Mathematical Sciences: Local and Global Techniques for the  Location of Periodic Solutions of Parameter Dependent       Systems with Time Delay
{'Code': '03040100', 'Directorate': {'Abbreviation': 'MPS', 'LongName': 'Direct For Mathematical & Physical Scien'}, 'Division': {'Abbreviation': 'DMS', 'LongName': 'Division Of Mathematical Sciences'}}
{'SignBlockName': 'Michael Steuerwalt', 'PO_EMAI': None, 'PO_PHON': None}
The purpose of this project is to develop new numerical techniques          for the study of periodic solutions of nonlinear differential equations         with time delay. The theory of such equations has been developed in the         last twenty years and applied to study of various dynamical systems in          engineering, biological sciences, and economics. These equations are            also called functional differential equations. One of the phenomena of          practical importance and theoretical interest is the existence of               oscillations, mathematically described as periodic solutions.                       Three sub-projects will be pursued by this investigator. First, he          will translate earlier known results on the so called Hopf bifurcation          (a change from non-periodic to periodic solutions due to a small change         of a parameter) into new computational methods. The investigator has            recently developed a technique to detect the existence of small periodic        solutions which avoids the analysis of equations in the infinite                dimensional state space. His method is based on the analysis of a scalar        "bifurcation function", which can be computed in a very direct manner.          He now proposes to extend the use of this technique to more general             equations such as Volterra integral equations, functional differential          equations of neutral type, and abstract functional differential                 equations.                                                                          Second, the investigator will develop algorithms to locate                  numerically the values of parameters for which the Hopf bifurcation             occurs. He will extend the existing methods to compute such bifurcation         points by analyzing the characteristic equation of a linearized system,         and by performing a numerical minimization of the norm of the left hand         side. He will also use curve tracking procedures, similar to those              developed for ordinary differential equations. He will also develop             methods for computing the Floquet multipliers for the equations under           study, by using collocation methods. Much of this work is technically           difficult and requires substantial computational skill and resources.           The investigator will use a Cray-2 supercomputer at the main campus of          the University of Minnesota in Minneapolis.                                         Third, the investigator will study a number of concrete examples            equations arising in applications, and will test his numerical methods          on these examples. The results of this work are expected to provide new         interpretations for dynamical phenomena encountered in a number of other        disciplines.                                                                        The University of Minnesota at Duluth is a primarily undergraduate          institution which has good traditions of student involvment in research.        In this project students will be employed to assist with programming and        computations. Their involvment in this research will have a strong              positive impact on their education, especially because of various               possible interpretations of the results on the grounds of applications          to other sciences.                                                              The project will be funded jointly with the Air Force Office of                 Scientific Research.
{'FirstName': 'Harlan', 'LastName': 'Stech', 'PI_MID_INIT': 'W', 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Harlan W Stech', 'EmailAddress': 'hstech@d.umn.edu', 'NSF_ID': '000166757', 'StartDate': '06/01/1987', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
{'Name': 'University of Minnesota Duluth', 'CityName': 'DULUTH', 'ZipCode': '558123031', 'PhoneNumber': '2187267582', 'StreetAddress': '1035 UNIVERSITY DR # 133', 'StreetAddress2': None, 'CountryName': 'United States', 'StateName': 'Minnesota', 'StateCode': 'MN', 'CONGRESSDISTRICT': '08', 'CONGRESS_DISTRICT_ORG': 'MN08', 'ORG_UEI_NUM': 'LPCTM8BS8NF3', 'ORG_LGL_BUS_NAME': 'REGENTS OF THE UNIV OF MN', 'ORG_PRNT_UEI_NUM': None}
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{'Code': '1266', 'Text': 'APPLIED MATHEMATICS'}
{'url': 'https://www.nsf.gov/awardsearch/download?DownloadFileName=1987&All=true', 'xml': '8701456.xml'}