The reduction to invariant cones for nonsmooth systems Tassilo Küpper Mathematical Institute University of Cologne kuepper@math.uni-koeln.de International Workshop on Resonance Oscillations and Stability of Nonsmooth Systems 16 June to 25 June, 2009, Imperial College London, United Kingdom.
Aims Determine invariant manifold for non-smooth system. Extend concept of center manifold approach to non-smooth system. Bifurcation analysis of reduced system. Ex. Alteration of stable/unstable systems(damping/ excitation) Ex. Mechanics - Cell interaction: Neural dynamics - Electrical circuits. Results Stable+Stable Unstable [Carmona,et al (2005)] Generalized Hopf Bifurcation: Generation of Periodic orbits n=2 : Küpper and Moritz [2002], Beyn, Küpper and Zou (2006) n > 2 : Küpper (2008), Küpper and Hosham (2009). Reduction techniques (center manifold approach).
1. Smooth systems: Center manifold approach. Review ξ = f(ξ, λ), (ξ R n, λ R p ), (1) ξ = 0, stationary solution linearization ξ = A(λ)ξ + g(ξ, λ) A = f (0, λ) = ξ ( ) A1 0 ξ = 0 A 2 λ Ev ofa 1 : Reλ < 0. λ Ev ofa 2 : Reλ = 0. ( ) x. y
ẋ = A 1 x + g 1 (x, y, λ) (2) ẏ = A 2 y + g 2 (x, y, λ). Center manifold reduction h = h(y), h(0) = 0, h (0) = 0, x = h(y). (3) y Reduced system ( locally equivalent to (2)) ẏ = A 2 (λ)y + g 2 (h(y), y, λ), (4) Remark: dim y n, once (4) has been established the dynamics, stability and bifurcation behavior of (1) can be obtained by studying (4). First Prev Next Last Go Back Full Screen Close Quit
2. Non-smooth system: general analysis R n = N M i, disjoint (5) i=1 f i = R n R p R n. ξ = f i (ξ, λ), (ξ M i ). (6) ξ = 0, (7) stationary solution in some M i, but no interior point. Local reduction to equivalent system?
3. Simplified setting ξ = f + (ξ, λ), e T 1 ξ > 0, (R n +) f 0 (ξ, λ), e T 1 ξ = 0, (R n 0) f (ξ, λ), e T 1 ξ < 0, (R n ) (8)
Special cases on M 0 Transversal crossing [e T 1 f + (ξ, λ)][e T 1 f (ξ, λ)] > 0. Sliding [e T 1 f + (ξ, λ)][e T 1 f (ξ, λ)] < 0. Higher order conditions [Dieci and Lopez (2008,2009)]. Jumps in phase space ξ(t + ) = Rξ(t ). Ex. Impact Pendulum Ex. Bells S. Köker: Zur Dynamik des Glockenläutens, Diploma Thesis (2009).
Approximation by PWLS, (0 = f + (0, λ) = f (0, λ)) f ± = A ± (λ)ξ + g ± (ξ, λ), g ± smooth, h.o.t. (9) 4. PWLS (transversal) a. Def. of PWLS For fixed λ set, ξ = { A + ξ, e T 1 ξ > 0, A ξ, e T 1 ξ < 0, W ± > = {ξ M e T 1 f ± ξ > 0}, W ± < = {ξ M e T 1 f ± ξ < 0}, W ± 0 = {ξ M e T 1 f ± ξ = 0}. transverse crossing if ξ W < W + < or ξ W > W + >, (10)
b. Set up of Poincarè map Flow: e t (ξ)a (λ) ξ e t+ A + (λ) η For ξ W < W + < : t (ξ) = inf{t > 0 e T 1 ϕ (t, ξ) = 0}, P (ξ) := e t (ξ)a ξ For η W > W + >, t + (η) = inf{t > 0 e T 1 ϕ + (t, η) = 0}, P + (η) := e t+ (η)a + η. Return map P (ξ) := P + (P (ξ)) = e t+ (η)a + e t (ξ)a ξ. P (ξ) nonlinear t + and t nonlinear constant on half-rays t (rξ) = t (ξ)(0 < r < ) and t + (rξ) = t + (ξ)(0 < r < ).
Theorem 1. (T. Küpper (2008)) Let ξ M be an eigenvector of the nonlinear eigenvalue problem, P ( ξ) = µ ξ with some real and positive eigenvalue µ. Then there is an invariant cone for (10) If µ > 1, then the stationary solution 0 is unstable. If µ = 1, then the cone consists of periodic orbits. If µ < 1, then the stability of 0 depends on the stability of P with respect to the complementary directions. Remark (i) ξ is an eigenvector of P ξ ( ξ) with eigenvalue µ. If we assume that the remaining (n 2) eigenvalues λ 1,...,λ n 2 of P ξ ( ξ) satisfy λ j < min{1, µ}, (j = 1,..., n 2) then the invariant cone is attractive under the flow of (10).
