Derivation of border-collision maps from limit cycle bifurcations

Transcription

1 Derivation of border-collision maps from limit cycle bifurcations Alan Champneys Department of Engineering Mathematics, University of Bristol Mario di Bernardo, Chris Budd, Piotr Kowalczyk Gabor Licsko,... Imperial 19/6/9 p. 1

2 Contents 1: discontinuity-induced bifurcation Imperial 19/6/9 p. 2

3 Contents 1: discontinuity-induced bifurcation 2: grazing bifurcation in impacting systems Ex. i. a rattling hydraulic valve Imperial 19/6/9 p. 2

4 Contents 1: discontinuity-induced bifurcation 2: grazing bifurcation in impacting systems Ex. i. a rattling hydraulic valve 3: grazing & corner bifurcation in PWS systems Ex. ii. extended model for stick-slip Ex. iii. DC-DC converter Imperial 19/6/9 p. 2

5 Contents 1: discontinuity-induced bifurcation 2: grazing bifurcation in impacting systems Ex. i. a rattling hydraulic valve 3: grazing & corner bifurcation in PWS systems Ex. ii. extended model for stick-slip Ex. iii. DC-DC converter 4: grazing-sliding bifurcation in Filippov systems Ex. iv. a simple dry friction oscillator Imperial 19/6/9 p. 2

6 Contents 1: discontinuity-induced bifurcation 2: grazing bifurcation in impacting systems Ex. i. a rattling hydraulic valve 3: grazing & corner bifurcation in PWS systems Ex. ii. extended model for stick-slip Ex. iii. DC-DC converter 4: grazing-sliding bifurcation in Filippov systems Ex. iv. a simple dry friction oscillator 5: Conclusion Imperial 19/6/9 p. 2

7 Contents 1: discontinuity-induced bifurcation 2: grazing bifurcation in impacting systems Ex. i. a rattling hydraulic valve 3: grazing & corner bifurcation in PWS systems Ex. ii. extended model for stick-slip Ex. iii. DC-DC converter 4: grazing-sliding bifurcation in Filippov systems Ex. iv. a simple dry friction oscillator 5: Conclusion Piecewise-Smooth Dynamical Systems: Theory & Applications dibernardo, Budd, C. & Kowalczyk Springer Jan SIAM review Dec 8 Imperial 19/6/9 p. 2

8 three types of nonsmoothness F Impacting systems: x Imperial 19/6/9 p. 3

9 three types of nonsmoothness F Impacting systems: x Imperial 19/6/9 p. 3

10 three types of nonsmoothness F Impacting systems: x Nonlinear Nonsmooth Piecewise smooth continuous systems: Imperial 19/6/9 p. 3

11 three types of nonsmoothness F Impacting systems: x Nonlinear Nonsmooth Piecewise smooth continuous systems: Y X Y X Y X a tlz b tgz T R P T P al Q Q ug ug Imperial 19/6/9 p. 3

12 three types of nonsmoothness F Impacting systems: x Nonlinear Nonsmooth Piecewise smooth continuous systems: F F 1 x Filippov systems: x= F 2 Imperial 19/6/9 p. 3

13 three types of nonsmoothness F Impacting systems: x Nonlinear Nonsmooth Piecewise smooth continuous systems: F F 1 x Filippov systems: x= F 2 Imperial 19/6/9 p. 3

14 (in)formalisms for nonsmooth system A piecewise smooth (PWS) system is set of ODEs ẋ = F i (x,µ), if x S i, Imperial 19/6/9 p. 4

15 (in)formalisms for nonsmooth system A piecewise smooth (PWS) system is set of ODEs ẋ = F i (x,µ), if x S i, discontinuity set Σ ij := S i S j is R (n 1) -dim manifold S j S i. Each F i smooth in S i generates flow Φ i (x,t) Σ ij S j x(t) S i Imperial 19/6/9 p. 4

16 (in)formalisms for nonsmooth system A piecewise smooth (PWS) system is set of ODEs ẋ = F i (x,µ), if x S i, discontinuity set Σ ij := S i S j is R (n 1) -dim manifold S j S i. Each F i smooth in S i generates flow Φ i (x,t) Σ ij S j x(t) S i Degree of smoothness of x Σ ij is order of 1st non-zero term in Taylor expansion of Φ i (x,t) Φ j (x,t) Imperial 19/6/9 p. 4

