Derivation of border-collision maps from limit cycle bifurcations

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

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

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

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

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

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

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

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

(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

(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

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

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

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

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

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

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

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

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

a motivating example Oscillations of a pressure relief valve. Licsko, C. & Hös noise at 375Hz at a range of flow speeds Amplitude 4 3 2 1 3 x 1 4 2 4 3 1 2 1 n [1/s] f [Hz] Amplitude 5 2 1.5 1.5 15 x 1 6 1 4 3 5 2 1 n [1/s] f [Hz] 5 Imperial 19/6/9 p. 9

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

brute force numerics κ = 1.25, β = 2, δ = 1 (representative of experiment) 8 7 6 5 pres 4 3 2 5 1 2 2 4 6 8 1 1 vel 1 2 pres 4 3 2 disp 1 1 4 35 3 1 2 3 4 disp 5 6 1 5 vel 5 25 pres 2 15 1 5 1 1 1 2 3 4 5 1 5 vel 5 disp q 2 6 15 5 pres 1 5 pres 4 3 2 4 1.5 2 2 2 1.5 2 2 4 1 4 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

discontinuity mapping (PDM) Π.1 x H x 4 x 5 x 2 x 3 x 6 Σ x 1.1.6.4 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

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

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

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

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

2D normal form Again everything depends on eigenvalues of N. WLOG use normal form with N = ( τ 1 δ a 2 1 τ.9.8.7.6.5.4 5 ), E = 3 ( 1 m = n=55 3 n=4 m = 4 n=3 m = 3 ) b, C T = [1 ], (1).3.2 5 1 1.1 4 2 1.5.5.5 1.5 a 2 1 δ 1 2 1 maximal period m of bifurcating orbit of type b n 1 a Imperial 19/6/9 p. 16

example N = (.4663 1.4337.227.1713 ), M = (.5337.2277 ), E = ( 1 ), C T = (1 ), D =. λ 1,2 =.1475 ±.4731i, τ =.295 and δ =.2456. Analysis m=3 maximal orbit bifurcates subcritically.6.35.5 (a) (b).3.4.3 x.2 1 x 1.1.1.2 x 1.25 x.2 1.15.1.5.3.5 µ.4.5.4.3.2.1.1.2.3.4.5 µ.5,.5.1.5 µ.15 µ.5.1.15 Imperial 19/6/9 p. 17

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 [ ].2.15.1.5.5.1 7.4 7.45 7.5 7.55 7.6 7.65 Imperial 19/6/9 p. 18

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

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

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

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 2 +... + ν k 1 ) < µ < 1 ν k 1. (ν 1) case γ = 2: 5 45 k=6 k=7 4 35 k=5 3 βµ 25 2 k=4 15 1 k=3 5 k=2.2.4 α Imperial 19/6/9 p. 22

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

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 1 1 + d. PWS continuous across discontinuity boundary y 4 = 1. Imperial 19/6/9 p. 24

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

grazing bifurcation analysis Dankowicz & Nordmark 2 Simulation (left) and iteration of DM (right) in local map co-ordinates y 4 1 4 Imperial 19/6/9 p. 25

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

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

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

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

Corner-collision V 12.4 12.3 12.2 12.1 12 11.9 11.8 11.7 (a) E=19.9786656 11.6.5.1.15.2.25 t V 13 12.8 12.6 12.4 12.2 12 11.8 (b) E=53.51 11.6.5.1.15.2.25 Comparison between numerical and theoretical maps y(5t).14.12.1.8.6.4.2 -.2 -.1 -.5.5.1.1 (a) E=E (b) E=E y() (c) E - E =-.25 x(5t).1.5 -.5 -.1 -.15 -.1 -.5.5.1.1 y() (d) E - E =.25 t y(5t).8.6.4.2 -.2 -.4 -.2.2.4 y() y(5t).8.6.4.2 -.2 -.4 -.2.2.4 y() Imperial 19/6/9 p. 3

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

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

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

e.g. N1 =.5 1.6 (a) x1.1 x 1 (c), N2 = τ2 1! µ. x 1 (b).2.2.1! µ.4.1.1.1 x 1 (d) Imperial 19/6/9 p. 34

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 = 1.777997 y (a) 1 (b) S2 y 2.25 1 2.251 2.25 2.2499 S1 2.2498 1.777 1.7774 ν 1.7778.94 1.7782 1.3 1.7782.94 1.3 1.5 y 1.5 1.7 1.7 Imperial 19/6/9 p. 35

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

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

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