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"Number of real roots and points of intersection". - Michael Shub.
"On the number of real roots of random polynomials". - Diego Armentano.
"Partial Hyperbolicity and Ergodic Theory". - Michael Shub.
"Some aspects of partially hyperbolic diffeomorphisms in dimension 3". - Rafael Potrie.
"Dynamical Systems and Cellular Biology" - Michael Shub.
Cusp Bifurcation in Metastatic Breast Cancer Cells
Brenda Delamonica, Gabor Balazsi, Michael Shub
Ordinary differential equations (ODEs) can model the transition of cell states over time. Bifurcation theory is a branch of dynamical systems which studies changes in the behavior of an ODE system while one or more parameters are varied. We have found that concepts in bifurcation theory may be applied to model metastatic cell behavior. Our results show how a specific phenomenon called a cusp bifurcation describes metastatic cell state transitions, separating two qualitatively different transition modalities. Moreover, we show how the cusp bifurcation models other genetic networks, and we relate the dynamics after the bifurcation to observed phenomena in commitment to enter the cell cycle.