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...yperplane xk = 0, then we define P̄ : V̄ ⊂ Rn−1 → Rn−1 as P̄ (x̄, λ) = Rk(P (R −1 k (x̄), λ)). (6) 4 Details on the explicit computation of the map Rk, and its inverse are given in [14]. There and in =-=[22]-=- periodic orbits of (5) were obtained by computing the fixed points of P̄ . kx = 0 x λP(x, ) x P(x,λ) R −1 xk Σ R P 1 a) k k λP(x, ) P(x,λ) λ),λ)f(P(x, R −1R x = 0k x xk Σ x x∆ ∆x y w z= P w 1 b) k k ...

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...s September 8, 2010 of subcritical or multicritical transitions. In these situations pseudo-arclength continuation methods [5, 6, 7], and the linear stability analysis of the time dependent solutions =-=[8, 9]-=- provide powerful instruments to clarify the dynamics. In addition, the computation of Liapunov exponents is used to assess if a system is chaotic. All these tools have been historically applied to lo...

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... each matrix product requires the time integration of a system of 2n equations. This formulation, together with Newton-Krylov methods, was used in thermal convection problems in [98] and [89], and in =-=[100]-=- to obtain a normal form of a periodic orbit with symmetries. The latter study also required the eigenfunctions of an adjoint operator at the specific periodic orbit. A multiple shooting variant using...

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...mail addresses: juan.j.sanchez@upc.edu, sanchez@fa.upc.edu (J. Sánchez). The calculation of coefficients of normal forms at fixed points and periodic orbits has also been considered, for instance, in =-=[16,17]-=-, and that of portions of two-dimensional unstable manifolds of periodic orbits in [18]. The method to compute invariant two-dimensional tori described in [11] consists in finding a single point of th...