Periodic solutions of the perturbed symmetric Euler top

Transcription

1 Periodic solutions of the perturbed symmetric Euler top Universitatea Babeş-Bolyai (Cluj-Napoca, Romania) and Universitat de Lleida (Lleida, Spain)

2 Plan of the talk 1 Our problem - the perturbed symmetric Euler top 2 The Euler top - short presentation 3 The time-dependent perturbation 4 The nonlinear time-dependent perturbation

3 1 Our problem - the perturbed symmetric Euler top 2 The Euler top - short presentation 3 The time-dependent perturbation 4 The nonlinear time-dependent perturbation

4 Our aim is to provide results on the existence/bifurcation of T -periodic solutions (for some fixed T > ) for systems of the form ẋ = yz +εp(t, x, y, z), ẏ = xz +εq(t, x, y, z), ż = εs(t, x, y, z), with ε > sufficiently small and p, q, s : R R 3 R continuous, T periodic in the time variable and locally Lipschitz in the space variables. Our approach is the study of the Poincaré translation map using the Implicit Function Theorem (in the additional condition that p, q, s are C 1 ) or the Brouwer Degree Theory. In the case that p, q, s does not depend on (x, y, z) we use direct methods from Linear Systems Theory.

5 1 Our problem - the perturbed symmetric Euler top 2 The Euler top - short presentation 3 The time-dependent perturbation 4 The nonlinear time-dependent perturbation

6 The unperturbed system is the symmetric Euler top ẋ = yz, ẏ = xz, ż =, a particular case (α + β = ) of the Euler top ẋ = αyz, ẏ = βxz, ż = (α + β)xy.

7 The Euler equations with parameters ẋ = αyz, ẏ = βxz, ż = γxy, (1) α := µ 2 µ 3 µ 2 µ 3, β := µ 3 µ 1 µ 1 µ 3, γ := µ 1 µ 2 µ 1 µ 2. (note that α + β + γ = ) describe the rotation of a rigid body with a fixed point and no torques acting on it. Here (x, y, z) R 3 denotes the three components of the angular momentum, and the constants (µ 1, µ 2, µ 3 ) are the moments of inertia about the coordinate axes of the rigid body.

8 The Euler top is an integrable system having the first integrals H(x, y, z) = 1 ( x 2 + y 2 ) + z2, D(x, y, z) = x 2 + y 2 + z 2, 2 µ 1 µ 2 µ 3 corresponding to the conservation of energy and, respectively, of the Euclidean norm of the angular momentum.

9 The symmetric Euler top corresponds to the case when two of the moments of inertia are equal, for example µ 1 = µ 2. When µ 1 = µ 2 < µ 3 the body is oblate whereas when µ 1 = µ 2 > µ 3 the body is prolate

10 Let (x, y, η) R 3. The solution of ẋ = yz, ẏ = xz, ż =. (2) satisfying x() = x, y() = y, z() = η is x(t) = x cos ηt y sin ηt, y(t) = x sin ηt + y cos ηt, z(t) = η.

11 The phase portrait in R 3 of (2) has the following features: - the plane z = and the three axes are fulfilled of equilibria; - each pane z = η, with η, is invariant and foliated by periodic solutions of minimal period 2π/ η ; - the nontrivial T -periodic solutions of (2) have the initial values (x, y, 2kπ/T ) for each k Z \ {} and (x, y ) R 2 \ {(, )}, that is, each plane z = 2kπ/T is foliated by T -periodic solutions and there are no other nontrivial T -periodic solutions.

12 1 Our problem - the perturbed symmetric Euler top 2 The Euler top - short presentation 3 The time-dependent perturbation 4 The nonlinear time-dependent perturbation

13 We consider the perturbed symmetric Euler top ẋ = yz + εp(t), ẏ = xz + εq(t), ż = εs(t), (3) with ε R and p, q, s : R R cont and T periodic. z(t) = η + εs(t), for all t R, (4) with z() = η R an arbitrary constant and S(t) = t s(τ)dτ. We assume that s := 1 T s(τ)dτ =. (5) ẋ = y (η + εs(t)) + εp(t), ẏ = x (η + εs(t)) + εq(t) (6) that is a linear differential system with T -periodic coefficients.

14 ẋ = y (η + εs(t)) + εp(t), ẏ = x (η + εs(t)) + εq(t) (7) Lemma Let p, q, S : R R be continuous and T -periodic, and η, ε R. (a) If η = 2kπ εs for some k Z(ε) then all solutions of T (7) are T -periodic. (b) If η = 2kπ εs for some k Z \ Z(ε) then (7) has no T T -periodic solution. { } 2kπ (c) If η R \ T εs : k Z then (7) has one and only one T -periodic solution.

15 p = 1 T S = 1 T p(t)dt, q = 1 T S(t)dt, Ş(t) = t q(t)dt, ( S(σ) S ) dσ, τ(t, η, ε) = εş(t) + (η + εs)t, τ(t, k, ε) = εş(t) + 2kπ T t. Z(ε) = {k Z : [p(t) cos τ(t, k, ε) + q(t) sin τ(t, k, ε)] dt = and } [ p(t) sin τ(t, k, ε) + q(t) cos τ(t, k, ε)] dt =.

