## Research

The chain fountain

Given a sufficiently long bead chain in a cup, if we pull the end of the chain over the rim of the cup, the chain tends to continuously “flow”out of the cup, under gravity, in a common siphon process. Surprisingly enough, under certain conditions, the chain forms a fountain in the air! This became known as the Mould effect, after Steve Mould who discovered this phenomenon and made this experiment famous on YouTube in a video that went viral. The reason for the emergence of this fountain remains unclear. This effect was shown  to be due to an anomalous reaction force from the top of the pile of beads [ref]; a possible origin for this force was proposed in the same paper.

In  we describe some experiments that contribute toward the clarification of the origin of this force, and show that the explanation goes far beyond the one proposed by Biggins and Warner.

Human perception of dimension

Did you ever see a hypersphere? I guess not. So… why not? This is the question we all, sooner or later, have posed to ourselves. Why do we blockade when passing from three to higher dimensions? Could this have been different? Is there hope that in the future we can overcome this condition? These were some questions we try to answer in .

Let us start by giving an attempt to clarify what we usually mean when we say that we are ‘‘seeing’’ or ‘‘visualizing’’ a geometrical object, for example, a sphere. On one hand, there is the pure mathematical object that we all know as the sphere, this sphere has a mathematical characterization and lives in an abstract space, in the Platonic sense if you will. On the other hand, there is our physical experience of a real sphere, something that we can perceive with our senses and spatial intuition. What do we mean by ‘‘seeing’’ a mathematical sphere? It is, in my opinion, the possibility to imagine the mathematical sphere in our three-dimensional physical world, something that could have been real, like a soccer ball. Even as a product of our imagination, we can imagine some physical interaction with it, holding it, rotating it, or changing its position. Somehow we can use our three-dimensional physical intuition to understand the sphere’s properties and the way it interacts with other geometrical objects. We use our three-dimensional physical space to understand the abstract Euclidean three-dimensional vector space.

It’s funny to think this way. Certainly the idea of a vector space, and the sphere, was created on the physically sensible experience of space. Anyway, the abstract idea of vector space and its geometric objects gained a life of their own and their properties were generalized to higher-dimensional spaces. Now we have a four-dimensional object, which we call a hypersphere, with similar properties to the three-dimensional counterpart, and we do not have the correspondent physical object to ‘‘see’’ it.

Evidence from neurobiology shows that our brain seems to have been originally created to manage our homeostatic mechanisms and physical interactions with our surroundings. Memory, conscience, planning, and deciding seem to be superimposed on the top of a cerebral structure originally built to deal with sensation and movement. Almost every part of our brain, even if it is responsible for some other cognitive function, has some sensory and motor signals. Apparently when we imagine a physical sensation, for example holding a sphere, we are partially activating the very same tactile responses that would be activated if the sphere were really in our hands. These ideas are also somehow connected with a philosophical theory called embodied cognition. Therefore, our physically sensible intuition must be incomparably stronger than our mathematical intuition in general; this is probably what gives us this sensation of ‘‘seeing’’ a sphere in opposition to comprehending a sphere. When we ‘‘see’’ we are using this strong spatial intuition. This is probably why we feel this powerlessness when moving from three to four dimensions.

This is usually the end of the story. We cannot see a hypersphere simply because we live in a three-dimensional physical space. In  we try to convince you that it could be different.

This is of course a question of mathematical relevance; this limits our daily mathematical activity, so at least we hope to clarify some ideas. With this in mind we will try to find some analogies with the way we perceive color and make some imaginary experiments that we hope will convince you that, even imprisoned in a three dimensional physical space, we could have been different.

We propose that this limitation is simply a characteristic of our species; it is given by our biology, which in turn was shaped to succeed in our environment, from an evolutionary point of view. A different sensory system or body characteristics could have equipped us with other perception capabilities of the mathematical geometric objects.

Synchronization of nonlinear coupled oscillators

The idea of synchronization of nonlinear systems weakly coupled had attracted a lot of attention recently, given the broad range of applications, from neuroscience to engineering. When the oscillators are not identical we can consider the generalizing synchronization framework, where a existence of an invariant manifold allows a reduction in dimension.

In  we consider the synchronization of a network of linearly coupled and not necessarily identical oscillators, where we present an approach to the existence of the synchronization manifold which is based on some result developed by R. Smith for the study of periodic solutions of ODEs. Our framework allows the study of a large class of systems and does not assume that the systems are small perturbations of linear systems. Moreover, it provides a practical way to compute estimations on the parameters of the system for which generalized synchronization occurs. Additionally, we give a new proof of the main result of R. Smith on invariant manifolds using Wazewski’s principle.

