Homoclinic Orbits and Bifurcations - Chaos Theory | Lecture 14

In this lecture, we uncover the surprising role of homoclinic orbits—trajectories that leave a fixed point only to return to it both forward and backward in time—and show how they act as an organizing backbone for chaotic dynamics in discrete-time systems. You’ll see how non-degenerate (structurally stable) homoclinic orbits force dynamics that are topologically conjugate to the shift map, guaranteeing chaos in a rigorous way. Then, using the logistic map as a concrete example, we explore what happens when homoclinic orbits become degenerate, triggering dramatic homoclinic bifurcations with infinitely many saddle-node and period-doubling bifurcations packed into every parameter neighborhood. By the end, you’ll see why homoclinic orbits offer one of the cleanest and most compelling ways to detect and understand chaos hiding inside a dynamical system. More information on the instructor: https://hybrid.concordia.ca/jbrambur/ Follow @jbramburger7 on Twitter for updates.