๐ก + ๐, ๐ฅ,๐ฆ, ๐ โก ๐(๐ก, ๐ฅ,๐ฆ, ๐) Consider a system of 2 o.d.e.s ๏ฟฝฬ๏ฟฝ = ๐ฆ; ๏ฟฝฬ๏ฟฝ = ๐(๐ก, ๐ฅ,๐ฆ, ๐) (1) Denote the set of corresponding right hand sides by ๐๐ = ๐๐(๐ฝ,๐,๐) Denote ๐ง = (๐ฅ,๐ฆ)
. . .
Eq. (1) is satisfied for ๐ฅ > 0. If ๐ฅ = 0 the following impact conditions take place: If ๐ฅ ๐ก0 โ 0 = 0 then ๐ฆ ๐ก0 + 0 = โ๐๐ฆ ๐ก0 โ 0 , ๐ฅ ๐ก0 + 0 = ๐ฅ(๐ก0 โ 0) (2) . Here ๐ = ๐ ๐ โ 0,1 is a ๐ถ2 smooth function. Denote by (*) the VIS defined by Eq. (1) and impact conditions (2) Let .
[ ] ( ) ( ){ }0,,,0,,0,0 111 >=โรรโ=ฮ yryyJ ยตยตR
A function ๐ง ๐ก = col(๐ฅ ๐ก ,๐ฆ(๐ก)) is called solution of the VIS (*) with a finite number of impacts on an interval (๐, ๐) if the following conditions are satisfied: there exist instants ๐ก0, โฆ , ๐ก๐+1 where ๐ก0 = ๐, ๐ก๐+1 = ๐ such that 1) The components ๐ฅ ๐ก is continuous, points ๐ก1, โฆ , ๐ก๐ are all points of discontinuity for ๐ฆ(๐ก). 2) The function ๐ฅ ๐ก is non-negative and ๐ก1, โฆ , ๐ก๐ are all its zeros. 3) ๐ฆ ๐ก๐ + 0 = โ๐ ๐ ๐ฆ ๐ก๐ โ 0 for all ๐ = 1, โฆ ,๐ 4) The function ๐ง ๐ก is a solution of system (1) on every segment ๐ก๐ , ๐ก๐+1 .
, , , .
Lemma 1. Let ๐ง ๐ก = col(๐ฅ ๐ก ,๐ฆ(๐ก)) - be a solution of (*), corresponding to the value ๐0 of the parameter and to the initial data (๐ก0, ๐ฅ0,๐ฆ0). Let ๐ฅ02 + ๐ฆ02 > 0 and let the considered solution be defined on ๐กโ, ๐ก+ . Let the function ๐ฅ(๐ก) have exactly ๐ zeros ๐ก1, โฆ , ๐ก๐ on the interval (๐กโ, ๐ก+). Suppose that ๐ฆ ๐ก๐ โ 0 โ 0 for all k. Then for any ๐ก โ (๐กโ, ๐ก+) โ {๐ก1, โฆ , ๐ก๐} there is a neighborhood ๐ of the point ๐ก0, ๐ฅ0,๐ฆ0, ๐0 such that the mapping ๐ง(๐ก, ๐กโฒ, ๐ฅโฒ,๐ฆโฒ,๐๐) is smooth with respect to (๐กโฒ, ๐ฅโฒ,๐ฆโฒ, ๐๐) from ๐. All corresponding solutions have exactly ๐ impacts ๐ก๐(๐ก๐ก, ๐ฅ๐ก,๐ฆ๐ก,๐๐ก) over (๐กโ, ๐ก+). Impact instants ๐ก๐(๐ก๐ก, ๐ฅ๐ก,๐ฆ๐ก, ๐๐ก) and corresponding velocities ๐ฆ(๐ก๐ ๐ก๐ก, ๐ฅ๐ก,๐ฆ๐ก, ๐๐ก โ 0) are smooth functions of ๐ก๐ก, ๐ฅ๐ก,๐ฆ๐ก,๐๐ก in ๐.
This is a space ๐, consisting of ๐ถ1smooth pairs ๐, ๐ . The topology is minimal one there for any
pair (๐0, ๐0) and any ๐ > 0 the set
๏ฟฝ ๐, ๐ : sup๐ก,๐ง,๐ โ๐
( ๐ ๐ก, ๐ง, ๐ โ ๐0 ๐ก, ๐ง, ๐ + ๏ฟฝ๐๐๐๐ง
๐ก, ๐ง, ๐
โ๐๐0๐๐ง
๐ก, ๐ง, ๐ ๏ฟฝ + ๏ฟฝ๐๐๐๐
๐ก, ๐ง, ๐
โ๐๐0๐๐
๐ก, ๐ง, ๐ ๏ฟฝ + |๐ ๐ โ ๐0(๐)|) < ๐ ๏ฟฝ
is open.
