Drawing the periodicity in a chemical oscillatory dynamical system

What is it about?

Chemical oscillations have been used as a mathematical paradigm for the analysis of dynamical systems presenting periodic behavior. Despite all knowledge accumulated over decades, the complexity exhibited by many dynamical systems is poorly understood, yielding a scenery where a theoretical model is non-existent. Our work shows that the periodicity in the Brusselator, an autocatalytic chemical oscillator, self-organizes in hierarchical structures responsible for the genesis of complex sets. Working as a “skeleton” for the periodic motion, two fundamental structures in the forced Brusselator support periodic oscillation, connecting local behavior to global one. We apply our ideas to a variety of situations in the Brusselator obtained from several combinations of its parameters, discussing how the skeletal formulation explains the large variety of behaviors found in the Brusselator system.

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Photo by Alexander Grey on Unsplash

Photo by Alexander Grey on Unsplash

Why is it important?

Complex phenomena involving periodic motion are found in a large class of dynamical systems used in many technological applications. Since periodic motion is frequently accompanied by chaotic motion, knowledge under what conditions periodicity occurs is vital to all the practical points of view, allowing control of a desired behavior or avoiding undesired behavior. Also, knowledge of where and how periodicity appears is essential for the construction of a theoretic formulation in dynamical systems, gathering advances in different areas such as multistability, fractal basins, and chaotic motion.

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