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We derive the global bifurcation diagram of a three-parameter family of cubic Li enard systems. This family seems to have a universal character in that its bifurcation diagram (or parts of it) appears in many models from applications. In fact, it is the 3-jet of a three-dimensional slice of the universal unfolding of a doubly degenerate Bogdanov-Takens point, that is, of the codimension-four singularity with nilpotent linear part and no quadratic terms in the normal form.

SUMMARYModels describing systems of coevolving populations often have asymptotically non-equilibrium dynamics (Red Queen dynamics (RQD)). We claim that if evolution is much slower than ecological changes, RQD arises due to either fast ecological processes, slow genetical processes, or to their interaction. The three corresponding generic types of RQD can be studied using singular perturbation theory and have very different properties and biological implications. We present simple examples of ecological, genetical, and ecogenetical RQD and describe how they may be recognized in natural populations. In particular, ecogenetical RQD often involve alternations of long epochs with radically different dynamics.

In this paper we present an oscillatory neural network composed of two coupled neural oscillators of the Wilson-Cowan type. Each of the oscillators describes the dynamics of average activities of excitatory and inhibitory populations of neurons. The network serves as a model for several possible network architectures. We study how the type and the strength of the connections between the oscillators affect the dynamics of the neural network. We investigate, separately from each other, four possible connection types (excitatory-->excitatory, excitatory-->inhibitory, inhibitory-->excitatory, and inhibitory-->inhibitory) and compute the corresponding bifurcation diagrams. In case of weak connections (small strength), the connection of populations of different types lead to periodic in-phase oscillations, while the connection of populations of the same type lead to periodic anti-phase oscillations. For intermediate connection strengths, the networks can enter quasiperiodic or chaotic regimes, and can also exhibit multistability. More generally, our analysis highlights the great diversity of the response of neural networks to a change of the connection strength, for different connection architectures. In the discussion, we address in particular the problem of information coding in the brain using quasiperiodic and chaotic oscillations. In modeling low levels of information processing, we propose that feature binding should be sought as a temporally coherent phase-locking of neural activity. This phase-locking is provided by one or more interacting convergent zones and does not require a central ¿top level¿ subcortical circuit (e.g., the septo-hippocampal system). We build a two layer model to show that although the application of a complex stimulus usually leads to different convergent zones with high frequency oscillations, it is nevertheless possible to synchronize these oscillations at a lower frequency level using envelope oscillations. This is interpreted as a feature binding of a complex stimulus.

We present the bifurcation analysis of Chua’s circuit equations with a smooth nonlinearity, described by a cubic polynomial. Our study focuses on phenomena that can be observed directly in the numerical simulation of the model, and on phenomena which are revealed by a more elaborate analysis based on continuation techniques and bifurcation theory. We emphasize how a combination of these approaches actually works in practice. We compare the dynamics of Chua’s circuit equations with piecewise-linear and with smooth nonlinearity. The dynamics of these two variants are similar, but we also present some differences. We conjecture that this similarity is due to the central role of homoclinicity in this model. We describe different ways in which the type of a homoclinic bifurcation influences the behavior of branches of periodic orbits. We present an overview of codimension 1 bifurcation diagrams for principal periodic orbits near homoclinicity for three-dimensional systems, both in the generic case and in the case of odd symmetry. Most of these diagrams actually occurs in the model. We found several homoclinic bifurcations of codimension 2, related to the so called resonant conditions. We study one of these bifurcations, a double neutral saddle loop.

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