Phase Plane Portraits Research Articles

Interactive computer graphics for the generation of phase plane portraits

This paper describes the use of computer graphics facilities for generating phase plane portraits by truly interactive means. The final portrait produced is completely governed by the user who chooses the starting points for any particular future trajectories using only the graphical information currently available to him on the screen.

Circularly Constrained Particle Motion in Spinning and Coning Bodies

The motion of a particle constrained to move on a circular path within a body that is spinning and coning is examined. The equation governing this motion is expressed as a nonlinear, nonautonomous ordinary differential equation, and the behavior of solutions of this equation is investigated using phase plane techniques. Several classes of phase plane portraits are identified, and boundaries that separate system parameters giving rise to portraits of each class are established. Practical applications for the information thus generated include the analysis and design of certain passive balancing devices used to attenuate wobble in spinning spacecraft and the study of simplified models of nutation dampers and thermal control devices consisting of circular tubes partially filled with fluid.

An analysis of models describing predator‐prey interaction

AbstractMathematical models of the interaction between predator and host populations have been expressed as systems of nonlinear ordinary differential equations. Solutions of such systems may be periodic or aperiodic. Periodic, oscillatory solutions may depend on the initial conditions of the system or may be limit cycles. Aperiodic solutions can, but do not necessarily, exhibit oscillatory behavior. Therefore, it is important to characterize predatory‐prey models on the basis of the possible types of solutions they may possess. This characterization can be accomplished using some well‐known methods of nonlinear analysis. Examination of the system singular points and inspection of phase plane portraits have proved to be useful techniques for evaluating the effect of various modifications of early predator‐prey models. Of particular interest is the existence of limit cycle oscillations in a model in which predator growth rate is a function of the concentration of prey.

Analysis of Series Generator Series Motor Drive

A drive is analyzed consisting of a series-field generator, driven at a constant speed and electrically connected to a series-field motor which provides the load. The magnetization curves of both machines are assumed to consist of a linear region and two saturated regions. Both machines saturate at the same current level. Hysteresis is neglected. This system constitutes a second-order system with three nonlinear terms. The behavior of this system can be described by phase-plane representation If current and motor speed are used as the co-ordinates of the phase plane, then the phase-plane portrait will be symmetrical to the motor-speed axis. Considering all possible variations of the system parameters, two basic cases of system operation are possible. They are: 1. Steady-state current-flow exists. 2. Steady-state current-flow cannot exist. Case 1 includes ordinary system-operation as a drive. In case 2, both the generator and the motor are driven, but no steady-state power transfer between generator and motor takes place. The two basic cases will be represented by the following singular-point arrangements on the phase plane with current and motor speed as the co-ordinates: 1. One saddle point on the speed axis (current equal to zero), plus one pair of singular points symmetrical to the speed axis, which, depending on the values of the system-parameters, may be one of the following: a. Stable nodes. b. Stable focuses. c.