Generic Hopf–Neĭmark–Sacker bifurcations in feed-forward systems

We show that generic Hopf–Neĭmark–Sacker bifurcations occur in the dynamics of a large class of feed-forward coupled cell networks. To this end we present a framework for studying such bifurcations in parametrized families of perturbed forced oscillators near weak resonance points. Our approach is based on fine-tuning existing normal form techniques. We then apply this framework to show that certain cells in the feed-forward networks exhibit this forced oscillator dynamics and, hence, undergo generic Hopf–Neĭmark–Sacker bifurcations. These bifurcations correspond to the occurrence of resonance tongues in parameter space with a ‘standard’ geometry.

Resonance and Singularities

Abstract The phenomenon of resonance will be dealt with from the viewpoint of dynamical systems depending on parameters and their bifurcations. Resonance phenomena are associated to open subsets in the parameter space, while their complement corresponds to quasi-periodicity and chaos. The latter phenomena occur for parameter values in fractal sets of positive measure. We describe a universal phenomenon that plays an important role in modelling. This paper gives a summary of the background theory, veined by examples. 1 What Is Resonance? A heuristic definition of resonance considers a dynamical system, usually depend-ing on parameters, with several oscillatory subsystems having a rational ratio of frequencies and a resulting combined and compatible motion that may be amplified as well. Often the latter motion is also periodic, but it can be more complicated as will be shown below. We shall take a rather eclectic point of view, discussing several examples first. Later we shall turn to a number of universal cases, these are context-free models that occur generically in any system of sufficiently high-dimensional state and parameter space. Part of these results were announced in [14]. Among the examples are the famous problem of Huygens’s synchronizing clocks and that of the Botafumeiro in the Cathedral of Santiago de Compostela, but also we briefly touch on tidal resonances in the planetary system. As universal models we shall deal with the Hopf–Neı̆mark–Sacker bifurcation and the Hopf saddle-node bifurcation for mappings. The latter two examples form ‘next cases ’ in th

MULTIPLE PURPOSE ALGORITHMS FOR INVARIANT MANIFOLDS

This paper deals with the numerical continuation of invariant manifolds, regardless of the restricted dynamics. Typically, invariant manifolds make up the skeleton of the dynamics of phase space. Examples include limit sets, co-dimension 1 manifolds separating basins of attraction (separatrices), stable/unstable/center manifolds, nested hierarchies of attracting manifolds in dissipative systems and manifolds in phase plus parameter space on which bifurcations occur. These manifolds are for the most part invisible to current numerical methods. The approach is based on the general principle of normal hyperbolicity, where the graph transform leads to the numerical algorithms. This gives a highly multiple purpose method. Examples of computations of both attracting and saddle-type (1D and 2D) manifolds will be given, with and without non-uniform adaptive re nement. A convergence result for the algorithm will be sketched