Volume 81, Issue 10, October 2013

The logistic map difference equation is encountered in the theoretical ecology literature as a mathematical model of population change for organisms with nonoverlapping generations and densitydependent dynamics influenced solely by intraspecific interactions. This article presents the logistic map as a simple model suitable for introducing students to the properties of dynamical systems including periodic orbits, bifurcations, and deterministic chaos. After a brief historical and mathematical introduction to models of population change and the logistic map, the article summarizes the logistic map activities I teach in my introductory physics laboratories for nonphysics majors. The logistic map laboratory introduces the many bioscience students in my courses to a foundational model in population ecology that has inspired ecologists to recognize the importance of nonlinear dynamics in real populations. Although I use this activity in courses for nonmajors, the logistic map model of population change could also be taught to physics majors to introduce properties of dynamical systems while demonstrating an application of mathematical modeling outside of traditional physics.
 PAPERS


Exploring dynamical systems and chaos using the logistic map model of population change
View Description Hide DescriptionThe logistic map difference equation is encountered in the theoretical ecology literature as a mathematical model of population change for organisms with nonoverlapping generations and densitydependent dynamics influenced solely by intraspecific interactions. This article presents the logistic map as a simple model suitable for introducing students to the properties of dynamical systems including periodic orbits, bifurcations, and deterministic chaos. After a brief historical and mathematical introduction to models of population change and the logistic map, the article summarizes the logistic map activities I teach in my introductory physics laboratories for nonphysics majors. The logistic map laboratory introduces the many bioscience students in my courses to a foundational model in population ecology that has inspired ecologists to recognize the importance of nonlinear dynamics in real populations. Although I use this activity in courses for nonmajors, the logistic map model of population change could also be taught to physics majors to introduce properties of dynamical systems while demonstrating an application of mathematical modeling outside of traditional physics.

The shape function of a freefalling laminar jet: Making use of Bernoulli's equation
View Description Hide DescriptionThe shape function of a laminar liquid jet issuing from a circular orifice and falling vertically in air under gravity is analyzed. The diameter of the jet is observed to decrease with the axial distance from the nozzle. The governing equation for variation of the jet radius with the axial coordinate is derived from a modified Bernoulli's law, including the interfacial energy density and viscous losses. The analytical solution found in terms of dimensionless group numbers agrees well with experimental data.

Recreating Gauss's method for nonelectrical absolute measurements of magnetic fields and moments
View Description Hide DescriptionIn 1832, Gauss made the first absolute measurements of magnetic fields and of magnetic moments in experiments that are straightforward and instructive to replicate. We show, using rareearth permanent magnets and a variation of Gauss's technique, that the horizontal component of the ambient geomagnetic field, as well as the size of the magnetic moments of such magnets, can be found. The method shows the connection between the SI and cgs emu unit systems for these quantities and permits an absolute realization of the Ampere with considerable precision.

A digital hysteresis loop experiment
View Description Hide DescriptionA toroid with primary and secondary windings is used as a transformer to generate magnetic hysteresis curves. The primary winding is driven by a signal generator, which induces an alternating voltage in the secondary winding. Both input and output voltages are captured using a digital storage oscilloscope and processed to generate and display a hysteresis curve. We show such curves are representative of the material used as the transformer core. Data acquisition and processing steps are presented in a manner suitable for use in an undergraduate laboratory or lecture demonstration.

Stationary points of the rigid body kinetic energy and the principal axes of inertia
View Description Hide DescriptionWe demonstrate how searching for the extrema of the rotational kinetic energy can be exploited to introduce the idea of the principal axes of inertia in a general and intuitive way. To that end, we propose a motivational starting point for a discussion with students, followed by a physical problem that, in its simplest form, can be handled using only basic calculus. Principal axes and moments of inertia arise seamlessly as solutions to the proposed problem. A numerical example is provided to emphasize key ideas. To encourage alternative ways of thinking, we also show how similar conclusions can be derived from the square of the angular momentum. A generalized procedure, appropriate for a classical mechanics course, is outlined in the Appendix.

Generalization of Torricelli's parabola to problems with a moving source
View Description Hide DescriptionAnalytical forms for the envelopes of trajectories for particles radially ejected from a moving source in uniform gravity are derived. These results generalize Torricelli's (safety) parabola for ballistic problems. As a motivating scenario we consider the spherical case shot, later named the shrapnel shell. Several cases that illustrate interesting features of the problem are presented along with an example inspired by the spherical case shot.

Air expansion in a water rocket
View Description Hide DescriptionWe study the thermodynamics of a water rocket in the thrust phase, taking into account the expansion of the air with water vapor, vapor condensation, and the corresponding latent heat. We set up a simple experimental device with a stationary bottle and verify that the gas expansion in the bottle is well approximated by a polytropic process PVβ = constant, where the parameter β depends on the initial conditions. We find an analytical expression for β that depends only on the thermodynamic initial conditions and is in good agreement with the experimental results.

A graphical description of optical parametric generation of squeezed states of light
View Description Hide DescriptionThe standard process for the production of strongly squeezed states of light is degenerate optical parametric amplification (OPA) below threshold in nonlinear dielectric media such as LiNbO3 or periodically poled potassium titanyl phosphate (KTP). Here, we present a graphical description of squeezedlight generation via OPA, visualizing the interaction between the nonlinear dielectric polarization of the medium and the electromagnetic quantum field. We explicitly focus on the transfer from the field's ground state to a squeezed vacuum state and from a coherent state to a bright squeezed state by the medium's secondorder nonlinearity, respectively. Our pictures illustrate the phasedependent amplification and deamplification of quantum uncertainties and give the phase relations between all propagating electromagnetic fields as well as the internally induced dielectric polarizations. The graphical description can also be used to describe the generation of nonclassical states of light via higherorder effects of the nonlinear dielectric polarization such as fourwave mixing and the optical Kerr effect.

About photon correlations
View Description Hide DescriptionSome general properties of photon correlations are discussed in a simple way through an analysis of the twodetector measurement scheme. It is shown that the assumption of the discreteness of the random process leads directly to the conclusion that the zerodelay value of the correlation function is only bound to be nonnegative. The adopted approach allows discussing in a more intuitive way the photon correlation properties of different optical fields, including nonclassical fields presenting an apparent violation of the CauchySchwarz inequality. The comparison between the two and the singledetector experiment clarifies the role of the operator ordering in the definition of the correlation function.