In  we consider the case of generalized synchronization of a system of several oscillators coupled by a medium. In particular we study the case of a concrete system modeling the dynamics of a chemical solution in two containers connected to a common third container.

Dynamics of dissipative systems on the cylinder

My PhD studies  were supervised by Professor Rafael Ortega from the University of Granada.

We studied the dynamics of periodic dissipative systems with a spatial angular coordinate. More precisely, systems of the form x’’= F(t,x) where x\in\R^n and F(t,x)=F(t,x+X)=F(t+T,x) for some constant vector X and positive T. Some examples are the pendulum equation or systems of coupled pendula. Our main goal was to study the attractor of the Poincaré time T-map of those systems. When the dissipation is strong enough we expect to have a Jordan curve as an attractor. This fact was proved for the forced pendulum by Mark Levi and simultaneously by Min, Xian, Jinyan.

In  we proved that the attractor associated to the equation x''+h(x)x'+g(t,x)=0, where g and h are periodic in both variables, is a Jordan curve provided that (g(t,x)-g(t,y))/(x-y) is bounded and have c^2/4 as an upper bound, where min h=c>0. This is a slight improvement on the results on the results mention above since it included the case where g is not differentiable. On the other hand, we proved  that the existence of inversely unstable solutions imply that the attractor is not a Jordan curve. The notion of inversely unstable solution was introduced by N. Levinson [ref.], we used a topological version of this concept. This result was used, in the same paper, to show that the above constant, c^2/4, is optimal.

Later on we proved the optimality of c^2/4 by means of an autonomous equation. This proof was based on some bifurcation diagrams constructed by Tricomi on the thirties for the pendulum equation driven by a constant torque 

In higher dimensions we can not apply the ideas used for the plane so we used a condition introduced by R. Smith to give conditions under which a system of the above type has a Jordan curve as an attractor. The above result was applied to systems of coupled oscillators and to a n'th order ordinary differential equation of the type

x^(n)+a_{n-1}x^(n-1)+…+a_2x''+a_1x'=g(t,x,x',…,x^(n-1)),

where g is periodic on t and x .

Dynamics of conservative systems in the plane

My research on this topic has to do with the Poincaré’s last geometric theorem and its generalizations. This theorem states that if an area-preserving homeomorphism from the annulus to itself rotates the two components of the boundary in opposite directions (twist condition) then there are at least two fixed points.

A generalization of this theorem, where it is assumed that only one point of the inner boundary of the annulus is rotated in opposite direction, was presented In [ref.]. We proved in  that the roles of the inner and outer boundary cannot be interchanged.

There is also a generalization due to W. Ding [ref.] where the homeomorphism is defined in an annular region and the boundary curves do not need to be invariant. In this result it is assumed that the inner boundary of the annular region is star-shaped. At first sight this seems to be technical condition and there were some efforts to remove it. We proved  that this condition is necessary, constructing an example of an area-preserving homeomorphism between two annular regions, without fixed points, and satisfying the twist condition.

Applied dynamics / Physics

We worked on some models arising in the theory of liquid crystals. The materials are submitted to an elliptic shear and to magnetic periodic fields. In  we describe the study of a system of non-autonomous equations with singularities on the plane, where we found by computational methods a cascade of subharmonics.

In  we do a numerical simulation study of the effect of an applied magnetic field on the stability of the flow of nematic slabs subjected to an elliptically polarized shear. The flow of a low molecular weight nematic and of a polymer nematic is studied for small excitations. For both materials, irregular motion of the director can develop, depending on the ellipticity of the shear. A strong enough applied magnetic field has a stabilizing effect on the director motion in such flow regime.

Nonlinear oscillations

During the Master thesis , supervised by Professor Maria do Rosário Grossinho, we studied some second order periodic differential equations of the type u’’+f(u)u’+g(t,u)=e(t).

In  we prove the existence of a sequence of periodic solutions of this equation with minimal period tending to infinity, assuming f=0 and some conditions on the growth of the non-linearity on infinity. On the same paper we also studied the case where a one-sided growth restriction is assumed for g and we prove the existence of infinitely many periodic oscillations.

The case of a non-linearity with a repulsive singularity on the origin was studied in . Since the work of Lazer and Solimini [ref.] on this problem that the strong force condition \int_0^1 g(u) du was assumed and considered necessary for the existence of periodic oscillations. We found a weaker condition, namely the existence of a sequence of intervals on a neighbourhood of the origin where the integral of -g is arbitrarily large. In this conditions we proved the existence of an infinite number subharmonics with minimal period tending to infinity when f=0, and the existence of periodic solutions when f is not equal to zero. This new condition, in some sense, unifies the singular with the non-singular case. The paper  won the Student Paper Prize, SIAM, 2001.