16 Proposition. - the existence of T -periodic solutions - ẋ = yz + εp(t), ẏ = xz + εq(t), ż = εs(t), (3) Let p, q, s : R R be cont. and T -periodic with s =, ε R. Case S =. { } 2kπ Given η R \ T : k Z, ε R there exists a unique T -periodic solution of (3) with the initial value in the plane z = η. Assume that Z(ε ). Then, for each j = j(ε ) Z(ε ) the plane z = 2jπ/T is filled with initial values of T -periodic solutions of (3), while for k Z \ Z(ε ) the plane z = 2kπ/T does not contain any initial value of T -periodic solutions of (3).

17 ẋ = yz + εp(t), ẏ = xz + εq(t), ż = εs(t), (3) Let p, q, s : R R be cont. and T -periodic with s, ε R. Case S. Fix η R. { ( ) } 1 2kπ For ε R \ S T η : k Z there exists a unique T -periodic solution of (3) with the initial value in the plane z = η. ( ) For those ε R such that Z(ε ) and ε = 1 2jπ S T η for some j Z(ε ) the plane z = η is filled with initial values of T -periodic solutions of (3).

18 Which of the planes z = 2kπ/T has T -periodic solutions? (a) (no one) When p(t) = t + t 2, q(t) = s(t) = system (3) has no T -periodic solution initiating in any of the planes z = 2kπ/T for k Z. (b) (only one) When p(t) = t + t 2, q(t) = T (1 + T )/π cos 2kπt/T T 2 /π 2 sin 2kπt/T, and s(t) = only the plane z = 2π/T if filled with initial conditions of T -periodic solutions.... (c) (all) When p(t) = sin t, q(t) = cos t and s(t) = system (3) has all the planes z = 2kπ/T for k Z \ {} filled with initial conditions of T -periodic solutions.

19 Remark in the case S The problem Find k Z \ {} such that the plane z = 2kπ/T εs is filled with initial conditions of T -periodic solutions of (3) for each < ε 1. reduces to Find k Z \ {} such that [ p(t) cos 2kπt T [ p(t) sin 2kπt T ] 2kπt + q(t) sin Ş m (t) dt =, T + q(t) cos 2kπt T ] Ş m (t) dt = for all m. This last problem is related to the Moment Problem, intensively studied in the last years, being connected also with the center-focus problem of Poincaré in planar polynomial systems.

20 Theorem. - the bifurcation of T -periodic solutions - Let p, q, s : R R be continuous and T -periodic such that s =. (i) From each equilibrium point (,, η) of (2) with η R \ {2kπ/T : k Z} emanates a branch of T -periodic solutions of (3) with z(, ε) = η. When S = this branch exists for all ε R. (ii) If p 2 + q 2 then, from each equilibrium point (x, y, ) of (2) with x p + y q = and (x, y ) (, ) emanates a branch of T -periodic solutions of (3) with z(, ε) = ε(p/y S).

21 (iii) If p 2 i + q 2 i for some i Z \ {} then, from each periodic solution of (2) with initial value (x, y, 2iπ/T ) with x p i + y q i = and (x, y ) (, ), emanates a branch of T -periodic solutions of (3) with z(, ε) = 2iπ/T + ε(p i /y S). where p k = 1 T q k = 1 T [ p(t) cos 2kπt T [ p(t) sin 2kπt T ] 2kπt + q(t) sin T + q(t) cos 2kπt T dt, ] dt.

22 (iv) If p = q = then, from the origin of coordinates emanates a family of branches of T -periodic solutions of (3). This family is parameterized by θ R \ S and we have z(, ε) = εθ. (v) If p i = q i = then, from the equilibrium point (,, 2iπ/T ) of (2) emanates a family of branches of T -periodic solutions of (3). This family is parameterized by θ R \ S and we have z(, ε) = 2iπ/T εθ.

23 1 Our problem - the perturbed symmetric Euler top 2 The Euler top - short presentation 3 The time-dependent perturbation 4 The nonlinear time-dependent perturbation

24 ẋ = yz +εp(t, x, y, z), ẏ = xz +εq(t, x, y, z), ż = εs(t, x, y, z), (8) with < ε 1 and p, q, s : R R 3 R are C 1 (or only locally Lipschitz) and T periodic in their first variable.

25 Theorem. (i) Assume that z s(t,,, z)dt has a zero η R \ {2kπ/T : k Z} of nonzero index. Then from the equilibrium point (,, η ) of (2) emanates a branch of T -periodic solutions of (8).

26 Proof. Denote F = (F 1, F 2, F 3 ) the displacement map, that is the difference between the Poincaré map at time T and the identity map. ( (i) Consider F = F 1, F 2, 1 ) ε F 3. We have F 1 ε= = x (cos ηt 1) y sin ηt F 2 ε= = x sin ηt + y (cos ηt 1) F 3 ε= = s(t, x cos ηt y sin ηt, x sin ηt + y cos ηt, η)dt. For η R \ {2kπ/T : k Z} a zero of z s(t,,, η) of nonzero index we have that (,, η ) is a zero of F ε= of nonzero index.

