Two scientists from Research Center of the Sea and the Atmosphere of Buenos Aires (Argentina) have created a mathematical tool that could help fully understand the chaos. Christened as “templex“, the system combines two branches of knowledge: algebraic topology and graph theory. It is the first system The advance is funded by the Ministry of Science, Technology and Innovation and has been the featured article in the specialized magazine chaos.
The researchers Denisse Sciamarella and Gisela Charó propose a new mathematical concept that will allow, in the short term, the design of a software capable of serving other scientific teams in Argentina and the world to solve very different problems: measure the dispersion of a contaminant in a fluid, create models of tumor cell development, estimate mixing processes in a reactor, as well as address different areas of the scientific knowledge.
A triangle and its deformed imitation in a circle are equivalent for the topology
“Topology was an invention of the mathematician Henri Poincare in the XIX century. He defined it as ‘a purely qualitative geometry whose theorems would be true if the figures, instead of being exact, were crudely imitated by a child’. The core idea is that two things can be different from each other, but if one is the deformation of the other, then they are topologically equivalent“, explains Sciamarella.
The scientist adds that “a child can draw a deformed circle to such an extent that it looks like a triangle, but for topology both figures are the same thing. In this sense, what we propose is a new mathematics that allows us to use topology to understand chaos.”
To understand it in a simpler way, “let’s suppose that until now, with the previous system, scientists had a network of highways. Our contribution, by creating the templexis that from now on they will have a guide that has arrows and signs that indicate how to circulate on the highways”, he clarifies. In addition, “according to the study of that network of highways with traffic direction, it is possible to notice what is the law that governs a certain phenomenon“.
Just give the templex any time series, so that it determines what type of dynamics it is: the observed variable can be the global temperature of the Earth, the salinity measured at a certain point in the ocean or a voice signal. Just as there is a law that determines that energy is conserved, the principle that governs in this case is that, for dynamic and deterministic systems, the topology does not change.
Helpful for understanding chaos
The definition of chaos is chaotic. Physics defines it as “the order within the disorder”, since, although it has a random appearance, in reality it is not. The point is that chaos, as a scientific problem, does not have a correct description with current mathematical tools.
Mathematics has failed to define chaos correctly
Those that exist use knot theory to put together a model called “template“, but knots are undone in a maximum of three dimensions and most chaotic problems require more. Christophe LetellierProfessor at the University of Rouen and co-author of this paper, is a specialist in calculating templates for chaotic three-dimensional systems.
Precisely, the templex is the invention of Gisela Charo —PhD in Engineering from the University of Buenos Aires (UBA) and Conicet postdoctoral fellow—, and Denise Sciamarella— doctor in physics (UBA), researcher at the Center National de la Recherche Scientifique (France) and deputy director of the Franco-Argentine Institute for Climate and Impact Studies (IFAECI)— with the aim of finding a solution to this problem.
“Some scientists call this problem the curse of dimensionality,” says Sciamarella. The templex is a mixture of an algebraic complex, also called complexand a template. “Until now, knot theory was used in chaos topology. The problem was that if the system had more than three variables, that is, more than three dimensions, it no longer applied,” he says.
Denisse Sciamarella began her specialization in this field in 2001 from her doctoral thesis, related to the analysis of the topological structure of chaotic flows. Over time, she became a specialist and a reference in the field of nonlinear dynamics (chaos) and fluid mechanics applied to biological and geophysical problems.
The difficulty of explaining what mathematics is for lies in its transversality. The general answer is that “it works for everything”, enthuses Sciamarella. The risk of saying that something is good for everything is that it might not be good for anything.. In this effort, the specialist provides some examples that can help to break down the importance of progress.
The risk of saying that something is good for everything is that it might not be good for anything.
From a time series, it is possible to infer the topology and from this, determine the type of law that governs the phenomenon. “In climate science, there are more than 20 climate models that are used to perform climate simulations and future climate projections. Although all of them have a formulation based on the equations of fluids, they differ in several aspects: modeling of aerosols, the dynamics of the ice sheet or the ocean, among other processes”, he states.
The author reasons that “there has been talk of a tower of Babel of models. How do you know which model to stay with? To what extent can a model be said to be a good representation of the observations? templexcan provide the solution.
The ‘templex’ can determine which model is better for climate projections
Another utility of templex lies in comparing data with each other. “For example, when you have a set of data measured by satellites and another set measured by a buoy in the ocean. Although they are different technologies with different scopes, if both time series have the same topology, it means that the information they provide is quite similar to each other. Assessing this is important if you are looking examine the effectiveness of both resources.
Also, it is used in a wide range of additional areas: to detect cardiac arrhythmiasto forecast the solar cyclefor monitoring the vegetation in arid regionsto determine the equivalence of electronic circuits, in the ecological model analysisin laser optics or in acoustics.
“When I started working on chaos topology, I used it to characterize spoken and sung voice signals, and we discovered that the topology allowed us to account for the fact that the phonatory system does not work in the same way in the different vocal registers, and that what singers call ‘passage’ from one register to another results in a topological change. Knowledge of these topological changes can be used when designing a prosthesis.”
Uses are also reported in an area as sensitive as oncology. “In medicine, Christophe Letellierfor example, used templates to study three cell populations (host, immune and tumor cells) and thus suggest new trends to understand the interactions of some tumor cells and their environment”, he adds. templex opens a path that extends applications to even more complex systems, freeing these methods from the curse of dimensionality.