Modeling Nonlinear Dynamics from Equations and Data with Applications to Solids, Fluids, and Controls

“George Haller provides a substantial literature overview which, in combination with his many contributions, results in the first book on spectral model reduction techniques. He covers basic theory for both linear and nonlinear problems and model- and data-driven techniques, and he demonstrates the method’s excellent behavior on nontrivial problems.” – Jan S. Hesthaven, Karlsruhe Institute of Technology (KIT) “This is an exciting topic and a powerful method that already has and will continue to have an impact on any field where model reduction is a key ingredient or a central concern. The content is presented in a logical order, from the equation-based to the data-driven version of the method to applications that progressively increase in complexity.” – Peter Schmid, KAUST This concise text presents an introduction to the emerging area of reducing complex nonlinear differential equations or time-resolved data sets to spectral submanifolds (SSMs). SSMs are ubiquitous low-dimensional attracting invariant manifolds that can be constructed systematically, building on the spectral properties of the linear part of a nonlinear system. The internal dynamics within SSMs then serve as exact, low-dimensional models with which the full system evolution synchronizes exponentially fast. SSM-based model reduction has a solid mathematical foundation and hence is guaranteed to deliver accurate and predictive reduced-order models under a precise set of assumptions. This book illustrates the power of SSM reduction on a large collection of equation- and data-driven applications in fluid mechanics, solid mechanics, and control.

George Haller is a professor of mechanical engineering at ETH Zürich, where he holds the Chair in Nonlinear Dynamics and heads the Institute for Mechanical Systems. His prior appointments include tenured faculty positions at Brown, McGill, and MIT. He also served as the inaugural director of Morgan Stanley’s fixed-income modeling center. Professor Haller is the recipient of a Sloan Fellowship in mathematics, an ASME Thomas Hughes Young Investigator Award, a School of Engineering Distinguished Professorship (McGill), the Stanley Corrsin Award of the APS, and the Lyapunov Award of the ASME. He is an external member of the Hungarian Academy of Science and a fellow of SIAM, APS, and ASME.