27 Theorem. (ii) Assume that ( (x, y) x p(t, x, y, )dt + y q(t, x, y, )dt s(t, x, y, )dt ) has a zero (x, y ) R2 \ {(, )} of nonzero index. Then from the equilibrium point (x, y, ) of (2) emanates a branch of T -periodic solutions of (8) with z(, ε) = εγ(ε). Moreover γ(ε) γ as ε and γ = 1 T p(t, x, y, )dt/y 1 t T s(τ, x, y, )dτdt.

28 (ii) Consider F (x, y, γ, ε) = 1 ε F (x, y, εγ, ε), where (x, y, η) F (x, y, η) is the displacement map. We have F 1 ε= = γy T y F 2 ε= = γx T + x F 3 ε= = t t s(t, x, y, )dt. s(τ, x, y, )dτdt + s(τ, x, y, )dτdt + p(t, x, y, )dt q(t, x, y, )dt We proved that, in our hypotheses, (x, y, γ ) is a zero of F ε= of nonzero index.

29 Theorem. (iii) Let i Z \ {}. Assume that the next map has a zero (xi, y i ) R \ {(, )} of nonzero index, ( xp(t, x, y, )dt + y q(t, x, y, )dt (x, y) s(t, x, y, )dt ) where x(t, x, y, i) = x cos 2iπt/T y sin 2iπt/T and y(t, x, y, i) = x sin 2iπt/T + y cos 2iπt/T. Then from the T -periodic solution with the initial value (xi, y i, 2iπ/T ) of (2) emanates a branch of T -periodic solutions of (8) with z(, ε) = 2iπ/T + εγ(ε). Moreover γ(ε) γ as ε and γ = [...].

30 Proof. (iii) Consider F (x, y, γ, ε) = 1 F (x, y, 2iπ/T + εγ, ε). ε

31 Further problems to be considered: Uniqueness and Stability When the perturbation is not C 1 we expect to need to add stronger condition than locally Lipschitz, as we needed in [1] A.B., O. Makarenkov, J. Llibre, Lipschitz generalization of Malkin-Loud result on the existence and uniqueness of periodic solutions, Preprint 29. [2] A.B., O. Makarenkov, J. Llibre, Asymptotic stability of periodic solutions for nonsmooth differential equations with application to the nonsmooth Van der Pol oscillator, SIAM J. Math. Anal. 4 (29), [3] A.B., A. Daniilidis, Stability of periodic solutions for Lipschitz systems obtained via the averaging method, Proc. Amer. Math. Soc. 135 (27),

32 Assume that f C 2 (R R n, R n ) and g C (R R n [, 1], R n ) are T -periodic in the first variable, and that g is locally uniformly Lipschitz with respect to the second variable. Consider the unperturbed system ẋ = f (t, x) (9) and its perturbation ẋ = f (t, x) + εg(t, x, ε). (1) Suppose that system (9) satisfies the following conditions. (A1) There exist an open ball V R k with k n, and a C 1 function β : V R n k such that any point of the set Z = { z α = (α, β (α)) : α V } is the initial condition of a T -periodic solution of (9).

33 (A2) For each z α Z there exists a fundamental matrix solution Y (, α) of the linearization around x(t, z α ) of (9), such that the matrix Y 1 (, α) Y 1 (T, α) has in the upper right corner the null k (n k) matrix, while in the lower right corner has the (n k) (n k) matrix (α) with det( (α)). We define the function M : V R k by M(α) = π Y 1 (t, α)g (t, x(t, z α, ), ) dt.

34 Then the following statement holds. (C1) If M has a zero α V of nonzero index then from the T -periodic solution x(t, z α, ) of (9) emanates a branch of T -periodic solutions ϕ ε of (1). Remarks. 1. The symmetric Euler top satisfies hypotheses (A1) and (A2). Hence, the proof of the existence results for the perturbed Euler top can be obtained as consequences. Except the estimations for the initial conditions. 2. The proof that we gave for the perturbation of the symmetric Euler top uses a different idea than the proof that we provided for this general result. Mainly, this other proof consists in applying the Lyapunov-Schmidt reduction method to the modified displacement map F (t, z, ε) = Y 1 (t, z) (x(t, z, ε) z).

35 In addition we assume: (A3) There exists δ 1 > and L M > such that M(α 1 ) M(α 2 ) L M α 1 α 2, for all α 1, α 2 B δ1 (α ), (A4) For δ > sufficiently small there exists I δ [, T ] Lebesgue measurable with mes(i δ ) = o(δ)/δ such that g(t, z 1 +ζ, ε) g(t, z 1, ) g(t, z 2 +ζ, ε)+g(t, z 2, ) o(δ)/δ z 1 z for all t [, T ] \ I δ and for all z 1, z 2 B δ (z ), ε [, δ] and ζ B δ (). Then the following conclusion holds. (C9) There exists δ 2 > such that for any ε (, ε 1 ], ϕ ε is the only T -periodic solution of (1) with initial condition in B δ2 (z ).

36 Thank you!