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Bifurcations, Center Manifolds, and Periodic Solutions



David E. Gilsinn


Nonlinear time delay differential equations are well known to have arisen in models in physiology, biology, and population dynamics. They have also arisen in models of metal cutting processes. Machine tool chatter, from a process called regenerative chatter, has been identified as self-sustained oscillations for nonlinear delay differential equations. The actual chatter occurs when the machine tool shifts from a stable fixed point to a limit cycle and has been identified as a Hopf bifurcation. This chapter develops the computational tools to determine whether a time-delay system satisfies the Hopf criteria and demonstrates the application of these tools on a model of a machine turning process.
Book chapter in Delay Differential Equations: Recent Advance


center manifolds, delay differential equations, exponential polynomials, Hopf bifurcation, limit cycle, machine tool chatter


Gilsinn, D. (2009), Bifurcations, Center Manifolds, and Periodic Solutions, Book chapter in Delay Differential Equations: Recent Advance, [online], (Accessed June 14, 2024)


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Created January 15, 2009, Updated June 2, 2021