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 – the most general model, Babitsky, 98 review,…

2. Grazing bifurcation. Nordmark, 91, Ivanov, 94, Dankowitz et al 02, Budd et al, 08,…

Fig. 1. An example of single degree-of-freedom vibro-impact system

1 – point mass, 2 – spring, 3 - delimiter, 4 – damping element

𝐽 = [0,𝜇∗] - values of parameter; 𝑓 𝑡, 𝑥, 𝑦, 𝜇 :𝑅3 × 𝐽 → 𝑅 - a 𝐶2 smooth function, . 𝑓 𝑡 + 𝑇, 𝑥,𝑦, 𝜇 ≡ 𝑓(𝑡, 𝑥,𝑦, 𝜇) Consider a system of 2 o.d.e.s �̇� = 𝑦; �̇� = 𝑓(𝑡, 𝑥,𝑦, 𝜇) (1) Denote the set of corresponding right hand sides by 𝑋𝑓 = 𝑋𝑓(𝐽,𝑛,𝑇) Denote 𝑧 = (𝑥,𝑦)

. . .

Eq. (1) is satisfied for 𝑥 > 0. If 𝑥 = 0 the following impact conditions take place: If 𝑥 𝑡0 − 0 = 0 then 𝑦 𝑡0 + 0 = −𝑟𝑦 𝑡0 − 0 , 𝑥 𝑡0 + 0 = 𝑥(𝑡0 − 0) (2) . Here 𝑟 = 𝑟 𝜇 ∈ 0,1 is a 𝐶2 smooth function. Denote by (*) the VIS defined by Eq. (1) and impact conditions (2) Let .

[ ] ( ) ( ){ }0,,,0,,0,0 111 >=−××∞=Λ yryyJ µµR

A function 𝑧 𝑡 = col(𝑥 𝑡 ,𝑦(𝑡)) is called solution of the VIS (*) with a finite number of impacts on an interval (𝑎, 𝑏) if the following conditions are satisfied: there exist instants 𝑡0, … , 𝑡𝑁+1 where 𝑡0 = 𝑎, 𝑡𝑁+1 = 𝑏 such that 1) The components 𝑥 𝑡 is continuous, points 𝑡1, … , 𝑡𝑁 are all points of discontinuity for 𝑦(𝑡). 2) The function 𝑥 𝑡 is non-negative and 𝑡1, … , 𝑡𝑁 are all its zeros. 3) 𝑦 𝑡𝑘 + 0 = −𝑟 𝜇 𝑦 𝑡𝑘 − 0 for all 𝑘 = 1, … ,𝑁 4) The function 𝑧 𝑡 is a solution of system (1) on every segment 𝑡𝑘 , 𝑡𝑘+1 .

, , , .

Lemma 1. Let 𝑧 𝑡 = col(𝑥 𝑡 ,𝑦(𝑡)) - be a solution of (*), corresponding to the value 𝜇0 of the parameter and to the initial data (𝑡0, 𝑥0,𝑦0). Let 𝑥02 + 𝑦02 > 0 and let the considered solution be defined on 𝑡−, 𝑡+ . Let the function 𝑥(𝑡) have exactly 𝑁 zeros 𝑡1, … , 𝑡𝑁 on the interval (𝑡−, 𝑡+). Suppose that 𝑦 𝑡𝑘 − 0 ≠ 0 for all k. Then for any 𝑡 ∈ (𝑡−, 𝑡+) ∖ {𝑡1, … , 𝑡𝑁} there is a neighborhood 𝑈 of the point 𝑡0, 𝑥0,𝑦0, 𝜇0 such that the mapping 𝑧(𝑡, 𝑡′, 𝑥′,𝑦′,𝜇𝜇) is smooth with respect to (𝑡′, 𝑥′,𝑦′, 𝜇𝜇) from 𝑈. All corresponding solutions have exactly 𝑁 impacts 𝑡𝑘(𝑡𝑡, 𝑥𝑡,𝑦𝑡,𝜇𝑡) over (𝑡−, 𝑡+). Impact instants 𝑡𝑘(𝑡𝑡, 𝑥𝑡,𝑦𝑡, 𝜇𝑡) and corresponding velocities 𝑦(𝑡𝑘 𝑡𝑡, 𝑥𝑡,𝑦𝑡, 𝜇𝑡 − 0) are smooth functions of 𝑡𝑡, 𝑥𝑡,𝑦𝑡,𝜇𝑡 in 𝑈.

This is a space 𝑋, consisting of 𝐶1smooth pairs 𝑓, 𝑟 . The topology is minimal one there for any

pair (𝑓0, 𝑟0) and any 𝑅 > 0 the set

� 𝑓, 𝑟 : sup𝑡,𝑧,𝜇 ∈𝑀

( 𝑓 𝑡, 𝑧, 𝜇 − 𝑓0 𝑡, 𝑧, 𝜇 + �𝜕𝑓𝜕𝑧

𝑡, 𝑧, 𝜇

−𝜕𝑓0𝜕𝑧

𝑡, 𝑧, 𝜇 � + �𝜕𝑓𝜕𝜇

𝑡, 𝑧, 𝜇

−𝜕𝑓0𝜕𝜇

𝑡, 𝑧, 𝜇 � + |𝑟 𝜇 − 𝑟0(𝜇)|) < 𝑅�

is open.

