Eduardo Colli

Infinitely many coexisting strange attractors

Abstract We prove that C ∞ diffeomorphisms of a two-dimension manifold M with a homoclinic tangency are in the closure of an open set of Diff ∞ ( M ) containing a dense subset of diffeomorphisms exhibiting infinitely many coexisting Henon-like strange attractors (or repellers). A similar statement is posed in terms of one-parameter C ∞ families of diffeomorphisms unfolding a homoclinic tangency. Moreover, we show the existence of infinitely many dynamical phenomena others than strange attractors.

Non-trivial wandering domains and homoclinic bifurcations

We prove that on any surface there is a C ∞ diffeomorphism exhibiting a wandering domain D with the following ergodic property: for any orbit starting in D the corresponding Birkhoff mean of Dirac measures converges to the invariant measure supported on a hyperbolic horseshoewhich is equivalent to the unique non-trivial Hausdorff measure in � . The construction is obtained by perturbation of a diffeomorphism such that the unstable and stable foliations of this horseshoeare relatively thick and in tangential position. We describe, in addition, the set of accumulation points of orbits starting in D.

Period adding cascades: Experiment and modeling in air bubbling.

Period adding cascades have been observed experimentally/numerically in the dynamics of neurons and pancreatic cells, lasers, electric circuits, chemical reactions, oceanic internal waves, and also in air bubbling. We show that the period adding cascades appearing in bubbling from a nozzle submerged in a viscous liquid can be reproduced by a simple model, based on some hydrodynamical principles, dealing with the time evolution of two variables, bubble position and pressure of the air chamber, through a system of differential equations with a rule of detachment based on force balance. The model further reduces to an iterating one-dimensional map giving the pressures at the detachments, where time between bubbles come out as an observable of the dynamics. The model has not only good agreement with experimental data, but is also able to predict the influence of the main parameters involved, like the length of the hose connecting the air supplier with the needle, the needle radius and the needle length.

Noise, transient dynamics, and the generation of realistic interspike interval variation in square-wave burster neurons.

First return maps of interspike intervals for biological neurons that generate repetitive bursts of impulses can display stereotyped structures (neuronal signatures). Such structures have been linked to the possibility of multicoding and multifunctionality in neural networks that produce and control rhythmical motor patterns. In some cases, isolating the neurons from their synaptic network reveals irregular, complex signatures that have been regarded as evidence of intrinsic, chaotic behavior. We show that incorporation of dynamical noise into minimal neuron models of square-wave bursting (either conductance-based or abstract) produces signatures akin to those observed in biological examples, without the need for fine tuning of parameters or ad hoc constructions for inducing chaotic activity. The form of the stochastic term is not strongly constrained and can approximate several possible sources of noise, e.g., random channel gating or synaptic bombardment. The cornerstone of this signature generation mechanism is the rich, transient, but deterministic dynamics inherent in the square-wave (saddle-node and homoclinic) mode of neuronal bursting. We show that noise causes the dynamics to populate a complex transient scaffolding or skeleton in state space, even for models that (without added noise) generate only periodic activity (whether in bursting or tonic spiking mode).

A starting condition approach to parameter distortion in generalized renormalization

We control parameter distortion in the generalized renormalization procedure provided a certain set of starting conditions is satisfied. This allows us to prove that for aC∞ — open set of unimodal families, almost all parameters inside an interval present either stochastic dynamics or a renormalization (in the classical sense). Moreover, easy consequences are that renormalization happens densely on this interval and stochastic behaviour with positive measure. A wide range use of this approach would rely mostly on proving that the starting conditions are satisfied for general families.