17 (in)formalisms for nonsmooth system A piecewise smooth (PWS) system is set of ODEs ẋ = F i (x,µ), if x S i, discontinuity set Σ ij := S i S j is R (n 1) -dim manifold S j S i. Each F i smooth in S i generates flow Φ i (x,t) Σ ij S j x(t) S i Degree of smoothness of x Σ ij is order of 1st non-zero term in Taylor expansion of Φ i (x,t) Φ j (x,t) Imperial 19/6/9 p. 4

18 impacting systems: deg. need reset map x R ij (x,µ), if x Σ ij Imperial 19/6/9 p. 5

19 impacting systems: deg. need reset map x R ij (x,µ), if x Σ ij PWS continuous systems: deg. 2 i.e. F i (x) = F j (x) but k 1 s.t. dk F i dx k dk F j dx k Imperial 19/6/9 p. 5

20 impacting systems: deg. need reset map x R ij (x,µ), if x Σ ij PWS continuous systems: deg. 2 i.e. F i (x) = F j (x) but k 1 s.t. dk F i dx k dk F j dx k Filippov systems deg. 1. Have possibility of sliding motion. E.g. if Σ ij := {H(x) = }, (H x F 1 ) (H x F 2 ) <. F 1 (a) (b) Σ F 2 Imperial 19/6/9 p. 5

21 impacting systems: deg. need reset map x R ij (x,µ), if x Σ ij PWS continuous systems: deg. 2 i.e. F i (x) = F j (x) but k 1 s.t. dk F i dx k dk F j dx k Filippov systems deg. 1. Have possibility of sliding motion. E.g. if Σ ij := {H(x) = }, (H x F 1 ) (H x F 2 ) <. F 1 (a) (b) Σ F 2 Imperial 19/6/9 p. 5

22 bifurcation All smooth bifurcations can occur in PWS systems (because Poincaré map is typically analytic!) Imperial 19/6/9 p. 6

23 bifurcation All smooth bifurcations can occur in PWS systems (because Poincaré map is typically analytic!) Also discontinuity induced bifurcations (DIB) where invariant sets have non-structurally stable interaction with a Σ ij. Imperial 19/6/9 p. 6

24 bifurcation All smooth bifurcations can occur in PWS systems (because Poincaré map is typically analytic!) Also discontinuity induced bifurcations (DIB) where invariant sets have non-structurally stable interaction with a Σ ij. Can lead to classical (topological) bifurcation or not (a) (b) S 1 S 1 S 2 S 2 S 3 S 3 S 4 S 4 Imperial 19/6/9 p. 6

25 bifurcation All smooth bifurcations can occur in PWS systems (because Poincaré map is typically analytic!) Also discontinuity induced bifurcations (DIB) where invariant sets have non-structurally stable interaction with a Σ ij. Can lead to classical (topological) bifurcation or not (a) (b) S 1 S 1 S 2 S 2 S 3 S 3 S 4 S 4 idea topological DIB PW structural stability Imperial 19/6/9 p. 6

26 this talk: periodic orbit DIBs Goal: Catalogue & unfold codim-1 possibilities. E.g. grazing bifurcation : [Nordmark] Imperial 19/6/9 p. 7

27 this talk: periodic orbit DIBs Goal: Catalogue & unfold codim-1 possibilities. E.g. grazing bifurcation : [Nordmark] Derive map close to DIB as composition of smooth Poincaré map P π and discontinuity mapping PDM Π P π PDM p(t) Σ : {H(x) = } Imperial 19/6/9 p. 7

28 this talk: periodic orbit DIBs Goal: Catalogue & unfold codim-1 possibilities. E.g. grazing bifurcation : [Nordmark] Derive map close to DIB as composition of smooth Poincaré map P π and discontinuity mapping PDM Π P π PDM p(t) Σ : {H(x) = } Use results on border collisions of maps to classify dynamics [Feigin] [Yorke, Banergee et al] Imperial 19/6/9 p. 7

29 this talk: periodic orbit DIBs Goal: Catalogue & unfold codim-1 possibilities. E.g. grazing bifurcation : [Nordmark] Derive map close to DIB as composition of smooth Poincaré map P π and discontinuity mapping PDM Π P π PDM p(t) Σ : {H(x) = } Use results on border collisions of maps to classify dynamics [Feigin] [Yorke, Banergee et al] Nb. piecewise linear (PWL) flow PWL map Imperial 19/6/9 p. 7