“Stiff” model with a perturbation �̇� = 𝑦; �̇� = 𝑓 𝑡, 𝑥,𝑦 + 𝑔 𝑡, 𝑥,𝑦 ; 𝑔 𝐶1 < 𝜀 We assume conditions (2) take place. “Soft” model with a perturbation. �̇� = 𝑦; �̇� = 𝑓 𝑡, 𝑥,𝑦 + 𝑔 𝑡, 𝑥,𝑦 + ℎ(𝜅, 𝑡, 𝑥,𝑦); 𝑔 𝐶1 < 𝜀 ℎ 𝜅, 𝑡, 𝑥,𝑦 = −2𝛼𝜅𝜒− 𝑥 𝑦 − 1 + 𝛼2 𝜅2𝜒− 𝑥 x, 𝛼 = −log 𝑟/𝜋. χ̲ =1 if x <0 and χ̲ =0 if x≥0; addition of functions h replaces the conditions of impact. Consider stroboscopic (period shift) mappings for both

of these models. Call them 𝑆𝑔,𝑟 and 𝑆𝑔,𝑟,𝜅 Hyperbolic invariant sets persist for small 𝜀 and big 𝜅 and

small changes in 𝑟.

1kδ

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 – periodic solutions 𝜑 𝑡, 𝜇 = (𝜑𝑥 𝑡,𝜇 ,𝜑𝑦 𝑡, 𝜇 ) of system (*), satisfying following properties: 1) For any 𝜇 ∈ 𝐽 the component 𝜑𝑥 𝑡,𝜇 has exactly 𝑁 + 1 zeros 𝑡0 𝜇 , … , 𝑡𝑁(𝜇) over the period [0,𝑇). 2) Velocities 𝑌𝑘(𝜇) = 𝑦(𝑡𝑘(𝜇) − 0) are such that 𝑌𝑘 𝜇 ≠ 0 for all 𝜇 ≠ 0, 𝑌0(0) = 0, 𝑌𝑘 0 ≠ 0 for 𝑘 = 1, … ,𝑁. 3) Instants 𝑡𝑘(𝜇) and impact velocities 𝑌𝑘(𝜇) continuously depend on the parameter 𝜇 ∈ 𝐽.

, ,

.

.

Condition 1.

Fig. 4. Curve Г. Homoclinic points arizing due to bent of unstable (а) or stable (b) manifold. Here we consider case 1 – presence of periodic motions with a

low impact velocity.

(a) (b)

Period shift map 𝑆𝜇,𝜃(𝑧0) = 𝑧(𝑇 − 𝜃,−𝜃, 𝑧0, 𝜇) Here θ > 0 is a small parameter, 𝑧𝜇,𝜃 = 𝜑(−𝜃, 𝜇) be the fixed point. Let

𝐴 = lim𝜇,𝜃→ 0

𝜕𝑧𝜕𝑧0

𝑇 − 𝜃,𝜃, 𝑧0,𝜇 =𝑎11 𝑎12𝑎21 𝑎22

be the matrix, corresponding for motion out of grazing Condition 2. 𝑎12 > 0 , Tr 𝐴 < −1.

𝐵𝜇,𝜃 =𝜕𝑧𝜕𝑧0

𝜃,−𝜃, 𝑧0, 𝜇

=−𝑟 0

−𝑟 + 1 𝑓 0,0,0

𝑌0 𝜇−𝑟 (𝐸 + 𝑂(µ+θ))

.

𝐷𝑆𝜇,𝜃 𝑧𝜇,𝜃 = 𝐴𝐵𝜇,𝜃(𝐸 + 0(𝜇 +\theta)) Trace is big if 𝑎12 ≠ 0, determinant is bounded. Eigenvalues 𝜆+ → ∞ as 𝜇 → 0 and 𝜆 − → 0 as 𝜇 → 0. Corresponding eigenvectors:

𝑢+ =𝑎21𝑎22 + O(𝜇 + 𝜃),

𝑢− = 01 + O(𝜇 + 𝜃).

Theorem 2. Let Conditions 1 and 2 be satisfied. Then there exist values 𝜇0,𝜃0 such that for all 𝜇 ∈ 0, 𝜇0 ,𝜃 ∈ (0,𝜃0) there exist a natural number m and a compact set 𝐾𝜇,𝜃, invariant with respect to 𝑆𝜇,𝜃 and such that the following statements are true. 1) There exists a neighborhood 𝑈𝜇,𝜃 of the set 𝐾𝜇,𝜃 such that the reduction 𝑆𝜇,𝜃|𝑈𝜇,𝜃 is a local diffeomorphism. The invariant set 𝐾𝜇,𝜃 is hyperbolic 2) The invariant set 𝐾𝜇,𝜃 of the diffeomorphism 𝑆𝜇,𝜃|𝑈𝜇,𝜃 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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