This thesis considers in detail the dynamics of two oscillators with weak nonlinear coupling. There are three classes of such problems: non-resonant, where the Poincar\u00e9 procedure is valid to the order considered; weakly resonant, where the Poincar\u00e9 procedure breaks down because small divisors appear (but do not affect the O(1) term)\r\nand strongly resonant, where small divisors appear and lead to O(1) corrections. A perturbation method based on Cole's two-timing procedure is introduced. It avoids the small divisor problem in a straightforward manner, gives accurate answers which are valid for long times, and appears capable of handling all three types of problems with no change in the basic approach.

\r\n\r\nOne example of each type is studied with the aid of this\r\nprocedure: for the nonresonant case the answer is equivalent to the Poincar\u00e9 result; for the weakly resonant case the analytic form of the answer is found to depend (smoothly) on the difference between the initial energies of the two oscillators; for the strongly resonant case\r\nwe find that the amplitudes of the two oscillators vary slowly with time as elliptic functions of \u03f5 t, where \u03f5 is the (small) coupling parameter.

\r\n\r\nOur results suggest that, as one might expect, the dynamical behavior of such systems varies smoothly with changes in the ratio of the fundamental frequencies of the two oscillators. Thus the pathological behavior of Whittaker's adelphic integrals as the frequency ratio is\r\nvaried appears to be due to the fact that Whittaker ignored the small divisor problem. The energy sharing properties of these systems appear to depend strongly on the initial conditions, so that the systems not ergodic.

\r\n\r\nThe perturbation procedure appears to be applicable to a wide variety of other problems in addition to those considered here.

", "doi": "10.7907/HXVR-7T87", "publication_date": "1969", "thesis_type": "phd", "thesis_year": "1969" }, { "id": "thesis:1309", "collection": "thesis", "collection_id": "1309", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-04082005-155822", "primary_object_url": { "basename": "Bluman_gw_1968.pdf", "content": "final", "filesize": 1694795, "license": "other", "mime_type": "application/pdf", "url": "/1309/1/Bluman_gw_1968.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Construction of solutions to partial differential equations by the use of transformation groups", "author": [ { "family_name": "Bluman", "given_name": "George W.", "clpid": "Bluman-G-W" } ], "thesis_advisor": [ { "family_name": "Lagerstrom", "given_name": "Paco A.", "clpid": "Lagerstrom-P-A" }, { "family_name": "Cole", "given_name": "Julian D.", "clpid": "Cole-J-D" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_eng" } ], "abstract": "NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.\n\nA systematic approach is given for finding similarity solutions to partial differential equations by the use of transformation groups.\n\nIf a one-parameter group of transformations leaves invariant a partial differential equation and its accompanying boundary conditions, then the number of variables can be reduced by one. In order to find the group of a given partial differential equation, the \"classical\" and \"non-classical\" methods are discussed. Initially no special boundary conditions are imposed since the invariances of the equation are used to find the general class of invariant boundary conditions.\n\nNew exact solutions to the heat equation are derived. In addition new exact solutions are found for the transition probability density function corresponding to a particular class of first order nonlinear stochastic differential equations. The equation of nonlinear heat conduction is considered from the classical point of view.\n\nThe conformal group in n \"space-like\" and m \"time-like\" dimensions, C(n, m), which is the group leaving invariant [...], is shown to be locally isomorphic to S O (n+l, m+l) for n + m >= 3. Thus locally compact operators, besides pure rotations, leave invariant Laplace's equation in n >= 3 dimensions. These are used to find closed bounded geometries for which the number of variables in Laplace's equation can be reduced.\n", "doi": "10.7907/X1E8-1E61", "publication_date": "1968", "thesis_type": "phd", "thesis_year": "1968" }, { "id": "thesis:7582", "collection": "thesis", "collection_id": "7582", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:04052013-154128630", "type": "thesis", "title": "Nearly free molecular heat transfer from a sphere", "author": [ { "family_name": "Schmulian", "given_name": "Robert Jay", "clpid": "Schmulian-R-J" } ], "thesis_advisor": [ { "family_name": "Cole", "given_name": "Julian D.", "clpid": "Cole-J-D" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_eng" } ], "abstract": "Consider a sphere immersed in a rarefied monatomic gas with\r\nzero mean flow. The distribution function of the molecules at infinity\r\nis chosen to be a Maxwellian. The boundary condition at the body is\r\ndiffuse reflection with perfect accommodation to the surface temperature.\r\nThe microscopic flow of particles about the sphere is modeled\r\nkinetically by the Boltzmann equation with the Krook collision term.\r\nAppropriate normalizations in the near and far fields lead to a perturbation\r\nsolution of the problem, expanded in terms of the ratio of\r\nbody diameter to mean free path (inverse Knudsen number). The distribution\r\nfunction is found directly in each region, and intermediate\r\nmatching is demonstrated. The heat transfer from the sphere is then\r\ncalculated as an integral over this distribution function in the inner\r\nregion. Final results indicate that the heat transfer may at first\r\nincrease over its free flow value before falling to the continuum level.", "doi": "10.7907/KZ16-1X08", "publication_date": "1968", "thesis_type": "phd", "thesis_year": "1968" }, { "id": "thesis:3576", "collection": "thesis", "collection_id": "3576", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-09172002-144944", "primary_object_url": { "basename": "Harstad_k_1967.pdf", "content": "final", "filesize": 4867770, "license": "other", "mime_type": "application/pdf", "url": "/3576/1/Harstad_k_1967.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Steady Laminar Compressible Magneto-Fluid-Dynamic Gas Flows in Channels.", "author": [ { "family_name": "Harstad", "given_name": "Kenneth Gunder", "clpid": "Harstad-Kenneth-Gunder" } ], "thesis_advisor": [ { "family_name": "Cole", "given_name": "Julian D.", "clpid": "Cole-J-D" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "GALCIT" }, { "literal": "div_eng" } ], "abstract": "Numerical computations are carried out for the core flow of subsonic MFD generator channels with a large length-to-height ratio and fine electrode segmentation. The working fluid is taken as potassium seeded argon. Variable transport properties and radiation effects are considered. It is shown that transverse variations in fluid properties are very important in Faraday generators; a one-dimensional analysis of the flow is not adequate. Axial currents in nonequilibrium flows can be kept low if the right value of the Hall parameter can be obtained; this also depends critically on the Mach number and load parameter. Mach numbers much less than one and high load parameters are to be avoided. Attainment of very large Hall parameters and fields cannot be expected.", "doi": "10.7907/5T8X-8S80", "publication_date": "1967", "thesis_type": "phd", "thesis_year": "1967" }, { "id": "thesis:4692", "collection": "thesis", "collection_id": "4692", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-11302005-135648", "primary_object_url": { "basename": "Zien_tf_1967.pdf", "content": "final", "filesize": 2985233, "license": "other", "mime_type": "application/pdf", "url": "/4692/1/Zien_tf_1967.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "A Class of Three-Dimensional Optimum Wings in Hypersonic Flow", "author": [ { "family_name": "Zien", "given_name": "Tse-Fou", "clpid": "Zien-Tse-Fou" } ], "thesis_advisor": [ { "family_name": "Cole", "given_name": "Julian D.", "clpid": "Cole-J-D" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "GALCIT" }, { "literal": "div_eng" } ], "abstract": "The idea of using streamlines of a certain known flow field to construct generally three-dimensional lifting surfaces together with the method of evaluating the aerodynamic forces on the surfaces, developed by Nonweiler, Jones and Woods, has been extended and applied to axisymmetric hypersonic flow fields associated with a class of slender power-law shock waves of the form r ~ \u03c4x