Remark (ii); Interpretation Cone: 2-dim invariant surface. Smooth system: flat cone 5. Numerical Illustration 3D PWLS [+] system [ - ]system ẋ = λ + x w + y, : ẋ = (λ w )x 2w y + α(λ w mu )z ẏ = w + x + λ + y, ẏ = w x + (λ + w )y + αw z, ż = µ + z. ż = µ z
Figure 1: Attractive/ periodic / repulsive invariant cone for case ii (1), µ = 1.0, w + = w = 1, λ = 0.7832, t + = π, t = 3π/4, δ = 0.0, α = 0.5. (a)λ + = 0.7768, µ + = 0.7165. (b) λ + = 0.8103, µ + = 0.7500. (c) λ + = 0.8406, µ + = 0.7803. First Prev Next Last Go Back Full Screen Close Quit
6. Nonlinear perturbation of PWLS f ± = A ± ξ }{{} Basic P W LS + g ± (ξ) }{{} h.o.t. consider truncated version of (11); i.e for same r > 0 (11) g ± (ξ) 0, ( ξ r). Theorem 2 (Küpper,Hosham,Weiß (2009)). Reduction to invariant manifold Assume Ex. invariant cone for (PWLS), i.e. µ, ξ P ( ξ) = µ ξ. Ev of P ξ, µ, λ 1, λ 2,...λ n 2 λ j < min{1, µ}, (j = 1,..., n 2) Then Poincarè map P for (11) h : [0, ) R n 1, L := (y, h(y)) y 0, invariant cone under P h(y) = ξy + O(y 2 ) tangent to cone. First Prev Next Last Go Back Full Screen Close Quit
Remark Flow of (11) applied to L gives invariant cone-like manifold C. Attractive if λ j < min{1, µ}, j = 1,..., n 1. Bifurcation analysis reduced to C res. L. 6.1. Numerical example (Nonlinear system) [+] system [ - ]system ẋ = λ + x w + y, ẋ = (λ w )x 2w y ẏ = w + x + λ + y + υy 3, ẏ = w x + (λ + w )y + υy 3 ż = (µ + υ)y. ż = δw x δ(λ +w µ )y+(µ w δ)z.
Figure 2: Branch of instable periodic orbits for nonlinear example, µ = 1.0, w + = w = 1, λ = 1.0, δ = 0.5, α = 0.0. λ + = 1.0, µ + = 1.5. First Prev Next Last Go Back Full Screen Close Quit
Remark Existence of multiple cones Example: 3D PWLS [+] system [ - ]system ẋ = λ + x w + y, : ẋ = (λ w )x 2w y + α(λ w mu )z ẏ = w + x + λ + y, ẏ = w x + (λ + w )y + αw z, ż = µ + z. ż = µ z
Figure 3: 2 attractive invariant cones,λ + = 0.5, µ + = 0.1751 w + = w = 1, λ = 0.5, µ = 1.0, t + = π, δ = 0.0, α = 0.5. where t = π for a 1 and t = 0.5505 for a 2. First Prev Next Last Go Back Full Screen Close Quit
7. Six-dimensional non-smooth Brake-system A brake pad 1 on a rigid frame acts on a brake disc 2. Between brake pad and brake disc there is a relative displacement with constant velocity v, thus the frictional forces depend only on the normal force and the constant friction µ 1. The brake pad is equipped with three mechanical degrees of freedom modelled. Vertical movement x 1. Horizontal movement x 2. Rotation φ.
The Mathematical Model mẍ 1 + (d 1 + d 2 ) x 1 + b 2 (d1 d2) φ + (c 1 + c 2 )x 1 + b 2 (c 1 c 2 )φ = µ 2 sgn( x 1 a φ)c 3 x 2, (12) mẍ 2 (d 1 + d 2 )µ 1 x 1 µ 1b 2 (d1 d2) φ (c 1 + c 2 )µ 1 x 1 +c 3 x 2 + µ 1b 2 (c 1 c 2 )φ = 0, (13) j φ + ( b 2 (d 1 d 2 ) (d 1 + d 2 )hµ 1 ) x 1 + ( b2 4 (d1 + d2) + bhµ 1 2 (d 2 d 1 )) φ +( b 2 (c 1 c 2 ) (c 1 + c 2 )hµ 1 )x 1 c 3 sx 2 + ( b2 4 (c 1 + c 2 ) + bhµ 1 2 (c 2 c 1 ))φ = µ 2 sgn( x 1 a φ)c 3 ax 2, (14) First Prev Next Last Go Back Full Screen Close Quit
The Covering is connected with a brake holder through a velocity depending friction force R(v rel ) (v rel is the relative velocity between the contacting surfaces) acting on the contact surfaces. Initially this dependance is modelled by simple Coulomb friction characteristic. This means R(v rel ) = Nµ 2 sgn(v rel ), with v rel = x 1 a φ. If a more realistic form of the friction force with a typically nonlinear characteristic will be taken, nonlinear terms have to be added.
