We study synchronisation properties of networks of coupled dynamical systems with interaction akin to diffusion. We assume that the isolated node dynamics possesses a forward invariant set on which it has a bounded Jacobian, then we characterise a class of coupling functions that allows for uniformly stable synchronisation in connected complex networks -in the sense that there is an open neighbourhood of the initial conditions that is uniformly attracted towards synchronisation. Moreover, this stable synchronisation persists under perturbations to non-identical node dynamics. We illustrate the theory with numerical examples and conclude with a discussion on embedding these results in a more general framework of spectral dichotomies. arXiv:1304.7679v3 [math.DS] 29 Aug 2013Coupling functions allowing persistent synchronisationwhere α is the overall coupling strength, and the matrix W = (W ij ) i,j∈{1,...,n} describes the interaction structure of the network, i.e. W ij measures the strength of interaction between the nodes i and j. The function f : R × R m → R m describes the isolated node dynamics, and the coupling function h : R m → R m describes the diffusion-like interaction between nodes. We make the following two assumptions for these functions.Assumption A1. The function f is continuous, and there exists an inflowing invariantFor instance, the Lorenz system has a bounded inflowing invariant ball, see Subsection 3.2. In general, smooth nonlinear systems with compact attractors satisfy Assumption A1. This assumption will be generalised in Section 5 to include also noncompact sets U .Assumption A2. The coupling function h is continuously differentiable with h(0) = 0. We define Γ := Dh(0) and denote the (complex) eigenvalues of Γ by β i , i ∈ {1, . . . , m}.The network structure plays a central role for the synchronisation properties. We consider the intensity of the i-th node V i = n j=1 W ij , and define the positive definite matrix V := diag(V 1 , . . . , V n ). Then the so-called Laplacian reads asCoupling functions allowing persistent synchronisation 3 Let λ i , i ∈ {1, . . . , n}, denote the eigenvalues of L. Note that λ 1 = 0 is an eigenvalue with eigenvector 1 √ n (1, . . . , 1). The multiplicity of this eigenvalue equals the number of connected components of the network.The following assumption incorporates the coupling and structural network properties.Assumption A3. We suppose thatwhere Re(z) denotes the real part of a complex number z.The dynamics of such a diffusive model can be intricate. Indeed, even if the isolated dynamics possesses a globally stable fixed point, the diffusive coupling can lead to instability of the fixed point and the system can exhibit an oscillatory behaviour [28].Note that due to the diffusive nature of the coupling, if all oscillators start with the same initial condition, then the coupling term vanishes identically. This ensures that the globally synchronised state x 1 (t) = x 2 (t) = . . . = x n (t) = s(t) is an invariant state for all coupling strengths α and all choices ...

Let M be an n-dimensional manifold; the assumed smoothness will be clear from the context. We are interested in uniformity properties of M when it is noncompact. These can be formulated in different ways, e.g. in terms of bounded geometry when a Riemannian metric g is present. If no such metric is (canonically) available, it may be more natural to express uniformity in terms of the atlas and its chart transition maps. We shall formulate various definitions of uniformity and investigate their relations.We follow [Eic91a] to define bounded geometry.Definition 1 (Bounded geometry). We say that a complete, finite-dimensional Riemannian manifold (M, g) has k-th order bounded geometry when the following conditions are satisfied:(B k ) the Riemannian curvature R and its covariant derivatives up to k-th order are uniformly bounded,with operator norm of ∇ i R(x) as an element of the tensor bundle over x ∈ M.Remark 2. Condition I already automatically implies that (M, g) is a complete metric space.

Abstract. In this paper we illustrate the potential role which relative limit cycles may play in biolocomotion. We do this by describing, in great detail, an elementary example of reduction of a lightly dissipative system modeling crawling-type locomotion in 3D. The symmetry group SE(2) is the set of rigid transformations of the horizontal (ground) plane. Given a time-periodic perturbation, the system will admit a relative limit cycle whereupon each period is related to the previous by a fixed translation and rotation along the ground. This toy model identifies how symmetry reduction and dissipation can conspire to create robust behavior in crawling, and possibly walking, locomotion.

Abstract. The classical question whether nonholonomic dynamics is realized as limit of friction forces was first posed by Carathéodory. It is known that, indeed, when friction forces are scaled to infinity, then nonholonomic dynamics is obtained as a singular limit.Our results are twofold. First, we formulate the problem in a differential geometric context. Using modern geometric singular perturbation theory in our proof, we then obtain a sharp statement on the convergence of solutions on infinite time intervals. Secondly, we set up an explicit scheme to approximate systems with large friction by a perturbation of the nonholonomic dynamics. The theory is illustrated in detail by studying analytically and numerically the Chaplygin sleigh as example. This approximation scheme offers a reduction in dimension and has potential use in applications.MSC2010 numbers: 37J60, 70F40, 37D10, 70H09

We study global properties of the global (center-)stable manifold of a normally attracting invariant manifold (NAIM), the special case of a normally hyperbolic invariant manifold (NHIM) with empty unstable bundle. We restrict our attention to continuous-time dynamical systems, or flows. We show that the global stable foliation of a NAIM has the structure of a topological disk bundle, and that similar statements hold for inflowing NAIMs and for general compact NHIMs. Furthermore, the global stable foliation has a C k disk bundle structure if the local stable foliation is assumed C k . We then show that the dynamics restricted to the stable manifold of a compact inflowing NAIM are globally topologically conjugate to the linearized transverse dynamics at the NAIM. Moreover, we give conditions ensuring the existence of a global C k linearizing conjugacy. We also prove a C k global linearization result for inflowing NAIMs; we believe that even the local version of this result is new, and may be useful in applications to slow-fast systems. We illustrate the theory by giving applications to geometric singular perturbation theory in the case of an attracting critical manifold: we show that the domain of the Fenichel Normal Form can be extended to the entire global stable manifold, and under additional nonresonance assumptions we derive a smooth global linear normal form. * No academic affiliation (jaap@jaapeldering.nl)