bifurcation theory and its applications: from biology to greek ornamental design

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Most complex physical, biological and other systems often depend on a limited number of parameters (i.e., growth, death, predation or competition rates in biological systems) that determine qualitative behavior of the system. The parameters may vary within specific ranges. Bifurcation theory provides methods to understand the qualitative behavior of a system and its variation as the parameters vary, especially when it comes to changes between equilibrium states of the system. It helps to formulate criteria in order to identify “critical boundaries” that one may wish to either approach, or stay away from.

In this seminar, the following topics (supplemented with numerous examples from biology) will be covered:

  • Introduction to non-linear dynamical systems

  • Introduction to bifurcation theory

  • Introduction to co-dimension of bifurcations and phase-parameter portraits of dynamical systems

  • Understanding the nature of equilibria, including bifurcations of equilibria in continuous and discrete time systems

  • Understanding the nature of oscillations, including bifurcations of equilibria and periodic orbits

  • Introduction to various types of bifurcations, such as homoclinic and heteroclinic bifurcations

  • Introduction to Hamiltonian systems

  • Application of Hamiltonian systems to understanding ancient Greek ornamental designs such as the one depicted in the photo above. 

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instructor: Faina Berezovskaya, howard university

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Dr. Faina Berezovskaya is a professor of mathematics at Howard University, Washington DC. She has significant teaching and research experience in various fields of applied mathematics and mathematical modeling in biology. Her main research interests include but are not limited to: (1) Development of analytical and numerical methods of the qualitative theory of dynamic systems and theory of bifurcations; (2) Applications to modeling of complex biological systems.