30 2. Grazing bifurcation in impact systems cf. Theory of impact oscillators: [Peterka] 197s, [Thompson & Ghaffari], [Shaw & Holmes] 198s, [Budd et al], [Nordmark] 199s. Consider single impact surface Σ := {H(x) = } with impact law: x + = R(x ) = x + W(x )H x F(x ) W is smooth function and H x F(x ) is velocity. e.g. W = (1 + r)h x Newton s restitution law More complex impact laws are possible, e.g. impact with friction (see later) Imperial 19/6/9 p. 8

31 a motivating example Oscillations of a pressure relief valve. Licsko, C. & Hös noise at 375Hz at a range of flow speeds Amplitude x n [1/s] f [Hz] Amplitude x n [1/s] f [Hz] 5 Imperial 19/6/9 p. 9

32 a simple (dimensionless) model ẏ 1 = y 2 ẏ 2 = κy 2 (y 1 + δ) + y 3 ẏ 3 = β (q y 3 y 1 ) y 1 > valve displacement; y 2 valve velocity, y 3 pressure β valve spring stiffness; δ valve pre-stress q, flow rate; κ, fluid damping at y 1 = apply a Newtonian restitution law: y 2 (t + ) = ry 2 (t ) Low κ limit cycles between 2 Hopf bifs q = q min, q max. Imperial 19/6/9 p. 1

33 brute force numerics κ = 1.25, β = 2, δ = 1 (representative of experiment) pres vel 1 2 pres disp disp vel 5 25 pres vel 5 disp q pres 1 5 pres vel 6 disp 8 1 vel disp Chaotic rattling due to Grazing events at q 7.54, 5.95 Imperial 19/6/9 p. 11

34 discontinuity mapping (PDM) Π.1 x H x 4 x 5 x 2 x 3 x 6 Σ x H x F PDM: x 1 x 5 maps Poincaré section Π = {H x F(x) = } to itself Computes correction to trajectory as if Σ were absent Imperial 19/6/9 p. 12

35 explicit form of PDM cf. [Fredrickson & Nordmark] { x if H(x) x x + β(x,y)y if H(x) < } where β = 2a where y = H and ( W (H xf) x W a ) F + O(y 2 ), a(x) = d 2 H/dt 2 = (H x F) x F = H xx FF + H x F x F square root map Imperial 19/6/9 p. 13

36 Proof is by Taylor expansion of flow in (x,y) and IFT Imperial 19/6/9 p. 14

37 Proof is by Taylor expansion of flow in (x,y) and IFT Use PDM to correct non-grazing Poincaré map P π : P N = P π P PDM P N (x,µ) = Nx + Mµ + O(x 2,µ 2 ) if H(x) > where H x (x)e = = Nx + Mµ + Ey + O(x 2,µ 2,y 2 ) H(x) < Imperial 19/6/9 p. 14

38 Proof is by Taylor expansion of flow in (x,y) and IFT Use PDM to correct non-grazing Poincaré map P π : P N = P π P PDM P N (x,µ) = Nx + Mµ + O(x 2,µ 2 ) if H(x) > where H x (x)e = = Nx + Mµ + Ey + O(x 2,µ 2,y 2 ) H(x) < Note since y = H(x), linear term for y < is not Nx has correction too. Imperial 19/6/9 p. 14

39 Proof is by Taylor expansion of flow in (x,y) and IFT Use PDM to correct non-grazing Poincaré map P π : P N = P π P PDM P N (x,µ) = Nx + Mµ + O(x 2,µ 2 ) if H(x) > where H x (x)e = = Nx + Mµ + Ey + O(x 2,µ 2,y 2 ) H(x) < Note since y = H(x), linear term for y < is not Nx has correction too. But in R n only have conditions on N, M, and E for given periodic orbit to exist [Nordmark 21] Imperial 19/6/9 p. 14

40 Quasi-1D behaviour Simplest case: if λ 1 > is real leading eigenvalue of N. then dynamics is determined by 1D map: f(x) = µ x + λ 1 µ x < µ, f(x) = λ 1 x x > µ, 1. 2/3 < λ 1 < 1: robust chaotic attractor size µ. 2. If 1/4 < λ 1 < 2/3 alternating series of chaos and period-n orbits, n as µ. 3. < λ 1 < 1/4: just period-adding cascade Imperial 19/6/9 p. 15

41 2D normal form Again everything depends on eigenvalues of N. WLOG use normal form with N = ( τ 1 δ a 2 1 τ ), E = 3 ( 1 m = n=55 3 n=4 m = 4 n=3 m = 3 ) b, C T = [1 ], (1) a 2 1 δ maximal period m of bifurcating orbit of type b n 1 a Imperial 19/6/9 p. 16

