• Abstract: 

The dynamics of (non-conservative) homeomorphisms of a compact metric space admits a canonical decomposition in chain recurrence classes (Conley). For C¹-generic diffeomorphisms, the chain recurrence class of a periodic point it is homoclinic class ([B–Crovisier], based on Hayashi connecting lemma). However there are locally C^r-generic diffeomorphisms having uncountably many aperiodic classes, i.e  chain recurrence classes without periodic orbits. Up to recently, all such aperiodic classes was adding machines. We present C¹-open sets of partially hyperbolic diffeomorphisms in dimension 3 present a wide variety of dynamical behaviors: minimal but not uniquely ergodic, or at the contrary uniquely ergodic but non-transitive, etc. This work consists in four long papers ending with [Invent. Math. 238 (2024), 63pages]. In this talk, I will present the motivation, the setting of our example, and the different steps of the construction, in particular the key notion of flexible periodic points, which may have many other applications.

• Bibliography:

[1] A periodic chain recurrence classes of C¹- generic diffeomorphisms. Bonatti Christian; Shinohara, Katsutoshi Invent. Math. 238 (2024), no. 2, 637-689. 

[2]  A mechanism for ejecting a horseshoe from a partially hyperbolic chain recurrence class
Bonatti, Christian; Shinohara, Katsutoshi Ergodic Theory Dynam. Systems 44 (2024), no.
8, 2080-2142.

[3] Volume hyperbolicity and wildness Bonatti, Christian; Shinohara, Katsutoshi Ergodic Theory
Dynam. Systems 38 (2018), no. 3, 886-920.

[4] Flexible periodic points Bonatti, Christian; Shinohara, Katsutoshi Ergodic Theory Dynam.
Systems 35 (2015), no. 5, 1394-1422. 

• Abstract:

The canard cycles considered in this talk are simple closed curves in a slow-fast system, consisting of a succession of fast orbits and curves of singularities, along which one has alternating normally hyperbolic attracting and normally hyperbolic repelling behavior. Canard cycles can be perturbed into limit cycles. A theoretical framework has been developed to study these limit cycles.

Besides its possible application to a thorough study of specific models, this theory is also very useful in the study of Hilbert’s 16th problem, asking for the maximal number of limit cycles a polynomial vector field can have, depending on its degree. Special attention will be given to Liénard equations. We will give a survey of the most important techniques that can be used in proving the existence of limit cycles near canard cycles. These canard cycles contain fast orbits, occurring in layers of fast orbits, and between these layers we restrict, along the curves of singularities, to generic Hopf breaking mechanisms and generic jump breaking mechanisms. The essential data can be described by the connection diagram. Based on the connection diagram we will provide some general results concerning these equations, especially related to totally balanced canard cycles. 

We prove that all possible abstract connection diagrams can be realized by a multi-layer canard cycle of a planar polynomial vector field. We also precisely state the conditions under which a connection diagram can be realized by a polynomial Liénard equation. 

• Abstract:

In the n-body problem, when bodies tend to a total collision, then its normalized shape curve converges to the set of normalized central configurations, which has SO(3)
symmetry in the planar case. This leaves a possibility that the normalized  shape curve  tends to the set obtained by rotations of some central configuration instead of a particular point on it. This is the infinite spin problem which concerns the rotational behavior of total collision orbits in the n-body problem. 
We show that the infinite spin is not possible if the limiting shape  is isolated from other connected components of the set of normalized central configurations. Our approach extends the method from recent work for total collision for the planar case by Moeckel and Montgomery.  The main tool is a full reduction SO(3)--symmetry in a context of vanishing angular momentum.

• Abstract: 

Spatially localised solutions are a characteristic phenomenon of nonlinear spatially extended systems. Classical examples in one spatial dimension are travelling waves such as interfaces in phase field models or pulses in driven-dissipative systems. Generally, a major theme is the existence and stability of such solutions on the real line or lattices. Beyond this, in the past decades also the interaction dynamics of sufficiently separated multiple localised structures has been broadly studied. The talk gives an introduction and then focusses on two directions in which much less is known, reporting results with various collaborators. The first concerns infinite time interaction that involves annihilation of localised states. Here, at the cost of losing spatial dispersion, more can be said for a class of cellular automata and their wave-related non-wandering sets and entropy. The second concerns self-organised motion of a localised state, which has been observed essentially numerically with evidence of chaotic dynamics. On a rigorous level such type of dynamics has recently been verified for a class of multi-component reaction diffusion systems through a singularity unfolding. 

• Abstract:

We study linear response for families of skew-product dynamical systems with contracting fibres. Our approach is based on a sectional transfer operator acting on families of probability measures along the fibres. This operator allows us to describe invariant measures of the skew-product in terms of sample measures over the base dynamics, irrespective of whether the base map is invertible. Under general assumptions, we prove their differentiability with respect to system parameters in suitable Banach spaces.

This is joint work with Wael Bahsoun.

• Abstract:

We consider delay differential equations of the form

  y'(t) = −ay(t) + bf(y(t−1)) (E_f)  

with parameters b>a>0 and Mackey–Glass-type nonlinear feedback functions

f(ξ) = ξ^k 1+ξⁿ ,   where   k>0   and   n>0.

The case k=1 corresponds to the classical Mackey–Glass equation, while the case k>1 appears in population models with an Allee effect. The case k∈(0,1) arises in certain economic growth models. The pointwise limit of f, as n tends to infinity, is the discontinuous g:[0,∞)→[0,1] with g(ξ)=ξ^k for ξ∈[0,1), g(1)=1/2, and g(ξ)=0 for ξ>1. The special structure of g makes it possible to analyze the dynamics of the discontinuous equation

  x'(t) = −ax(t) + bg(x(t−1)) (E_g)

It is then shown that several dynamical properties of (E_g) persist for the original equation (E_f) when n is large.

For certain parameter values, stable periodic orbits with complicated structures are constructed. Heteroclinic connections between periodic orbits, as well as between equilibria and periodic orbits, are described. A threshold value b^*=b^*(a)>a is obtained such that, for b<b^*, the dynamics is simple, whereas for b>b^*, the equation exhibits bistability. For the critical parameter value b=b^*, there exists a homoclinic orbit to a saddle point. In this case, a Shilnikov-type chaos is expected in a small neighborhood of the homoclinic orbit.

• Abstract: 

We study the problem of the orbital dynamics of the innermost exoplanet of the υ-Andromedæ system (i.e., υ-And b) into the framework of a Secular Quasi-Periodic Restricted Hamiltonian model. This means that we preassign the orbits of the planets that are expected to be the biggest ones in that extrasolar system (namely, υ-And c and υ-And d). The Fourier decompositions of their secular motions are injected in the equations describing the orbital dynamics of υ-And b under the gravitational effects exerted by those two exoplanets. By a computer-assisted procedure, we prove the existence of KAM tori corresponding to orbital motions that we consider to be very robust configurations, according to the analysis and the numerical explorations that will be (briefly) discussed and summarized.

Based on a joint work with Rita Mastroianni.