By Jaume Llibre, Richard Moeckel, Carles Simó

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 strategies of independent differential structures 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 equipment for learning valuable Configurations. The final one, by way of Carles Simó, describes the most mechanisms resulting in a reasonably worldwide description of the dynamics in conservative systems.

The publication is directed in the direction of graduate scholars and researchers attracted to dynamical platforms, 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 e-book comprise a wide variety of purposes.

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55). In what follows we consider the case where n2 and n3 are positive and m2 = m1 < n1 + n2 . 15. For this reason we present these proofs together in order to avoid repetitive arguments. Moreover, in what follows we consider T K= U (s)ds. 15. 59) with n2 and n3 positive and m2 = m1 < n1 + n2 . So we have dX = −εn2 AX + εn3 BY + ε−m1 +n1 +n2 AU (t), dt dY = −Y + XZ, dt dZ = 1 − Z − ε2m1 XY. 60). 5, we have x= (X, Y, Z)T and ⎛ ⎞ 0 F0 (t, x) = ⎝ −Y + XZ ⎠ . 61) 1−Z We start considering the system x˙ = F0 (t, x).

By continuity of the solution x(t, z, ε) and by compactness of the set [0, T ]× V × [−ε1 , ε1 ], there exits a compact subset K of D such that x(t, z, ε) ∈ K for all t ∈ [0, T ], z ∈ V and ε ∈ [−ε1 , ε1 ]. Now, by continuity of the function R, |R(s, x(s, z, ε), ε)| ≤ max{|R(t, x, ε)|, (t, x, ε) ∈ [0, T ] × K × [−ε1 , ε1 ]} = N . Then t T |R(s, x(s, z, ε), ε)| ds = T N, R(s, x(s, z, ε), ε)ds ≤ 0 0 which implies that t R(s, x(s, z, ε), ε)ds = O(1). 110) F0 (s, ϕ(s, z))ds. 0 Moreover, x(t, z, ε) = ϕ(t, z) + O(ε).

X= x , ε Y = y , ε2 h(t) , ε2 H(t) = B(s) = εb(s), A(s) = a(s) , ε becomes the polynomial q(x0 ). Hence the theorem is proved. 1(i). 4) on the time-scale 1/ε. We introduce t [F (s, x) − f 0 (x)]ds. , ||u(t, x)|| ≤ 2M T, t ≥ 0, x ∈ D. 36 Chapter 1. The Averaging Theory for Computing Periodic Orbits We now introduce a transformation near the identity x(t) = z(t) + εu(t, z(t)). 3). 3) yields x˙ = z˙ + ε ∂ ∂ u(t, z) + ε u(t, z)z˙ = εF (t, z + εu(t, z)) + ε2 R(t, z + εu(t, z), ε). 73), we write this equation in the form I +ε ∂ u(t, z) z˙ = εf 0 (z) + S, ∂z with I the n × n identity matrix, and where S = εF (t, z + εu(t, z)) − εF (t, z) + ε2 R(t, z + εu(t, z), ε).