42 example N = ( ), M = ( ), E = ( 1 ), C T = (1 ), D =. λ 1,2 =.1475 ±.4731i, τ =.295 and δ = Analysis m=3 maximal orbit bifurcates subcritically (a) (b) x.2 1 x x 1.25 x µ µ.5, µ.15 µ Imperial 19/6/9 p. 17

43 return to Ex.i: valve rattle Two grazing bifurcation events q = 7.54, q = 5.95 q = 5.95: λ 1 < m = 2, subcritical q = 7.54: λ 1 =.8537 jump to chaos. Iterate map.25 y [ ] Imperial 19/6/9 p. 18

44 3. DIBs in PWS continuous systems Simplest case: grazing bifurcation Analyse using discontinuity mapping: PDM Π 1 2 ε Π Π S S + Σ - -x δ x f εx 1 11 ts t 1 t 2 1 x^ Discontinuity Map tf Imperial 19/6/9 p. 19

45 general results for PDM Map discontinuity degree F jump Uniform Case Non-uniform δ-function x square-root 1 bounded F - O(1/2) 2 C F x O(3/2) O(3/2)-type 3 C 1 F xx O(5/2) O(3/2)-type F(x) continuous at Σ no immediate jump in attractor Imperial 19/6/9 p. 2

46 1D map with O(3/2) singularity Consider [Halse, di Bernardo et al] { νx µ x x νx + ηx 3/2 µ x > < ν < 1 simple fixed point. No bifurcation at µ =. but with η < get nearby fold at µ = 4(1 ν)3 3η 2 closer than smooth fold if ν 1) (much Imperial 19/6/9 p. 21

47 Also, get period-adding cascades. E.g. for γ = 3/2, η = 1. Then stable L k 1 R orbits exist for 8(ν k + 1) 3 12(1 ν k )(1 + ν k ) 2 ( ν k ) 2 27ν 2(k 1) (1 + ν + ν ν k 1 ) < µ < 1 ν k 1. (ν 1) case γ = 2: 5 45 k=6 k= k=5 3 βµ 25 2 k= k=3 5 k=2.2.4 α Imperial 19/6/9 p. 22

48 Ex.ii: A realistic stick-slip oscillator Dankowicz 1999 y 3 y 1 y 5 U = 1 y 1 - horizontal displacement; y 2 = ẏ 1 y 3 - vertical displacement; y 4 = ẏ 3 y 5 - shear deformation of asperities belt velocity U = 1 Imperial 19/6/9 p. 23

49 equations of motion ẏ 1 = y 2, ẏ 2 = 1 + ẏ 3 = y 4, ẏ 4 = sy 3 + [ 1 γu 1 y 4 y 2 + βu 2 (1 y 4 ) 2 ] K(y 1 ) e y1 d, ẏ 5 = 1 τ [(1 y 4) 1 y 4 y 5 ], gσ U e d [ µ(y 5 e y 1 1) + αu 2 S(y 1,y 4 ) ], where K(y 1 ) = 1 y 1 d, S(y 1,y 4 ) = (1 y 4 ) 1 y 4 K(y 1 )e y d. PWS continuous across discontinuity boundary y 4 = 1. Imperial 19/6/9 p. 24

50 grazing bifurcation analysis Dankowicz & Nordmark 2 3 successive zooms of bifurcation diagram: Imperial 19/6/9 p. 25

51 grazing bifurcation analysis Dankowicz & Nordmark 2 Simulation (left) and iteration of DM (right) in local map co-ordinates y Imperial 19/6/9 p. 25

52 Another DIB: boundary intersection (a) (b) Σ 1 F 1 Σ 2 S 1 C F 2 F 2 F 1 F 3 F 4 Σ 1 S 2 Σ 2 C (a) Boundary-intersection crossing (b) Corner collision - special case Imperial 19/6/9 p. 26

53 Analysis via discontinuity mapping H 2 = H 1 = F 1 x = F 2 PDM F 4 PDM F 3 H 2,x H 1,x locally piecewise-linear continuous map PDM(x) = ( ) x + x 2 F F 12 F 11 2 F 21 F 1 + O ( x 2) if x 2 > ( ) x + x 2 F F 32 F 31 4 F 41 F 3 + O ( x 2) if x 2 <, Imperial 19/6/9 p. 27

