Discrete & Continuous Dynamical Systems - S
December 2009 , Volume 2 , Issue 4
A special issue on Bifurcation Delay
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The theory of slow-fast systems is a challenging field both from the viewpoint of theory and applications. Advances made over the last decade led to remarkable new insights and we therefore decided that it is worthwhile to gather snapshots of results and achievements in this field through invited experts. We believe that this volume of DCDS-S contains a varied and interesting overview of different aspects of slow-fast systems with emphasis on 'bifurcation delay' phenomena. Unfortunately, as could be expected, not all invitees were able to sent a contribution due to their loaded agenda, or the strict deadlines we had to impose.
Slow-fast systems deal with problems and models in which different (time- or space-) scales play an important role. From a dynamical systems point of view we can think of studying dynamics expressed by differential equations in the presence of curves, surfaces or more general varieties of singularities. Such sets of singularities are said to be critical. Perturbing such equations by adding an $\varepsilon$-small movement that destroys most of the singularities can create complex dynamics. These perturbation problems are also called singular perturbations and can often be presented as differential equations in which the highest order derivatives are multiplied by a parameter $\varepsilon$, reducing the order of the equation when $\varepsilon\to 0$.
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In this paper we consider singular perturbation problems occuring in planar slow-fast systems $(\dot x=y-F(x,\lambda),\dot y=-\varepsilon G(x,\lambda))$ where $F$ and $G$ are smooth or even real analytic for some results, $\lambda$ is a multiparameter and $\varepsilon$ is a small parameter. We deal with turning points that are limiting situations of (generalized) Hopf bifurcations and that we call slow-fast Hopf points. We investigate the number of limit cycles that can appear near a slow-fast Hopf point and this under very general conditions. One of the results states that for any analytic family of planar systems, depending on a finite number of parameters, there is a finite upperbound for the number of limit cycles that can bifurcate from a slow-fast Hopf point.
The most difficult problem to deal with concerns the uniform treatment of the evolution that a limit cycle undergoes when it grows from a small limit cycle near the singular point to a canard cycle of detectable size. This explains the title of the paper. The treatment is based on blow-up, good normal forms and appropriate Chebyshev systems. In the paper we also relate the slow-divergence integral as it is used in singular perturbation theory to Abelian integrals that have to be used in studying limit cycles close to the singular point.
A singularly perturbed planar system of differential equations modeling an autocatalytic chemical reaction is studied. For certain parameter values a limit cycle exists. Geometric singular perturbation theory is used to prove the existence of this limit cycle. A central tool in the analysis is the blow-up method which allows the identification of a complicated singular cycle which is shown to persist.
We study the organization of mixed-mode oscillations (MMOs) in the Olsen model for the peroxidase-oxidase reaction, which is a four-dimensional system with multiple time scales. A numerical continuation study shows that the MMOs appear as families in a complicated bifurcation structure that involves many regions of multistability. We show that the small-amplitude oscillations of the MMOs arise from the slow passage through a (delayed) Hopf bifurcation of a three-dimensional fast subsystem, while large-amplitude excursions are associated with a global reinjection mechanism. To characterize these two key components of MMO dynamics geometrically we consider attracting and repelling slow manifolds in phase space. More specifically, these objects are surfaces that are defined and computed as one-parameter families of stable and unstable manifolds of saddle equilibria of the fast subsystem. The attracting and repelling slow manifolds interact near the Hopf bifurcation, but also explain the geometry of the global reinjection mechanism. Their intersection gives rise to canard-like orbits that organize the spiralling nature of the MMOs.
Acker et al (J. Comp. Neurosci., 15, pp.71-90, 2003) developed a model of stellate cells which reproduces qualitative oscillatory patterns known as mixed mode oscillations observed in experiments. This model includes different time scales and can therefore be viewed as a singularly perturbed system of differential equations. The bifurcation structure of this model is very rich, and includes a novel class of homoclinic bifurcation points. The key to the bifurcation analysis is a folded node singularity that allows trajectories known as canards to cross from a stable slow manifold to an unstable slow manifold as well as a node equilibrium of the slow flow on the unstable slow manifold. In this work we focus on the novel homoclinic orbits within the bifurcation diagram and show that the return of canards from the unstable slow manifold to the funnel of the folded node on the stable slow manifold results in a horseshoe map, and therefore gives rise to chaotic invariant sets. We also use a one-dimensional map to explain why many homoclinic orbits occur in "clusters'' at exponentially close parameter values.
