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Christopher Golé

Bio: Christopher Golé is an academic researcher from State University of New York System. The author has contributed to research in topics: Rotation number & Hilbert cube. The author has an hindex of 1, co-authored 1 publications receiving 25 citations.

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Journal ArticleDOI
TL;DR: A variational principle is proposed and implemented for determining unstable periodic orbits of flows as well as unstable spatiotemporally periodic solutions of extended systems by an initial loop approximating a periodic solution by a minimization of local errors along the loop.
Abstract: A variational principle is proposed and implemented for determining unstable periodic orbits of flows as well as unstable spatiotemporally periodic solutions of extended systems. An initial loop approximating a periodic solution is evolved in the space of loops toward a true periodic solution by a minimization of local errors along the loop. The "Newton descent" partial differential equation that governs this evolution is an infinitesimal step version of the damped Newton-Raphson iteration. The feasibility of the method is demonstrated by its application to the Henon-Heiles system, the circular restricted three-body problem, and the Kuramoto-Sivashinsky system in a weakly turbulent regime.

91 citations

Journal ArticleDOI
TL;DR: In this paper, an Aubry-Mather-type theory for solutions of non-linear, possibly degenerate, elliptic PDEs and other pseudo-differential operators is discussed.
Abstract: We discuss an Aubry–Mather-type theory for solutions of non-linear, possibly degenerate, elliptic PDEs and other pseudo-differential operators. We show that for certain PDEs and ΨDEs with periodic coefficients and a variational structure it is possible to find quasi-periodic solutions for all frequencies. This results also hold under a generalized definition of periodicity that makes it possible to consider problems in covers of several manifolds, including manifolds with non-commutative fundamental groups. An abstract result will be provided, from which an Aubry–Mather-type theory for concrete models will be derived.

53 citations

Book
01 May 2003
TL;DR: In this paper, the authors provide a survey of the literature on one-dimensional variational problems with global extremal fields and global minimals, as well as a generalization of the Hamiltonian formulation.
Abstract: 1 One-dimensional variational problems.- 1.1 Regularity of the minimals.- 1.2 Examples.- 1.3 The accessory variational problem.- 1.4 Extremal fields for n=1.- 1.5 The Hamiltonian formulation.- 1.6 Exercises to Chapter 1.- 2 Extremal fields and global minimals.- 2.1 Global extremal fields.- 2.2 An existence theorem.- 2.3 Properties of global minimals.- 2.4 A priori estimates and a compactness property.- 2.5 Ma for irrational a, Mather sets.- 2.6 Ma for rational a.- 2.7 Exercises to chapter II.- 3 Discrete Systems, Applications.- 3.1 Monotone twist maps.- 3.2 A discrete variational problem.- 3.3 Three examples.- 3.4 A second variational problem.- 3.5 Minimal geodesics on T2.- 3.6 Hedlund's metric on T3.- 3.7 Exercises to chapter III.- A Remarks on the literature.- Additional Bibliography.

44 citations

Journal ArticleDOI
TL;DR: A generalization of the celebrated criterion of existence of invariant circles if and only iff the Peierls–Nabarro barrier vanishes is provided.
Abstract: We consider several models of networks of interacting particles and prove the existence of quasi-periodic equilibrium solutions. We assume (1) that the network and the interaction among particles are invariant under a group that satisfies some mild assumptions; (2) that the state of each particle is given by a real number; (3) that the interaction is invariant by adding an integer to the state of all the particles at the same time; (4) that the interaction is ferromagnetic and coercive (it favors local alignment and penalizes large local oscillations); and (5) some technical assumptions on the regularity speed of decay of the interaction with the distance. Note that the assumptions are mainly qualitative, so that they cover many of the models proposed in the literature. We conclude (A) that there are minimizing (ground states) quasi-periodic solutions of every frequency and that they satisfy several geometric properties; (B) if the minimizing solutions do not cover all possible values at a point, there is another equilibrium point which is not a ground state. These results generalize basic results of Aubry–Mather theory (take the network and the group to be ℤ). In particular, we provide with a generalization of the celebrated criterion of existence of invariant circles if and only iff the Peierls–Nabarro barrier vanishes.

36 citations

Journal ArticleDOI
TL;DR: In this article, complete ordered invariant ghost circles are found for the gradient of the energy flow in the state space, containing the critical sets corresponding to the Birkhoff orbits of all rotation number.

33 citations