Slow divergence integral and its application to classical Liénard equations of degree 5

Chengzhi Li¹ and Kening Lu²

¹School of Mathematical Sciences, Peking University, Beijing 100871, China

²Department of Mathematics, Brigham Young University, Provo, UT 84602, USA

The slow divergence integral is a crucial tool to study the cyclicity of a slow-fast cycle for singularly perturbed planar vector fields. In this talk, we first introduce a useful form of this integral, then use it to prove that the slow divergence integral along any non-degenerate slow-fast cycle for singular perturbations of classical Li´enard equations of degree 5 has at most one zero, and the zero is simple if it exists; hence its cyclicity in this class of equations is at most two. Up to now there are many interesting results about Li´enard equations of degree 3, 4 and 6, but almost nothing is known about degree 5. This result can be seen as a first step to study the uniform property for classical Li´enard equations of degree 5.

Singular perturbations and ionchannel problem

Weishi Liu

University of Kansas, USA

I channel problems concern macroscopic properties of ionic flow through nano-scale ion channels .It is no coincidence that singularly perturbedsystemsserve as suitablemodels for ananlyzing these multi-reveals special structures (idealized physcial situations) of multi-scale phenomena and allows one to extract concrete informations for specific problems .this is the case for the poisson-Nernst-Plank(PNP)systems as primitive models for ionic flows. In this talk ,wewill describethe geometric singular perturbation framework for ananalysis of PNP systems and report a number of concrete results that are directly relevant to central topics of ion channel problems. the talk is based on works with several collaborations.

A Step-type Solution for the Affine Singularly Perturbed Optimal Control Problem

Mingkang Ni

Department of Mathematics, East China Normal University, Shanghai 200241

Limeng Wu

School of Mathematics and Information Technology, Hebei Normal University of Science and Technology, Qinhuangdao  Hebei 066004

In this paper, we consider the boundary layer and internal layer for a class of affine singularly perturbed optimal control problem. Based on the geometrical theory, we study the dynamical behavior of the solution for the optimal control system. Through the geometrical analysis, we obtain that the internal layer solution for the singularly perturbed optimal control problem exists. Then, we construct the uniformly valid formal asymptotic solution for the optimal control problem by the direct scheme method. Finally, a numerical example is given to show the main result.

Slow-fast Bogdanov-Takens bifurcations in an application

Peter de Maesschalck

 Hasselt University, Belgium

We study a more degenerate version of the well-known slow-fast Van der Polsystem, this time with a singularity that singularly unfolds as a Bogdanov-Takens bifurcation. We base ourselves on the local study performed in [1], complement it with a global geometric singular perturbation analysis(in [2]) to give a thorough view of all involved bifurcations such as Hopf, Homoclinic, SNIC bifurcations.

[1] De Maesschalck, P. and Dumortier, F., Slow-fast Bogdanov-Takens bifurcations, bifurcations, Journal of Differential Equations 250(2)(2011),1000-1025.

[2] De Maesschalck, P. and Wechselberger, M., Neural excitability and singular bifurcations, Journal of Mathematical Neuroscience 5(16)(2015),1-32.

On periodic orbits in non-smooth and singularly perturbed differential equations, with applications

Rafel Prohens

Universitat de les Illes   Balears, Spain

In this talk we present a review of some results concerning limit cycles in non-smooth differential equations and also on singularly perturbed systems. This review is not intended to be exhaustive, but rather to show some problems on which we have been working in this field. Regarding neuroscience some results on which we are currently working on, controlling periodic orbits and their period used in estimating synaptic conductances, will also be presented.

Canard cycles with several breaking parameters

Robert Roussarie

University   of Burgundy, France

In 2-dimensional slow-fast systems, existence of canard cycle is a non generic phenomenon. In order to exhibit a canard cycle, the system must be included in a parameter family. Among other parameters, there are breaking parameters: a canard cycle only exists when the breaking parameters are equal to zero. Then, each canard cycle is related to the number of its breaking parameters.

This talk is about bifurcations of canard cycles and particularly the study of theircyclicity, i.e. the upper bound for the number of bifurcating limit cycles. These limit cycles are related to the solutions of a system of equations which are expressed through smooth slow divergence integrals, depending on layer variables.

Bifurcations of canard cycles with a single breaking parameter are now well understood and the cyclicity is equal to 1 plus the multiplicity of a single smooth 1-variable function, difference of the two slow divergence integrals.

When = 2; it is possible to show that the cyclicity is equal to 2 plus the multiplicity of intersection of two smooth functions with two variables, each difference of two slow divergence integrals.

This reduction to a system of smooth functions, difference of slow divergence integrals, is not known for : Nevertheless, under a generic assumption, it is possible to reduce the system of equations to a single one, inside any rescaled layer : in such a rescaled layer the bifurcating limit cycles are given by the fixed points of a map, composition of translated power functions of the form .By a general argument of Khovanskii, it is known that the upper bound of the number offixed points isfinite and independent of the parameters. It is easy to obtain that and . I shall explain how to prove that . For , a very rough estimation of can be deduced from the Khovanskii theory of fewnomials.

This talk is based on several papers written in collaboration with Freddy Dumortier, Lilia Mahmoudi and Magdalena Caubergh.

Quasi-steady state - A mathematical characterization

Sebastian Walcher

Mathematik A, RWTH Aachen, Germany

We discuss parameter dependent polynomial ordinary differential equations that model chemical reaction networks. By classical quasi-steady state (QSS) reduction we understand the following heuristic: Set the rate of change for certain (a priori chosen) variables equal to zero and use the resulting algebraic equations to obtain a system of smaller dimension for the remaining variables. This procedure will generally be valid only within certain parameter ranges: The reduction is accurate if and only if the corresponding parameter is what we call a QSS parameter value, and the reduction is approximately accurate if and only if the corresponding parameter is close to a QSS parameter value. In turn, the QSS parameter values can be characterized by polynomial equations and inequations, hence parameter ranges for which QSS reduction is valid are accessible in an algorithmic-algebraic manner. A closer investigation of QSS parameter values and the associated systems shows the existence of certain invariant sets; here singular perturbations turn up in a natural manner. We compare QSS reduction and singular perturbation reduction, and show that, while they do not agree in general, they do, up to lowest order in a small parameter, for a quite large and relevant class of examples. Some applications are given.

Singular Perturbed Monotone & Competitive Systems

Yi Wang

Univ. of Sci & Tech of China

In this talk, I would like to present a brief survey on the dynamics of monotone and competitive systems. Based on this, certain dynamics are also considered for the singular perturbed competitive systems.

Dynamics of singularly perturbed differential systems via rotated vector fields and averaging methods

Xiang Zhang

School of Mathematical Sciences 

 Shanghai Jiao Tong University, Shanghai 200240, P.R. China

In this talk we report our recent results on singularly perturbed differential systems. One is on the persistence of homoclinic orbits and of heteroclinic orbits of the reduced systems on normally hyperbolic critical manifolds. Another one is on the persistence of periodic orbits inside a family of periodic orbits of the layer systems.

For the first problem, the usual way is to compute the Melnikov functions which determine the transversal intersection of the stable and unstable manifolds. As we know, it is in general a hard work. Here we provide a simple characterization on the transversal intersection in case that the reduced system forms a generalized rotated vector field.

For the second problem, as we know the family of periodic orbits of layer systems do not satisfy any hyperbolic conditions. How to get their persistence to the original singular perturbation systems? Here we show that the averaging methods are good to treat this problem. Moreover, our techniques are new in application of the averaging methods.