The FitzHugh-Nagumo equation has been investigated with a wide array of different methods in the last three decades. Recently a version of the equations with an applied current was analyzed by Champneys, Kirk, Knobloch, Oldeman and Sneyd  using numerical continuation methods. They obtained a complicated bifurcation diagram in parameter space featuring a C-shaped curve of homoclinic bifurcations and a U-shaped curve of Hopf bifurcations. We use techniques from multiple time-scale dynamics to understand the structures of this bifurcation diagram based on geometric singular perturbation analysis of the FitzHugh-Nagumo equation. Numerical and analytical techniques show that if the ratio of the time-scales in the FitzHugh-Nagumo equation tends to zero, then our singular limit analysis correctly represents the observed CU-structure. Geometric insight from the analysis can even be used to compute bifurcation curves which are inaccessible via continuation methods. The results of our analysis are summarized in a singular bifurcation diagram.
Mixed mode oscillations (MMO's) composed of subthreshold oscillations (STO's) and spikes appear via a variety of mechanisms in models of neural dynamics. Two key elements that can influence the prominence of the STO's are multiple time scales and time varying parameters near critical points. These features can lead to dynamics associated with bifurcation delay, and we consider three systems with this behavior. While it is well known that bifurcation delay related to a slow time scale is sensitive to noise, we compare other aspects of the noise-sensitivity in the context of MMO's, where not only bifurcation delay, but also coherence resonance and dynamics in the interspike interval play a role. Noise can play a role in amplifying the STO's but it can also drive the system into repetitive spiking without STO's. In particular we compare integrate and fire models with models that capture both spike and STO dynamics. The interplay of the underlying bifurcation structure and the modeling of the return mechanism following the spike are major factors in the robustness and noise sensitivity of the STO's in the context of multiple time scales.
In the classical bifurcation theory, behavior of systems depending on a parameter is considered for values of this parameter close to some critical, bifurcational one. In the theory of dynamical bifurcations a parameter is changing slowly in time and passes through a value that would be bifurcational in the classical static theory. Some arising here phenomena are drastically different from predictions derived by the static approach. Let at a bifurcational value of a parameter an equilibrium or a limit cycle loses its asymptotic linear stability but remains non-degenerate. It turns out that in analytic systems the stability loss delays inevitably: phase points remain near the unstable equilibrium (cycle) for a long time after the bifurcation; during this time the parameter changes by a quantity of order 1. Such delay is not in general found in non-analytic (even infinitely smooth) systems. A survey of some background on stability loss delay phenomenon is presented in this paper.
We consider a two dimensional family of real vector fields. We suppose that there exists a stationary point where the linearized vector field has successively a stable focus, an unstable focus and an unstable node. It is known that when the parameter moves slowly, a bifurcation delay appears due to the Hopf bifurcation. The main question investigated in this article is the continuation of the delay in the region of the unstable node. Another problem is to determine the input-output relation which characterizes all the possible delays.
We give a non-exhaustive overview of the problem of bifurcation delay from its appearance in France at the end of the eighties to the most recent contributions. We present the bifurcation delay for differential equations as well as for discrete dynamical systems.
The construction of orbits with specific asymptotic properties, such as orbits that are heteroclinic or homoclinic to certain invariant sets, involves tracking stable and unstable manifolds around the system's phase space. This work addresses how, in some generality, the tracking can be achieved during the passage near a distinguished invariant manifold in the phase space. This leads to a very general form of the Exchange Lemma and it is further shown how the lemma can be used in the construction of distinguished homoclinic and heteroclinic orbits.
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