8. Simplification and reduction of the problem by setting some parameters and transformation The equations of motion (1-3), contain six unknown variables (x 1, x 1, x 2, x 2, φ, φ) and 13 parameters. It is clear, the exact analytic solutions are unavailable. That s why we fix some parameters to simplify the problem for a start. We take the following: (i)- c := c 1 = c 2, (ii)- d := d 1 = d 2. The Dynamical system (1-3) reduces to the following mẍ 1 + 2dx 1 + 2cx 1 = µ 2 sgn( x 1 a φ)c 3 x 2, (15) mẍ 2 2dµ 1 x 1 2cµ 1 x 1 + c 3 x 2 = 0, (16) j φ 2dhµ 1 x 1 + db2 2 φ 2chµ 1 x 1 c 3 sx 2 + cb2 2 φ = µ 2sgn( x 1 a φ)c 3 ax 2, (17) First Prev Next Last Go Back Full Screen Close Quit
After the transformation z 1 := x 1 z 2 := x 2 z 3 := x 1 aφ and the scaling z 4 := µ 1 x 1 z 5 := x 2 z 6 := x 2 a φ t maµ 1 t.
We Obtain ż = { A + z, z 6 > 0, A z, z 6 < 0, with the simple form of the matrices 0 0 0 ã 0 0 0 0 0 0 b 0 A ± = 0 0 0 0 0 b c α 0 d 0 0 c γ 0 d 0 0 ẽ β g h 0 f (18) (19) where A ± are constant matrices containing various parameters, (Note that the elements of matrices are constant functions of eleven parameters i.e. ã = ma,...). The general structure is given by the observation that A + and A only differ in two entrances due to the simple (piecewise constant) form of the friction force.
Summary of analysis (α = 0) We can determine the eigenvalues and eigenvectors in simple form as: Eigenvalues Eigenvectors, λ 1,2 = l ± im, ν 12 = u 1 iw 1, λ 3,4 = ±in, ν 34 = u 2 iw 2, λ 5,6 = k ± ip, ν 56 = u 3 iw 3, where the λ 1,...,6 and ν 1,...,6 are to be represented in terms of 11 parameters in the Brake-system. So, we can write the general solution as: z = e lt (c 1 (cos(mt)u 1 + sin(mt)w 1 ) + c 2 ( sin(mt)u 1 + cos(mt)w 1 )) +c 3 (cos(nt)u 2 + sin(nt)w 2 ) + c 4 ( sin(nt)u 2 + cos(nt)w 2 ) + e kt (c 5 (cos(pt)u 3 + sin(pt)w 3 ) + c 6 ( sin(pt)u 3 + cos(pt)w 3 )), (20) where this solution corresponds to the system ż = A + z, z > 0 β = β + and ż = A z, z < 0 t = t with β = β. t = t + with
The constants c 1,...,6 are determined form the initial data z(0) = c 1 u 1 + c 2 w 1 + c 3 u 2 + c 4 w 2 + c 5 u 3 + c 6 w 3. The Poincarè map is of the form. c 11 0 0 c 14 0 c P = P + P 21 c 22 0 c 24 c 25 = c 31 c 32 c 33 a 34 c 35 c 41 0 0 c 44 0. (21) c 51 c 52 0 c 54 c 55 Note that, c 11,..., c 55 are functions of the parameters of t + and t, so the Poincare map depends on the quantities t + and t in a non-linear way. Both times are determined by the following equations: H 1 (t + ) = {e T 6 z = 0, z > 0} := a 16 z 1 + a 26 z 2 + a 36 z 3 + a 46 z 4 + a 56 z 5 = 0,(22) H 2 (t ) = {e T 6 z = 0, z < 0} := b 16 z 1 + b 26 z 2 + b 36 z 3 + b 46 z 4 + b 56 z 5 = 0,(23) The equations (22) and (23) can only be solved numerically.
The eigenvalues of P as: µ 1 = c 33, µ 2,3 = 1 2 (c 11 + c 44 ± c 2 11 + 4c 14 c 41 2c 11 c 44 + c 2 44) µ 4,5 = 1 2 (c 22 + c 55 ± c 2 22 + 4c 25 c 52 2c 22 c 55 + c 2 55) Some Results In order to obtain an invariant cone consisting of periodic orbits, we solve equations (22) and (23) for µ = 1, we determine the eigenvector (corresponding to µ = 1) of the Poincare map (21), which defines the cone, Fig. (5); The periodic orbit is stable when all eigenvalues µ 1,...,5 1 In Fig. (6) and (7) we see the influence of the friction parameter for various values of µ 1 (µ 1 = 0.01 and 0.9).
Figure 4: sets of stable periodic orbits where all eigenvalues of P µ 1,...,5 1 First Prev Next Last Go Back Full Screen Close Quit
Figure 5: sets of stable periodic orbits where all eigenvalues of P µ 1,...,5 1 First Prev Next Last Go Back Full Screen Close Quit
Figure 6: sets of stable periodic orbits where all eigenvalues of P µ 1,...,5 1 First Prev Next Last Go Back Full Screen Close Quit
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