By Jaume Llibre, Richard Moeckel, Carles Simó

ISBN-10: 3034809328

ISBN-13: 9783034809320

The notes of this ebook originate from 3 sequence of lectures given on the Centre de Recerca Matemàtica (CRM) in Barcelona. the 1st one is devoted to the examine of periodic options of independent differential platforms in R^{n} through the Averaging conception and was once introduced by way of Jaume Llibre. the second, given by means of Richard Moeckel, focusses on tools for learning crucial Configurations. The final one, via Carles Simó, describes the most mechanisms resulting in a pretty international description of the dynamics in conservative systems.

The e-book is directed in the direction of graduate scholars and researchers drawn to dynamical structures, particularly within the conservative case, and goals at facilitating the certainty of dynamics of particular types. the consequences offered and the instruments brought during this ebook comprise a wide range of functions.

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**Example text**

111) Let (λk ) = ∂ k−1 G t, 1 0 k ◦ k−1 ◦ ···◦ 1 (x(t, z, ε)) . So, (λk )dλk = (1) − (0) = ∂ k−1 G t, k−1 ◦ k−2 ◦ ···◦ 1 (x(t, z, ε)) − ∂ m G(t, ϕ(t, z)). 3. Averaging theory for arbitrary order and dimension 53 The derivative of (λk ) can be easily obtained as (λk ) = λk−1 λk−2 · · · λ1 ∂ k G t, k ◦ k−1 ◦ ··· ◦ 1 (x(t, z, ε)) ∂ k G t, k (x(t, z, ε) − ϕ(t, z)). Hence, 1 (λk )dλk = λk−1 λk−2 · · · λ1 0 1 0 ◦ k−1 ◦ ···◦ 1 (x(t, z, ε)) − ∂ k G(t, ϕ(t, z)) dλk · (x(t, z, ε) − ϕ(t, z)) + λk−1 λk−2 · · · λ1 ∂ k G(t, ϕ(t, z))(x(t, z, ε) − ϕ(t, z)).

3. 50 Chapter 1. 6 when F0 = 0. 6, is used. 5. Suppose that F0 = 0. 100), we assume the following conditions: (i) for each t ∈ R, Fi (t, ·) ∈ C k−i for i = 1, 2, . . , k; ∂ k−i Fi is locally Lipschitz in the second variable for i = 1, 2, . . , k; and R is continuous and locally Lipschitz in the second variable; (ii) fi = 0 for i = 1, 2, . . , r − 1 and fr = 0, where r ∈ {1, 2, . . , k} (here, we are taking f0 = 0). Moreover, suppose that for some a ∈ D with fr (a) = 0, there exists a neighborhood V ⊂ D of a such that fr (z) = 0 for all z ∈ V \ {a}, and that dB (fr (z), V, a) = 0.

82) are the zeros of the characteristic polynomial det(A − λId). If these eigenvalues λk are diﬀerent, with eigenvectors ek for k = 1, . . , n, then ek eλk t , for k = 1, . . 82). Assume now that not all eigenvalues are diﬀerent, thus suppose that the eigenvalue λ has multiplicity m > 1. 82) of the form P0 eλt , P1 (t)eλt , . . , Pm−1 (t)eλt , where Pi (t) for i = 0, 1, . . , m − 1 are polynomial vectors of degree at most i. With n independent solutions x1 (t), . . 82) we form a matrix Φ(t) = (x1 (t), .