โStiffโ model with a perturbation ๏ฟฝฬ๏ฟฝ = ๐ฆ; ๏ฟฝฬ๏ฟฝ = ๐ ๐ก, ๐ฅ,๐ฆ + ๐ ๐ก, ๐ฅ,๐ฆ ; ๐ ๐ถ1 < ๐ We assume conditions (2) take place. โSoftโ model with a perturbation. ๏ฟฝฬ๏ฟฝ = ๐ฆ; ๏ฟฝฬ๏ฟฝ = ๐ ๐ก, ๐ฅ,๐ฆ + ๐ ๐ก, ๐ฅ,๐ฆ + โ(๐ , ๐ก, ๐ฅ,๐ฆ); ๐ ๐ถ1 < ๐ โ ๐ , ๐ก, ๐ฅ,๐ฆ = โ2๐ผ๐ ๐โ ๐ฅ ๐ฆ โ 1 + ๐ผ2 ๐ 2๐โ ๐ฅ x, ๐ผ = โlog ๐/๐. ฯฬฒ =1 if x <0 and ฯฬฒ =0 if xโฅ0; addition of functions h replaces the conditions of impact. Consider stroboscopic (period shift) mappings for both
of these models. Call them ๐๐,๐ and ๐๐,๐,๐ Hyperbolic invariant sets persist for small ๐ and big ๐ and
small changes in ๐.
1kฮด
Fig. 2. A ยซsoftยป model of impact, ยซsuturedยป from two linear systems. A delimiter is replaced with a very stiff spring.
Fig. 3. A grazing bifurcation. Here the case ยต>0 corresponds to existence of a low-velocity impact, we can select ยต equal to this velocity. ยต=0 corresponds to a tangent motion (grazing), the case ยต<0 corresponds to passage near delimiter without impact.
There exists a continuous family of ะข โ periodic solutions ๐ ๐ก, ๐ = (๐๐ฅ ๐ก,๐ ,๐๐ฆ ๐ก, ๐ ) of system (*), satisfying following properties: 1) For any ๐ โ ๐ฝ the component ๐๐ฅ ๐ก,๐ has exactly ๐ + 1 zeros ๐ก0 ๐ , โฆ , ๐ก๐(๐) over the period [0,๐). 2) Velocities ๐๐(๐) = ๐ฆ(๐ก๐(๐) โ 0) are such that ๐๐ ๐ โ 0 for all ๐ โ 0, ๐0(0) = 0, ๐๐ 0 โ 0 for ๐ = 1, โฆ ,๐. 3) Instants ๐ก๐(๐) and impact velocities ๐๐(๐) continuously depend on the parameter ๐ โ ๐ฝ.
, ,
.
.
Condition 1.
Fig. 4. Curve ะ. Homoclinic points arizing due to bent of unstable (ะฐ) or stable (b) manifold. Here we consider case 1 โ presence of periodic motions with a
low impact velocity.
(a) (b)
Period shift map ๐๐,๐(๐ง0) = ๐ง(๐ โ ๐,โ๐, ๐ง0, ๐) Here ฮธ > 0 is a small parameter, ๐ง๐,๐ = ๐(โ๐, ๐) be the fixed point. Let
๐ด = lim๐,๐โ 0
๐๐ง๐๐ง0
๐ โ ๐,๐, ๐ง0,๐ =๐11 ๐12๐21 ๐22
be the matrix, corresponding for motion out of grazing Condition 2. ๐12 > 0 , Tr ๐ด < โ1.
๐ต๐,๐ =๐๐ง๐๐ง0
๐,โ๐, ๐ง0, ๐
=โ๐ 0
โ๐ + 1 ๐ 0,0,0
๐0 ๐โ๐ (๐ธ + ๐(ยต+ฮธ))
.
๐ท๐๐,๐ ๐ง๐,๐ = ๐ด๐ต๐,๐(๐ธ + 0(๐ +\theta)) Trace is big if ๐12 โ 0, determinant is bounded. Eigenvalues ๐+ โ โ as ๐ โ 0 and ๐ โ โ 0 as ๐ โ 0. Corresponding eigenvectors:
๐ข+ =๐21๐22 + O(๐ + ๐),
๐ขโ = 01 + O(๐ + ๐).
Theorem 2. Let Conditions 1 and 2 be satisfied. Then there exist values ๐0,๐0 such that for all ๐ โ 0, ๐0 ,๐ โ (0,๐0) there exist a natural number m and a compact set ๐พ๐,๐, invariant with respect to ๐๐,๐ and such that the following statements are true. 1) There exists a neighborhood ๐๐,๐ of the set ๐พ๐,๐ such that the reduction ๐๐,๐|๐๐,๐ is a local diffeomorphism. The invariant set ๐พ๐,๐ is hyperbolic 2) The invariant set ๐พ๐,๐ of the diffeomorphism ๐๐,๐|๐๐,๐ 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.
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