54 Ex.iii DC-DC Buck power converter [di Bernardo, C. & Budd 98], To overcome sliding use relay control on ramp signal v r Vr Ramp generator S open if vin1 c > v r S closed if in2 v c < v r S E D + v c Vc + + g i Ii Ii L Vv C g v V v R desired V Output V battery OFF battery ON t ẋ = a 1 + b 1 x c 1 y, ẏ = a 2 c 2 y + dh(σ(t) x), Imperial 19/6/9 p. 28

55 output voltage against input Imperial 19/6/9 p. 29

56 Corner-collision V (a) E= t V (b) E= Comparison between numerical and theoretical maps y(5t) (a) E=E (b) E=E y() (c) E - E =-.25 x(5t) y() (d) E - E =.25 t y(5t) y() y(5t) y() Imperial 19/6/9 p. 3

57 4. Sliding DIBs in Filippov systems Kowalczyk, Nordmark, dibernardo Four possible DIB involving collision of limit cycle with sliding boundary ˆΣ ; see Mike Jeffrey s talk S + C B A S + A B S (a) crossing sliding (b) grazing sliding C S + S + A C B (c) S switching sliding B (d) C adding sliding A Imperial 19/6/9 p. 31

58 Unfold with discontinuity mapping Di Bernardo, Kowalczyk, Nordmark Bifurcation type DM leading-order term Map singularity crossing sliding ε 2 + O(ε 3 ) 2 grazing sliding ε + O(ε 3/2 ) 1 switching sliding ε 3 + O(ε 4 ) 3 adding sliding ε 2 + O(ε 5/2 ) 2 Maps are non-invertible on one side Only grazing sliding jump in attractor Imperial 19/6/9 p. 32

59 Poincaré maps at grazing-sliding Kowalczyk 25 Sliding non-invertible Poincaré map. 2D case: ( x 1 x 2 ) ( ( τ 1 1 δ 1 τ 2 1 ) ( ) ( x 1 x 2 x 1 x 2 Robust chaotic attractors on lines ) ) + + ( ( 1 1 ) ) µ, if x 1, µ, if x 1. Imperial 19/6/9 p. 33

60 e.g. N1 = (a) x1.1 x 1 (c), N2 = τ2 1! µ. x 1 (b).2.2.1! µ x 1 (d) Imperial 19/6/9 p. 34

61 Ex.iv: a simple stick-slip oscillator u + u = C(1 u ) + A cos(νt), C(v) = α sgn(v) α1 v + α2 v 3 grazing-sliding of period-2 cycle for α = alpha1 = 1.5, α2 =.45, A =.1, ν = y (a) 1 (b) S2 y S ν y Imperial 19/6/9 p. 35

62 6. Conclusion periodic-orbit DIBs analysed via discontinuity mappings nonsmooth global Poincaré maps Imperial 19/6/9 p. 36

63 6. Conclusion periodic-orbit DIBs analysed via discontinuity mappings nonsmooth global Poincaré maps degree of smoothness of flow same degree of smoothness of map Imperial 19/6/9 p. 36

64 6. Conclusion periodic-orbit DIBs analysed via discontinuity mappings nonsmooth global Poincaré maps degree of smoothness of flow same degree of smoothness of map specific DIBs different forms of map, e.g: - multiple collision impact map with a jump Budd - grazing in impact systems square root map - grazing in PWS continuous 3/2-singularity - boundary-intersection crossing in Filippov PWL - grazing-sliding in Filippov singular PWL Kowalczyk Imperial 19/6/9 p. 36

65 6. Conclusion periodic-orbit DIBs analysed via discontinuity mappings nonsmooth global Poincaré maps degree of smoothness of flow same degree of smoothness of map specific DIBs different forms of map, e.g: - multiple collision impact map with a jump Budd - grazing in impact systems square root map - grazing in PWS continuous 3/2-singularity - boundary-intersection crossing in Filippov PWL - grazing-sliding in Filippov singular PWL Kowalczyk provides natural explanation of observed behaviour Imperial 19/6/9 p. 36

66 6. Conclusion periodic-orbit DIBs analysed via discontinuity mappings nonsmooth global Poincaré maps degree of smoothness of flow same degree of smoothness of map specific DIBs different forms of map, e.g: - multiple collision impact map with a jump Budd - grazing in impact systems square root map - grazing in PWS continuous 3/2-singularity - boundary-intersection crossing in Filippov PWL - grazing-sliding in Filippov singular PWL Kowalczyk provides natural explanation of observed behaviour lots of ongoing work; - catastrophic bifurcations in Filippov systems Jeffrey - DIBs in impacts with frictions Dankowicz & Nordmark Imperial 19/6/9 p. 36