Deterministic Chaos: an approach with Insight Maker software focused on discussions about determinism and predictability

Demonstrating the existence of events that are hypersensitive to initial conditions, non-linear dynamics shook the bases of philosophy of Science, mainly in what refers to the predictive power of scientific knowledge. In educational context, however, deterministic chaos continues to be little explored, even in undergraduate courses, despite its potential to favor the learning of both scientific knowledge and central elements of philosophy of science. Proposing an alternative approach to the subject in teacher training courses, this article presents a didactic approach for teaching nonlinear dynamics using an online and open source application called Insight Maker, which allows the solving of differential equations by numerical integration and automated generation of graphs, requiring little programming knowledge. We discuss the notions of causality, determinism and predictability by analyzing two theoretical models of pendulums: the simple pendulum and the forced damped one. In the first situation, numerical integration of the equations of motion in Insight Maker shows that the phenomenon is, in principle, predictable, since its solutions have little sensitivity to small variations in the imposed initial conditions. The same did not occur with the forced damped pendulum. Using the same tool, we show that this phenomenon can present chaotic behavior due to variations in a system parameter. This was done both by applying a sensitivity test considering variations in the initial conditions and the indirect calculation of the Lyapunov exponent. We showed the so-called “islands of order” and highlighted the route to chaos via the period doubling cascade, conceptually exposing the role of the Feigenbaum number in nonlinear systems.

Keywords:
Chaos; Nonlinear dynamics; Predictability; Computer simulation; Insight Maker; Forced damped pendulum