The one-dimensional ray equation, the differential description of Fermat’s principle, is deduced directly from Snell’s law using two methods. In the first method, we obtain the ray equation from a differential equation relating the spatial coordinates derivative with the index of refraction field. In the second method, the ray equation is deduced from the proper generalization of Snell’s law for a refractive field, that is, a differential equation relating the index of refraction field and the refraction angle. Additionally, we used an intermediate expression of the first method to find a straightforward analytical solution of the ray path to an inferior mirage.

We examine how stochastic time-delayed negative feedback affects the dynamical behavior of a model oscillatory reaction. We apply constant and stochastic time-delayed negative feedbacks to a point Field-Körös-Noyes photosensitive oscillator and compare their effects. Negative feedback is applied in the form of simulated inhibitory electromagnetic radiation with an intensity proportional to the concentration of oxidized light-sensitive catalyst in the oscillator. We first characterize the system under nondelayed inhibitory feedback; then we explore and compare the effects of constant (deterministic) versus stochastic time-delayed feedback. We find that the oscillatory amplitude, frequency, and waveform are essentially preserved when low-dispersion stochastic delayed feedback is used, whereas small but measurable changes appear when a large dispersion is applied.

We analyse the velocity of a particle propelled by a constant power and subjected to a dissipative force. We consider three cases of dissipative processes: dry friction and viscous drag in the regime of linear and quadratic dependence on velocity. We derive the equation of motion and then, based on a rational and graphical analysis, determine the qualitative shape of the velocity curve as a function of time. Finally, we solve the equation of motion for the particle’s velocity and confirm the previous result. One of our motivations for this work is the ubiquitous phenomenon of vehicles reaching a nearly constant velocity when propelled by a fairly constant power.