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Equivalent Differential Equations for Nonlinear Dynamical Systems


Patula, Edward John (1970) Equivalent Differential Equations for Nonlinear Dynamical Systems. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/4PEG-2Q54.


A technique for obtaining approximate periodic solutions to nonlinear ordinary differential equations is investigated. The approach is based on defining an equivalent differential equation whose exact periodic solution is known. Emphasis is placed on the mathematical justification of the approach. The relationship between the differential equation error and the solution error is investigated, and, under certain conditions, bounds are obtained on the latter. The technique employed is to consider the equation governing the exact solution error as a two point boundary value problem. Among other things, the analysis indicates that if an exact periodic solution to the original system exists, it is always possible to bound the error by selecting an appropriate equivalent system.

Three equivalence criteria for minimizing the differential equation error are compared, namely, minimum mean square error, minimum mean absolute value error, and minimum maximum absolute value error. The problem is analyzed by way of example, and it is concluded that, on the average, the minimum mean square error is the most appropriate criterion to use.

A comparison is made between the use of linear and cubic auxiliary systems for obtaining approximate solutions. In the examples considered, the cubic system provides noticeable improvement over the linear system in describing periodic response.

A comparison of the present approach to some of the more classical techniques is included. It is shown that certain of the standard approaches where a solution form is assumed can yield erroneous qualitative results.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:(Applied Mechanics)
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Applied Mechanics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Iwan, Wilfred D.
Thesis Committee:
  • Unknown, Unknown
Defense Date:8 May 1970
Other Numbering System:
Other Numbering System NameOther Numbering System ID
Caltech Dynamics LaboratoryDYNL-101
Additional Information:Also published as Caltech Dynamics Laboratory technical report DYNL-101.
Record Number:CaltechTHESIS:08182015-085203745
Persistent URL:
Related URLs:
URLURL TypeDescription ItemCaltechAUTHORS EERL Dynamics Laboratory technical report
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:9104
Deposited On:18 Aug 2015 22:41
Last Modified:17 May 2024 21:27

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