Peter B. Kahn

Energy level spacing distributions

Abstract The probability function P 1 ( s ) for the next-nearest neighbour spacing in the infinite dimensional limit, is computed and compared with Porters result for 3 × 3 random real symmetric matrices. The probability function for Dysons unitary ensemble β = 2; P 0 ( s ; 2), for the nearest neighbour spacing, is also computed. It is compared with results for 2 × 2 random hermitian matrices.

Averaging methods in predator-prey systems and related biological models

We use the averaging techniques of Krylov-Bogoliubov-Mitropolsky (K-B-M) to solve several nonlinear models with oscillatory behavior. Specifically, we study in detail a class of one predator-one prey systems which incorporate a functional response of the predator. The K-B-M method allows us to calculate, under certain conditions, the radius of the limit cycle, and the renormalized frequency of the systems response. The limitations of weak nonlinearity are not crucial to our qualitative results, although care must be taken to check the strength of the nonlinearity whenever quantitative results are obtained. Finally, we show how to find, near equilibrium points, an “equivalent linear” model, by renormalizing our coefficients of interaction. The method, widely used in control theory, gives indications of being useful for the study of many species models.

Solidification in finite bodies with prescribed heat flux: bounds for the freezing time and removed energy

Abstract One-dimensional, conduction-controlled solidification of initially overheated slabs, and cylindrical and spherical shells with insulated inner walls is considered. The heat flux from the outer face is taken to be constant or monotonically decreasing in time. We derive sufficiently simple and tight upper and lower bounds for the full freezing time and the extracted energy. These bounds are compared with some known approximate solutions, the accuracy of which is thereby established.

Qualitative behavior of predator-prey communities☆

Abstract We examine a set of n -species predator-prey models which incorporate functional responses of the predators to their prey and non-linear intraspecific interactions. We review the limitations of a linear analysis around equilibrium states and provide an extension of the Routh-Hurwitz criteria to the non-linear regime by using Birkhoffs normal forms of differential equations. Qualitative properties like the orbit structure around isolated singularities become clear in this method. It is possible to obtain the radius of the torus and the renormalized frequencies when the eigenvalues of the community matrix have small positive real parts, and to classify different topological structures near bifurcation values of a convenient set of control parameters. Examples of two, three, and four species are analyzed in the context of normal forms. We conclude with some suggestions concerning the coupling of a small subsystem to a large community, and the relations between the graphical method of isoclines and normal forms. This correspondence indicates a road to generalize the study of many-dimensional systems when the intuition provided by the graphical method fails.

Weakly nonlinear oscillations: A perturbative approach

The perturbative analysis of a one-dimensional harmonic oscillator subject to a small nonlinear perturbation is developed within the framework of two popular methods: normal forms and multiple time scales. The systems analyzed are the Duffing oscillator, an energy conserving oscillatory system, the cubically damped oscillator, a system that exhibits damped oscillations, and the Van der Pol oscillator, which represents limit-cycle systems. Special emphasis is given to the exploitation of the freedom inherent in the calculation of the higher-order terms in the expansion and to the comparison of the application of the two methods to the three systems.

Nonlinear dynamics: A tutorial on the method of normal forms

We consider a variety of nonlinear systems, described by linear differential equations, subjected to small nonlinear perturbations. Approximate solutions are sought in terms of expansions in a small parameter. The method of normal forms is developed and shown to be capable of constructing a series expansion in which the individual terms in the series correctly incorporate the essential aspects of the full solution. After an extensive introduction, we discuss a series of examples. Most of our attention is given to autonomous systems with imaginary eigenvalues for the unperturbed problem. But, we also analyze a system of equations with negative eigenvalues; one zero and one negative eigenvalue; two nonautonomous problems and phase locking in a coupled-oscillator system. We conclude with a brief section on an integral formulation of the method.

Qualitative dynamics of three species predator-prey systems

SummaryA qualitative analysis of some two and three species predator-prey models is achieved by application of the method of averaging in conjunction with a Lyapunov function constructed from the appropriate Volterra-Lotka model. We calculate the limit cycle solution for a two-species model with a Holling type functional response of the predator to its prey by means of a time-scaled transformation. The existence of a bifurcation of steady states for a community of three species is discussed and the periodic solution around one of the unstable steady states is calculated to the lowest approximation. Several comments are made regarding the behavior of these systems under changes of some control parameters.

Consistent perturbation expansion for phase‐changing material in a finite domain

A self‐consistent perturbation expansion is developed for the case of a finite slab of phase‐changing material with constant boundary conditions corresponding to a ‘‘two‐phase’’ problem. Consistency is achieved by a power‐series expansion in functions of both solid and liquid Stefan numbers. Terms through third order are calculated.

Averaging methods in the delayed-logistic equation

The delayed logistic equation is analyzed using the averaging method. Using the transformation of coordinates v=ln N/K it is shown that the first order term in perturbation theory yields N=K exp(r* cos πt/2) when the delay time T exceeds some critical value Tc. The amplitude r* is equal to (40μ/3π − 2)1/2 and μ is an expansion parameter that is proportional to (T − Tc). Comparison of the exponential solution of N and numerical results for the ratio Nmaximum/Nminimum provides a good fit for values of μ larger than the results using the N coordinate as the perturbed coordinate.

Level Density of a Fermi System: Nonperiodic Perturbations of the Energy‐Level Scheme

The level density of a degenerate Fermi system is modified by the perturbation in the position of a single‐particle level. To study this effect, miscellaneous exact relations between the level densities of the perturbed and unperturbed system are derived. For the special case of the perturbed uniform model, these connections become a set of recursion relations which lead to a complete solution of the problem. Results are also obtained in the saddle‐point approximation, and these have a simple interpretation in terms of the usual Fermi occupation probabilities. If a single‐particle level is deleted from the scheme, the resultant diminution in the level density persists indefinitely with increasing excitation energy. Information about the adequacy of the saddle‐point approximation is obtained by comparison with some exact solutions.