mathematics

John Guckenheimer

John Guckenheimer is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand John Guckenheimer rather than just read about it. In short: John Mark Guckenheimer (born 1945) joined the Department of Mathematics at Cornell University in 1985. He was previously at the University of California, Santa Cruz (1973-1985).

Key takeaways

John Guckenheimer belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect John Guckenheimer to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of John Guckenheimer from memory before moving on to harder problems.

Reference excerpt

John Mark Guckenheimer (born 1945) joined the Department of Mathematics at Cornell University in 1985. He was previously at the University of California, Santa Cruz (1973-1985). He was a Guggenheim fellow in 1984, and was elected president of the Society for Industrial and Applied Mathematics (SIAM), serving from 1997 to 1998. Guckenheimer received his A.B. in 1966 from Harvard and his Ph.D. in 1970 from Berkeley, where his Ph.D. thesis advisor was Stephen Smale. His book Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields (with Philip Holmes) is an extensively cited work on dynamical systems.

Research Guckenheimer's research has focused on three areas — neuroscience, algorithms for periodic orbits, and dynamics in systems with multiple time scales.

Neuroscience Guckenheimer studies dynamical models of a small neural system, the stomatogastric ganglion of crustaceans — attempting to learn more about neuromodulation, the ways in which the rhythmic output of the STG is modified by chemical and electrical inputs.

Algorithms for periodic orbits Employing automatic differentiation, Guckenheimer has constructed a new family of algorithms that compute periodic orbits directly. His research in this area attempts to automatically compute bifurcations of periodic orbits as well as "generate rigorous computer proofs of the qualitative properties of numerically computed dynamical systems".

Dynamics in systems with multiple time scales Guckenheimer's research in this area is aimed at "extending the qualitative theory of dynamical systems to apply to systems with multiple time scales". Examples of systems with multiple time scales include neural systems and switching controllers.

DsTool Guckenheimer's research has also included the development of computer methods used in studies of nonlinear systems. He has overseen the development of DsTool, an interactive software laboratory for the investigation of dynamical systems.

Awards and honors He became a SIAM Fellow in 2009. In 2012 he became a fellow of the American Mathematical Society. He won a Leroy P. Steele Prize in 2013 for his book (coauthored with Philip Holmes), and he gave the Moser Lecture in May 2015.

Selected publications Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields (with Philip Holmes), Springer-Verlag, 1983, 453 pp. Phase portraits of planar vector fields: computer proofs, Journal of Experimental Mathematics 4 (1995), 153–164. An improved parameter estimation method for Hodgkin-Huxley model (with A. R. Willms, D. J. Baro and R. M. Harris-Warrick), J. Comp. Neuroscience 6 (1999), 145–168. Computing periodic orbits and their bifurcations with automatic differentiation (with B. Meloon), SIAM J. Sci. Stat. Comp. 22 (2000), 951–985. The forced van der Pol equation I: the slow flow and its bifurcations (with K. Hoffman and W. Weckesser), SIAM J. App. Dyn. Sys. 2 (2002), 1–35.

Notes

References John Guckenheimer. "John Guckenheimer, Professor of Mathematics and Theoretical and Applied Mechanics, Cornell University". Cornell University. Archived from the original on 2007-10-23. Retrieved 2008-03-31. John Guckenheimer. "John M. Guckenheimer, Professor of Mathematics". Cornell University. Retrieved 2008-03-31.

External links John Guckenheimer - Home page Department of Mathematics home page Theoretical & Applied Mechanics home page DsTool FTP download area Dr. John Guckenheimer user page on Scholarpedia

Illustrations

Worked examples

Example 1 — a first encounter with John Guckenheimer

Start with the simplest possible case. Write down what John Guckenheimer claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to John Guckenheimer before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about John Guckenheimer ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of John Guckenheimer

In research: John Guckenheimer appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses John Guckenheimer in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: John Guckenheimer is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1945 births, 20th-century American mathematicians, 21st-century American mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for John Guckenheimer outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study John Guckenheimer in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what John Guckenheimer means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain John Guckenheimer out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is John Guckenheimer in simple terms?

John Mark Guckenheimer (born 1945) joined the Department of Mathematics at Cornell University in 1985. He was previously at the University of California, Santa Cruz (1973-1985).

Why does John Guckenheimer matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study John Guckenheimer?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on John Guckenheimer.