{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T13:31:04Z","timestamp":1787232664344,"version":"build-2736575974"},"reference-count":27,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Dyn. Syst."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>This contribution extends the normal vector method for the optimization of parametrically uncertain dynamical systems to a general class of nonlinear discrete time systems. Essentially, normal vectors are used to state constraints on dynamical properties of fixed points in the optimization of discrete time dynamical systems. In a typical application of the method, a technical dynamical system is optimized with respect to an economic profit function, while the normal vector constraints are used to guarantee the stability of the optimal fixed point. We derive normal vector systems for flip, fold, and Neimark\u2013Sacker bifurcation points, because these bifurcation points constitute the stability boundary of a large class of discrete time systems. In addition, we derive normal vector systems for a related type of critical point that can be used to ensure a user-specified disturbance rejection rate in the optimization of parametrically uncertain systems. We illustrate the method by applying it to the optimization of a discrete time supply chain model and a discretized fermentation process model.<\/jats:p>","DOI":"10.1137\/09075696x","type":"journal-article","created":{"date-parts":[[2010,5,7]],"date-time":"2010-05-07T18:28:08Z","timestamp":1273256888000},"page":"357-390","source":"Crossref","is-referenced-by-count":14,"title":["Robust Optimization of Fixed Points of Nonlinear Discrete Time Systems with Uncertain Parameters"],"prefix":"10.1137","volume":"9","author":[{"given":"Darya","family":"Kastsian","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Martin","family":"M\u00f6nnigmann","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,5,7]]},"reference":[{"key":"R1","unstructured":"M. E. Alexander and S. M. Moghadas,\n                      $O(l)$ shift in Hopf bifurcations for a class of non-standard numerical schemes\n                      , in Proceedings of the 2004 Conference on Differential Equations and Applications in Mathematical Biology, Electronic J. Differ. Equ. Conf. 12, Texas State University-San Marcos, San Marcos, TX, 2005, pp. 9\u201319."},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.2006.1724"},{"key":"R3","doi-asserted-by":"crossref","unstructured":"W.J. Beyn, A. Champneys, E. Doedel, W. Govaerts, Yu. A. Kuznetsov, and B. Sandstede,\n                      Numerical continuation, and computation of normal forms\n                      , in Handbook of Dynamical Systems, Vol. 2, B. 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